Smolin on mathematics

ineffectivemby Massimo Pigliucci

I have continued on with my critical reading of Roberto Unger and Lee Smolin’s thought provoking The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy [1], about which I have already published one essay here at Scientia Salon [2], focused on the general premise of the book and on the first half of the volume, which presents the more obviously philosophical argument in support of the authors’ theses, and is written by Unger.

My original idea was to eventually publish a second commentary, focused on Smolin’s half of the book, which is written from a more overtly scientific perspective. While I have not actually finished the book yet, I decided otherwise. Smolin’s contribution is definitely worth reading in its entirety, but some of his general points, of course, are the same as Unger’s, complemented by his knowledge of physics and cosmology. So I am instead focusing here on one of Smolin’s most intriguing individual chapters: his treatment of mathematics (chapter II.5 of the book).

Before we proceed, however, a brief reminder of the three fundamental theses that Unger and Smolin present and defend in their book. Quoting from pp. x-xii, where they first lay out what they are up to:

“The first idea is the singular existence of the universe. … There is only one universe at a time, with the qualifications that we discuss. The most important thing about the natural world is that it is what it is and not something else. This idea contradicts the notion of a multiverse — of a plurality of simultaneously existing universes — which has sometimes been used to disguise certain explanatory failures of contemporary physics as explanatory successes. … The second idea is the inclusive reality of time. Time is real. Indeed, it is the most real feature of the world, by which we mean that it is the aspect of nature of which we have most reason to say that it does not emerge from any other aspect. Time does not emerge from space, although space may emerge from time. … The third idea is the selective realism of mathematics. (We use realism here in the sense of relation to the one real natural world, in opposition to what is often described as mathematical Platonism: a belief in the real existence, apart from nature, of mathematical entities.) Now dominant conceptions of what the most basic natural science is and can become have been formed in the context of beliefs about mathematics and of its relation to both science and nature. The laws of nature, the discerning of which has been the supreme object of science, are supposed to be written in the language of mathematics.”

This essay is concerned precisely with this third point [3].

Smolin begins by acknowledging that some version of mathematical Platonism — the idea that “mathematics is the study of a timeless but real realm of mathematical objects,” is common among mathematicians (and, I would add, philosophers of mathematics), though by no means universal or uncontroversial. The standard dichotomy here is between mathematical objects (a term I am using loosely to indicate any sort of mathematical construct, from numbers to theorems, etc.) being discovered (Platonism) vs being invented (nominalism [4]).

Perhaps the most original and intriguing contribution by Smolin to this debate is to reject the above choice as a case of false dichotomy: it is simply not the case that either mathematical objects exist and are therefore discovered, or that they do not exist prior to the intervention of human minds and are therefore invented. Smolin presents instead a table with four possibilities:

existed prior? yes existed prior? no
has rigid properties? yes discovered evoked
has rigid properties? no fictional invented

By “rigid properties” here Smolin means that the objects in question present us with “highly constrained” choices about their properties, once we become aware of such objects.

Let’s begin with the obvious entry in the table: when objects exist prior to humans thinking about them, and they have rigid properties. All scientific discoveries fall into this category: planets, say, exist “out there” independently of anyone being able to verify this fact (pace extreme postmodernists and radical skeptics), so when we become capable of verifying their existence and of studying their properties we discover them.

Objects that had no prior existence, and are also characterized by no rigid properties include, for instance, fictional characters. Sherlock Holmes did not exist until the time Arthur Conan Doyle invented (the appropriate term!) him, and his characteristics are not rigid, as has been (sometimes painfully) obvious once Holmes got into the public domain and different authors could pretty much do what they wanted with him (and I say this as a fan of both Robert Downey Jr. and Benedict Cumberbatch).

Smolin, unfortunately, doesn’t talk about the “fictional” category, comprising objects that had prior existence and yet are not characterized by rigid properties. Anyone wishs to submit examples?

The crucial entry in the table, of course, is that of “evoked” objects: “Why could something come to exist, which did not exist before, and, nonetheless, once it comes to exist, there is no choice about how its properties come out? Let us call this possibility evoked. Maybe mathematics is evoked” (p. 422).

Smolin goes on to provide an uncontroversial class of evocation: “For example, there are an infinite number of games we might invent. We invent the rules but, once invented, there is a set of possible plays of the game which the rules allow. We can explore the space of possible games by playing them, and we can also in some cases deduce general theorems about the outcomes of games. It feels like we are exploring a pre-existing territory as we often have little or no choice, because there are often surprises and incredibly beautiful insights into the structure of the game we created. But there is no reason to think that game existed before we invented the rules. What could that even mean?”

Interestingly, Smolin includes forms of poetry and music into the evoked category: once someone invented haiku, or blues, then others were constrained by certain rules if they wanted to produce something that could reasonably be called haiku poetry, or blues music.

The obvious example that is most close to mathematics (and logic?) itself is provided by board games: “When a game like chess is invented a whole bundle of facts become demonstrable, some of which indeed are theorems that become provable through straightforward mathematical reasoning. As we do not believe in timeless Platonic realities, we do not want to say that chess always existed — in our view of the world, chess came into existence at the moment the rules were codified. This means we have to say that all the facts about it became not only demonstrable, but true, at that moment as well … Once evoked , the facts about chess are objective, in that if any one person can demonstrate one, anyone can. And they are independent of time or particular context: they will be the same facts no matter who considers them or when they are considered” (p. 423).

This struck me as very powerful. Smolin isn’t simply taking sides in the old Platonist / nominalist debate, he is significantly advancing that debate by showing that there are two other cases missing from the pertinent taxonomy, and that moreover one of those cases provides a positive account of mathematical (and similar) objects, rather than just a rejection of Platonism.

But in what sense is mathematics analogous to chess? Here is Smolin again: “There is a potential infinity of formal axiomatic systems (FASs). Once one is evoked it can be explored and there are many discoveries to be made about it. But that statement does not imply that it, or all the infinite number of possible formal axiomatic systems, existed before they were evoked. Indeed, it’s hard to think what belief in the prior existence of an FAS would add. Once evoked, an FAS has many properties which can be proved about which there is no choice — that itself is a property that can be established. This implies there are many discoveries to be made about it. In fact, many FASs once evoked imply a countably infinite number of true properties, which can be proved” (p. 425).

But Smolin’s positive argument doesn’t end there. He recognizes that he has to come up with an alternative account for what has been called the “unreasonable effectiveness of mathematics” [5], or with an answer to the closely related “no miracles” argument for mathematical realism put forth by Quine and Putnam [6]. It does so by a dual, in my mind compelling, strategy: he wants to show that the effectiveness of mathematics in physics is actually somewhat overrated, and then proceeds to propose a multiple-stage account of the development of mathematics as a discipline.

In terms of the first point, Smolin observes that mathematical objects are actually seldom, if ever, a perfect match with objects in the real world, which is to be expected if one thinks of mathematics as dealing in part with abstractions from the real world. Also, mathematical models are grossly underdetermined by physical systems, in the sense that most mathematical laws do not actually have a physical counterpart, or do not uniquely model the physical systems they are intended to account for [7].

As for the second point, I can provide only the highlights here, but the chapter is well worth a full reading. According to Smolin, we can think of mathematics as having developed along the following stages:

“At the first stage, there is the study of the structure of our world, by examination of examples and relations between them, coming from the properties of physical objects or processes and their relations … The second stage is the organization of the knowledge acquired in the naturalistic phase. One makes the discovery that all the knowledge gathered by examination of cases in nature can be reproduced by deduction from a small set of axioms. This is the phase of the formalization of natural knowledge … At the next, or third, stage in the development of mathematics, several mechanisms of growth of mathematical knowledge come into effect which are internal to mathematics, as they no longer require the study of examples in nature to proceed … [then] More non-trivial examples of varying the natural case are found by altering one of the postulates. Famously , modification of the fifth postulate gave rise to the non-Euclidean geometries. This is the fourth stage, that of the evocation and study of variations on the natural case … A fifth stage of development is the invention and development of new modes of thought, new concepts and new methodologies in the study of an area. These can greatly progress an area as new kinds of facts become definable and discussable … Once there are a variety of cases developed by variation of the natural case, a sixth stage of development can play a role, which is to define new kinds of objects by unification of diverse cases. For example, the different Euclidean and non-Euclidean geometries are all unified within Riemannian geometry … mathematics [further] develops through two more kinds of discoveries, one external and one internal. The first is that a construction, example or case developed in the path flowing out of one of the core concerns can turn out to illuminate or apply to knowledge in another stream of development. Developments in geometry can illuminate problems in number theory and vice versa … Lastly, examples, cases or modes of reasoning invented due to the internal development of mathematics can surprisingly turn out to be applicable to the study of nature” (pp. 432-441).

And here is Smolin’s conclusion for that chapter: “the main effectiveness of mathematics in physics consists of these kinds of correspondences between records of past observations or, more precisely, patterns inherent in such records, and properties of mathematical objects that are constructed as representations of models of the evolution of such systems … Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that, once they come to exist, they don’t change” (pp. 445-446).

As should be clear by now, I find Smolin’s view intriguing, but not because it answers all the questions about the nature of mathematics and its relationship with the natural sciences. Frankly, nobody else has come even close to providing such a comprehensive account anyway, so it would be asking a bit too much of Smolin (and Unger) within the context of the much broader project with which they are primarily concerned.

But reading chapter II.5 of The Singular Universe and the Reality of Time did something that rarely happens to me: it provided me both with a fresh perspective on an old problem, and it sketched out tantalizing new answers to that problem. That chapter is worth the price of the book in and of itself, and the rest of the volume ain’t a slacker either.

_____

Massimo Pigliucci is a biologist and philosopher at the City University of New York. His main interests are in the philosophy of science and pseudoscience. He is the editor-in-chief of Scientia Salon, and his latest book (co-edited with Maarten Boudry) is Philosophy of Pseudoscience: Reconsidering the Demarcation Problem (Chicago Press).

[1] The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy, by R.M. Unger and L. Smolin, Cambridge University Press, 2014.

[2] The Singular Universe and the Reality of Time, by M. Pigliucci, Scientia Salon, 24 March 2015.

[3] For a look at my changing opinions about mathematical Platonism, see: On mathematical Platonism, Rationally Speaking, 14 September 2012. / Mathematical Universe? I ain’t convinced, Rationally Speaking, 11 December 2013. / My philosophy, so far — part I, Scientia Salon, 19 May 2014.

[4] Nominalism in the Philosophy of Mathematics, by O. Bueno, Stanford Encyclopedia of Philosophy.

[5] The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by E. Wigner, Communications in Pure and Applied Mathematics, 1960.

[6] Indispensability Arguments in the Philosophy of Mathematics, by M. Colyvan, Stanford Encyclopedia of Philosophy.

[7] Yes, one could go with the Max Tegmark’s mathematical universe hypothesis, or with David Lewis’s modal realism, but that’s just crazy talk. (Yeah, I know, this is going to be controversial, so bring it on!)

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75 thoughts on “Smolin on mathematics

  1. Let’s consider the inverse-square law about the force of gravity. Was it “invented” or “discovered”? It was invented, but it was invented as a way of modelling reality, thus it was discovered that the inverse-square law models the world.

    That is not saying that the inverse-square law has Platonic existence, but nor is it saying that it is an arbitrary invention as, say, the game of chess is.

    From Massimo’s summary of Smolin’s view, I think I largely agree with it. In Smolin’s “first stage” maths is invented, but is invented as a model of the natural world, and thus we discover mathematical ideas as being a good model of the world. That mathematical system then gets formalised and developed as time goes on.

    It is then the case that, given the axioms adopted, one can then reason internally to the system and develop constructs within the system. That is also the case with physics, where one often takes known physical laws, and develops constructs based on them. For example physicists, by that process, predicted the existence of black holes and neutron stars before they were later observed in nature.

    In this view the “unreasonable effectiveness of mathematics” is explained, since maths was developed to model the world in the same way that physics was. Indeed maths and physics are much the same thing really (the main difference being that one uses the term “axioms” whereas the other uses the term “laws” 🙂 ).

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  2. Hi Massimo,

    I appreciated this article, and I think it’s a good idea to have separate essays on some of the different points raised by Smolin and Unger. I can’t disagree with how you have presented their views and I think you’ve done an excellent job of summarising them.

    I think for you and Smolin, the “uncontroversial class of evocation” exemplified by board games is an important point. The problem is that this is not uncontroversial at all, but a kind of begging of the question. Actual Platonists will not accept that “there is no reason to think that game existed before we invented the rules.” As such, I do not agree with you that this is not “simply taking sides in the old Platonist / nominalist debate”, or “significantly advancing that debate by showing that there are two other cases missing from the pertinent taxonomy”.

    By assuming that chess didn’t exist until it was invented, Smolin takes sides against Platonism. This “new” case is not missing from the pertinent taxonomy as far as Platonists are concerned because such games count as mathematical objects. As such I am far less impressed by his contribution than you, although he is to be credited for highlighting these cases which are compelling for some such as yourself.

    What does it even mean to suggest that chess existed prior to its invention? Smolin is happy to discuss the space of possible plays for any particular game ruleset. The set of possible plays is evoked by the invention of that game, so these plays are real in some sense, even if they have never actually been played. He also mentions that there are an infinite number of games we might invent. To a Platonist, this space of possible game rulesets is just as real as the space of possible chess matches. It is in this sense, that of a space of possibilities, that all possible games exist before they are invented.

    I hold this view because it resolves various conundrums Smolin appears not to consider. Take for example a simple game such as tic-tac-toe. Such a game could be “invented” independently in different cultures. If each culture has identical rules for the game, then I see no reason to deny that they are in fact the same game. But the same object cannot be created twice. If it is the same game, perhaps one instance was invented and the other was (re-)discovered? This is unsatisfactory to me because ex hypothesi, aside from chronological order there was no significant difference between the development of the games in the two cultures. I prefer to say they were both discovered and the game always existed as a possibility in the space of possible games.

    This is not to make an extravagant claim because no mystical or mysterious connotations are implied. Rather, this is just an attitude regarding whether we should take a possibility space as something real or not and so is largely a semantic difference.

    Regarding whether Tegmark is crazy, well if he is then so am I, so I didn’t particularly appreciate your sideswipe.

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  3. I’ve followed Leo Smolin’s writing over many years, but I haven’t read this book (by Unger and Smolin) yet. But just from the excerpts displayed above, the point of view about the nature of mathematics and its relationship to nature expressed (“came into existence at the moment the rules were codified”) matches mine. I will need to check out this book.

    A note: Leo got his Ph.D. (Harvard, theoretical physics, “Studies in Quantum Gravity”) the same year (1979) that got mine (Brown, applied mathematics, “Autoregression in Homogeneous Gaussian Configurations”), but he’s two years younger. 🙂

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  4. “When a game like chess is invented a whole bundle of facts become demonstrable, some of which indeed are theorems that become provable through straightforward mathematical reasoning. As we do not believe in timeless Platonic realities, we do not want to say that chess always existed …. This means we have to say that all the facts about it became not only demonstrable, but true, at that moment as well

    So, soon after the development of the modern game of chess, Mehmet thinks to himself “I wonder what it would be like if the Queen could also move like a knight”. Doesn’t say anything, just thinks it. So according to Smolin’s theory he has just evoked a set of mathematical truths that go with this hypothetical rule change – these facts all became demonstrable and true at the moment he thought of the change.

    A few moments later Aisha thinks to herself “I wonder what it would be like if the Queen could also move like a knight” but no evocation of new facts, nothing becomes demonstrable and true because they had already become true with Mehmet’s prior evocation a few moments ago, although they had not exchanged a word on the subject. But what has happened differently when Aisha thought of that rule, than when Mehmet thought of it? Nothing as far as I can see.

    How would I describe this situation if I were to treat it with the same charity that Smolin and Unger reserve for mathematical Platonism? I would probably say that Smolin is proposing some sort of dynamic Platonic realm with a mystical psychic link to people’s minds.

    Buy perhaps it is unworthy to pay them back in their own coin.

    Let me suggest a new game which is similar to chess, but there are no rules before you start. Instead there are a set of ‘chesslike’ criteria for rules and the players must negotiate a set of rules before they start, using those criteria.

    Would the invention of such a game immediately evoke every mathematical fact associate with chesslike games (under my particular criteria for chesslike)? But surely by defining ‘chesslike’ rules I have also define ‘unchesslike’ rules, so have I evoked the facts about them too?

    This question is important, Massimo gave the example of the relaxation of the parallel postulate for evoking non-Euclidean geometries.

    But as Patrice pointed out a little while back, the ancient Greeks knew of non-Euclidan geometries even before Euclid. For example Aristotle (who is unlikely to have originated the idea) says in Physics.

    Since a straight line is what it is, it is necessary that the angles of a triangle
    should equal two right angles. But not conversely; though if the angles are not equal to two right angles, then the straight line is not what it is either.

    So did the Ancient Greeks evoke facts about non-Euclidean geometry before Euclid (although they did not see a use for it and did not pursue it)?

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  5. I truncated Aristotle’s quote in the last post. He was, of course, referring to triangles.

    Further to my point above, suppose X is a rule change to chess and F is a demonstrable mathematical fact that is not true of standard chess but is true of chess with rule change X.

    Suppose at time t1 X has not yet been thought of or formulated and at t2 X has been formulated but F is not yet known or formulated.

    Smolin says of when the rules of chess were formulated “This means we have to say that all the facts about (chess) became not only demonstrable, but true, at that moment as well”

    So, according to Smolin, F becomes demonstrable and true at t2, before it is known and formulated.

    So what could he mean by ‘demonstrable’ in this case, since it is trivially true that you cannot demonstrate a fact that is unknown and has never been formulated?

    So all Smolin could mean by ‘demonstrable’ in this case is to say that it was possible at t2 that F could be demonstrated some time in the future.

    But that was also true at t1 if it was true at t2.

    So in any sense that F was demonstrable after rule change X had been formulated, it was also demonstrable before rule change X had been formulated. And if Smolin would say that F was true because it was demonstrable in this sense, then it must have been also true before the formulation of rule change X.

    And since the invention of chess is, in effect, a set of such rule formulations, then if the facts about chess were true at the moment when the modern game had become completely formulated, then those facts must have been true before the rules of chess had been formulated.

    S&U could suggest, as Aristotle does, that ‘demonstrable’ means ‘has been demonstrated’, but it should be obvious why that fails. They could suggest that F only becomes demonstrable after F itself has been formulated. But this, too, is problematic. As Smolin himself suggests, there was never a chance that F was going to turn out to be a false statement about chess with rule change X.

    Now I no more want to say that chess existed before it was invented than Smolin and Unger do.

    But this suggests that Disagreeable Me’s point about possibility space is more on the right track, and certainly more parsimonious. Smolin’s suggestion appears to raise many more questions than it answers.

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  6. An interesting post. Thanks.

    As a mathematician, I consider myself a fictionalist. Platonism never made sense to me. But my view is actually fairly similar to that of Smolin (as you describe it). I wonder, though, whether it matters. As far as I can tell, platonists and fictionalists do their mathematics in about the same way. The really significant difference is between platonism/fictionalism on one hand, and intuitionism/constructivism on the other.

    I agree with Smolin, that the effectiveness of mathematics in the sciences is not as unreasonable as has been made out. However, I’m not sure that I agree with Smolin’s reasons for that.

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  7. The first mathematical model of the world probably preceded the evolution of man or even our close ancestors.

    It would have come the first time that some animal saw three bugs crawl into a hole and two come out and reason that there must be at least one still inside.

    Like all mathematical models this was not perfect. The two bugs could have eaten the third, or there could have been another exit.

    But the arithmetic was sound and we know that there are a range of animals which can do such arithmetic.

    So, if this was the case, how much would have been ‘evoked’ by this simple arithmetic? The rules for all addition and subtraction are implicit in simple additions and subtractions. Multiplication and whole number division are implicit in addition and subtraction. Factors and prime numbers are implicit in these. So were all of these evoked with that creature’s simple arithmetic? Does it matter whether the being doing this arithmetic could not, itself, demonstrate these other facts?

    Did the builders before Pythagoras (or whoever discovered the Pythagorean theorem) who knew the 3,4,5 rule for making an angle square evoke any facts about plane geometry? Did early agrarian societies swapping shells for goods evoke number theory?

    If not, why not. What is the upper limit of facts evoked from a particular forumlation?

    What range of facts, exactly, is evoked by any given formulation of mathematics (and how)? Could there be a mathematical expression for this?

    This ‘evocation’ seems like an undeveloped idea which, as I said earlier, raises many more questions than it answers. In fact it does not seem to answer any question that I can see.

    I think that many people are rather afraid of this word ‘timeless’ as it is linked in their minds to ideas of the supernatural. But there is no reason why it should be. There is nothing inherently supernatural about the concept. It seems much more parsimonious and more in tune with the facts if we simply say that mathematical truths are timelessly true, rather than tie ourselves in knots trying to deny the fact.

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  8. For anyone interested in fiction about fictionalism: I recommend Dan Simmons’ twin novels ILIUM and OLYMPOS. He marries Homer, Shakespeare, Proust, Browning, and more with future Earth/Mars/Jovian moons . . . the story comes alive (so to speak). They are entertaining.

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  9. “….he is significantly advancing that debate by showing that there are two other cases missing from the pertinent taxonomy, and that moreover one of those cases provides a positive account of mathematical (and similar) objects…”
    I may be missing something but Smolin’s argument to me does not seem all that different from e.g. Maddy’s claim that mathematicians are engaged in ‘language games’ and that the rules and methods are for those games are whatever the mathematical community finds convenient. Its an important argument but not, I think, a decisive or novel one. (Though there is certainly merit in finding a more powerful way to present it .)
    “There is a potential infinity of Formal Axiomatic Systems”. Yes, but this is to confuse syntax and semantics. One can prove a great deal about groups from the axioms , but that’s an entirely separate question of what groups exist – not just what physical systems instantiate group properties (e.g. display n-fold symmetry) or what groups we can generate from using various representations but what groups in fact, actually, really, no kidding, exist. I don’t think Smolin’s line of reasoning, as intriguing as it is, gets us any closer to an answer.

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  10. This argument by Smolin has a hidden assumption – that the reality is not deterministic and as such things like “inventions” make sense. Otherwise, in a deterministic reality, our actions are predetermined and as such the word “invention” does not make sense.

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  11. Note of service: I will respond to a number of comments later today or tomorrow (and, hopefully, so will Lee, who is monitoring this discussion).

    However, some of you have submitted comments that did not pass our filters, for one reason or the other. But we have been unable to contact you because you are logging in via Twitter, without using an actual email address. Please use an email address if you want to the option of being contacted by the Editors about your comments.

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  12. “There is a potential infinity of formal axiomatic systems (FASs). Once one is evoked it can be explored and there are many discoveries to be made about it. But that statement does not imply that it, or all the infinite number of possible formal axiomatic systems, existed before they were evoked. Indeed, it’s hard to think what belief in the prior existence of an FAS would add. Once evoked, an FAS has many properties which can be proved about which there is no choice — that itself is a property that can be established. This implies there are many discoveries to be made about it. In fact, many FASs once evoked imply a countably infinite number of true properties, which can be proved” (p. 425).

    But Smolin’s positive argument doesn’t end there. He recognizes that he has to come up with an alternative account for what has been called the “unreasonable effectiveness of mathematics” [5], or with an answer to the closely related “no miracles” argument for mathematical realism put forth by Quine and Putnam [6]. It does so by a dual, in my mind compelling, strategy: he wants to show that the effectiveness of mathematics in physics is actually somewhat overrated[…]

    Hallelujah!! 😀 A voice of reason! The above quote contains the whole crux of the matter, IMO.

    Massimo,

    I find Smolin’s view intriguing, but not because it answers all the questions about the nature of mathematics and its relationship with the natural sciences. Frankly, nobody else has come even close to providing such a comprehensive account anyway, so it would be asking a bit too much of Smolin[…]

    I don’t agree that nobody came close to providing such a comprehensive account. Perhaps nobody before has summarized these views in a chapter of a book for general public (although I’d be surprised that there are no priors to that too), or more plausibly — perhaps this is the first time for you to read a nice exposition of these views. But everything Lee says here (at least in the parts you quoted) is very well known, ever since the serious advent of mathematical logic, somewhere at the beginning of 20th century.

    Basically, everyone who has ever invested a moderately serious effort to understand the foundations of mathematics, learned enough about formal systems and mathematical logic, and read at least some history regarding, say, Brouwer-Hilbert controversy, non-Euclidean geometry, Banach-Tarski paradox, or some such — already knows everything Smolin seems to be saying in that chapter. In other words, only people who are ignorant of all the advances of 20th century mathematical logic seem to be bent on the ancient debate of Platonism vs. nominalism. Math has moved on from that debate a whole century ago (hint: both Platonism and nominalism suffered heavy casualties, neither won).

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  13. Great read Massimo.

    I like what Smolin is attempting to do here, and it’s something I’ve heard Mario Livio (astrophysicist whose written several books on science and mathematics) over the years as well. The problem with the entire “evoked” image Smolin subscribes to is that it doesn’t seem to be the case at all.

    I’ve pointed out before that the crux of mathematics deals with structures/patterns (of groups, geometries, fields and rings in number theory, etc.), not the symbols, equations, or “brain patterns” that we humans utilize to understand mathematics. That thought really has to sink in, because Smolin (and dare I throw in Massimo and a number of others as well) continues to think of math as a human activity. The “understanding” of mathematical content most certainly is. The “content” of mathematics most certainly is not.

    Now if there is a mathematical structure that we come to understand and write down, and that structure later seems to govern the particle interactions in our Universe, it’s gonna be pretty hard for me to swallow the logic that that structure didn’t actually exist out there in the universe until some pure mathematician wrote some stuff down in the late 19th century. Sorry, that structure was there long before we came along. A favorite retort of nominalists at this point is to harp on the idea that: “Well even though the mathematical group SU(3) X SU(2) X U(1) forms the basis of the Standard Model, that’s only a representation! It’s not really out there!!”

    What the hell then, may I ask, is out there? Do these subatomic particles grab drinks together and decide to interact based on a couple particular Lie groups that they pick from a hat? Or are those particles in a sense “forced” to interact that way because that particular structure happens to be embedded in the Universe itself? If you ask me which one is more palatable, I think the vast majority of us would agree it’s the latter. And if that’s the case, then you are by definition a mathematical realist. The structure is out there, and we’ve developed a certain syntax and axiomatic system that helps us understand the structure. But those lines on the paper aren’t where the structure comes from, the structure is already out there, instantiated/embedded in reality. The same goes for the number groupings and geometric relations that we all know so well from math class.

    One more thing I’d like to point out to is in relation to whether all mathematical structures are instantiated/embedded in reality. I can’t answer this question, we just don’t know. But if one wanted to one could indeed subscribe to a form of Tegmark’s MUH or Lewis’s Modal Realism. And no folks, contrary to Massimo using the term “crazy talk,” it just isn’t so. We’re talking about a tenured professor of cosmology at MIT and arguably one of the most influential philosophers of the last century. Modal realism especially has plenty of advocates and arguments in favor of it (hell Lewis himself mentions several criticisms you could level against it and responds to them in turn). Come on now, let’s get a grip.

    I’ll try to respond to other points as they come up, but this should get us started…

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  14. DM,

    “But the same object cannot be created twice.”

    Of course it can! As I remarked on the similar claim you made on another thread, concerning Newton and Leibniz both developing calculus at the same time, that two people having a similar knowledge base come up with roughly the same idea interdependently, is not at all surprising – indeed, it would be surprising if this never happened!

    Two problems with your assertion and its implied argument – it assumes that individuals are so unique that, where two individuals have similar ideas, it must be because the ideas themselves exist independently of the individuals. Apparently the process of reasoning is then merely a discovery of the pre-existent. I prefer to think of reasoning as an inevitable process of a human mind, which we all have, with considerable potential for innovation.

    The second problem is the evident assumption that reasoning arriving at ideas is largely independent of social and historical realities. That Baudhayana in India should elucidate the Pythagorean theorem within 200 years of Pythagoras himself, is not surprising, given that their different cultures had developed to effectively the same level of inquiry with similar problems to solve. It would actually be more persuasive if Pythagoras had developed calculus through pure reasoning, while Baudhayana remained stuck with geometry trying to describe how to construct altars for the fire ritual. But in fact, they lived in very similar worlds, and being human responded to it as humans would.

    Robin Herbert,

    Your argument is considerably more sophisticated, so I will need time to consider it more fully. However, your initial example, Mehmet and Aisha, did stick in my craw somewhat. Here’s the problem: you can’t have a game unless there are players. (Even in solitaire, the player is playing against the person who shuffled the deck, even if that person and the player happen to be the same person, just at different times.) It doesn’t matter what Mehmet or Aisha think until they actually express this and construct the new game and play it together. But once this happens, the play is firmly established pretty much in the way Smolin seems to claim. Indeed, it could not be played otherwise.

    Your example thus invokes the ‘private language argument,’ and AFAIK, Wittgenstein’s critique still holds against it.

    Your ‘game similar to chess’ example suffers a similar difficulty. Until rules are set and play begins, there isn’t any game; but once the rules are set, the matrix of possible moves can be fairly well determined.

    In general:

    I have not read the book, but so far, from what has been presented here, it seems to me that Unger and Smolin see science and mathematics as socially and historically embedded. Thus, I think there’s a danger in abstracting their claims into detached theoretical space. The universe they are attempting to elucidate is the universe in which we live. ‘Possible worlds’ are only tools for constructing hypotheses.

    Like

  15. DM, and also Robin,

    Take for example a simple game such as tic-tac-toe. Such a game could be “invented” independently in different cultures. If each culture has identical rules for the game, then I see no reason to deny that they are in fact the same game. But the same object cannot be created twice. […] I prefer to say they were both discovered and the game always existed as a possibility in the space of possible games.

    Why cannot the same game be invented twice? I think the phrase “reinventing the wheel” has a very well-defined meaning — its purpose is precisely to emphasize that something can be invented twice (with the unspoken implication that this can be redundant).

    Also, there is something rather important that my late logic professor used to teach us about the meaning of the word “exist” in math. Namely, when a mathematician says “X exists”, their words are a shorthand for a phrase “I am able to think about X in a consistent way” (with some footnotes about consistency). The lesson to learn from this is that mathematical objects do not “exist out there” in the real world, but rather exist in an abstract world of our thoughts. Two people (who belong to two different cultures) could both imagine the ruleset for tic-tac-toe, independently of each other. The ruleset for the game exists (in the math sense) in their thoughts, and it can obviously be invented multiple times, once per each human head.

    George,

    Smolin’s argument to me does not seem all that different from e.g. Maddy’s claim that mathematicians are engaged in ‘language games’ and that the rules and methods are for those games are whatever the mathematical community finds convenient.

    I’d say that is right, and I know a whole bunch of people (primarily from mathematical logic and set theory communities) that take this as the only viable point of view on math. I’m also in that camp, and I guess here among SciSal readers Coel is too (given his position that axioms always have empirical origin). Btw, who is Maddy?

    One can prove a great deal about groups from the axioms , but that’s an entirely separate question of what groups exist – not just what physical systems instantiate group properties (e.g. display n-fold symmetry) or what groups we can generate from using various representations but what groups in fact, actually, really, no kidding, exist.

    What do you mean by “in fact, actually, really, no kidding, exist”? Can a set of axioms (say, of a group) exist in any way other than abstractly, as thoughts in our heads? See my response to DM above, about the definition o the word “exist” in math community.

    Why is life…,

    Otherwise, in a deterministic reality, our actions are predetermined and as such the word “invention” does not make sense.

    In a fully deterministic reality, if our actions are predetermined, another thing that doesn’t make sense is to do science to begin with. This view is so-called “cognitively unstable”, like solipsism, brain-in-a-vat scenarios, Hume’s problem of induction stuff, etc. It is not very fruitful to take such a position about the world.

    Like

  16. Wonderful post Massimo. Furthermore I’m sure most appreciate your “Bring it on!” challenge to Tegmark disciples, since you’re obviously not actually questioning anyone’s sanity here. I hope you don’t mind my own help in this regard.

    My physical determinism is as “hard” as it gets, and therefore I don’t consider “platonic” mathematics to be a useful description of reality. The “physical” is all that’s truly real, I think, with the alternative being a “supernatural” reality. I see mathematics as a human language, categorically no different from English, sign language, and other such tools. We invent them in order to potentially help us describe reality, as well as for our function in general. Just as games of chess which aren’t played do not actually exist, the “infinite” arguments which are never thought up in English and mathematics, must also not exist.

    So then how does my “language” explanation seem? Does it not address “…the nature of mathematics and its relationship with the natural sciences”?

    Like

  17. Pete187 wrote:

    What the hell then, may I ask, is out there? Do these subatomic particles grab drinks together and decide to interact based on a couple particular Lie groups that they pick from a hat? Or are those particles in a sense “forced” to interact that way because that particular structure happens to be embedded in the Universe itself? If you ask me which one is more palatable, I think the vast majority of us would agree it’s the latter. And if that’s the case, then you are by definition a mathematical realist. The structure is out there.

    —————————————–

    The problem is that every key word in your account, here, is a metaphor; a word with invisible “scare quotes” around it, so it doesn’t really *say* anything.

    Some exhibits:

    1. How can an abstract object, which has no spatio-temporal qualities be “embedded” in something physical?

    2. How can an abstract object, which has no spatio-temporal qualities “force” physical things to interact in a certain way (or at all)?

    3. Given that nothing in nature actually has mathematical properties, in the sense that there are no actual lines, triangles, circles, etc., in nature, but only linear, triangular, circular things (which do not have the same properties as their mathematical counterparts), what reason is there to think that the mathematical objects exist, beyond nature?

    4. How can abstract objects, which have no spatio-temporal qualities, be “out there” or anywhere?

    Liked by 1 person

  18. Pete1187,

    > What the hell then, may I ask, is out there?

    The problem is that there are many mathematical structures that don’t describe anything but themselves (as far as we know now). U(1) is “out there” but are all groups therefore out there? Do all Calabi-Yau spaces exist? Is mathematical infinity out there too?

    I’m a bit of a cynic, but I get the impression that Lee Smolin is actually saying that the debate about Platonism etc.simply isn’t relevant for physics, and that he gives a – very intruiging – outline of a way to deal with this irrrelevance.

    Like

  19. Hi Marko and EJWinner,

    > That two people having a similar knowledge base come up with roughly the same idea interdependently, is not at all surprising

    > Why cannot the same game be invented twice?

    Both of you have really missed my point, I feel, for which I must take responsibility. Actually, my intended argument is basically the same as Robin’s (which I endorse and liked a great deal).

    When I say the same object cannot be created twice, I do not mean that reinvention of the same thing is impossible or that each of us are so unique that there is no chance that two people will invent the same thing. In fact my argument depends on exactly the opposite idea: that two people (such as Mehmet and Aisha) may think the same thought.

    Evocation as defined by Smolin is the bringing into existence of a mathematical reality by an act of invention. For evocation to be evocation, a mathematical object must transition from nonexistence to existence. My point is that if ever this same mathematical reality is independently discovered a second time, it can no longer be an act of true invention in this sense because nothing is evoked: nothing is brought into existence that did not exist already. This means that only the first act can be an act of true invention, the second would in Smolin’s own classification be an act of discovery. This seems absurd to me.

    I want to answer EJ’s answer to Robin’s sophisticated formulation of this argument, since I think the argument is the same. EJ’s argument is that if Mehmet or Aisha keep the rule change to themselves without playing chess, then nothing is evoked.

    The answer to this is really simple. Suppose Mehmet plays this rule variant with Mahmoud, and Aisha plays this rule variant with Aaliyah, without any communication of this idea between Mehmet and Aisha. Now each has evoked the new game variant, and we’re back where we started. As Robin says “But what has happened differently when Aisha thought of that rule, than when Mehmet thought of it? Nothing as far as I can see.”

    Marko:

    > Namely, when a mathematician says “X exists”, their words are a shorthand for a phrase “I am able to think about X in a consistent way”

    This is not news to me. Platonists feel this is actually a legitimate definition of existence.

    Exist: verb
    1) To be physically present in the universe
    2) To entail no contradictions with respect to a given mathematical framework

    Since nobody is claiming that mathematical objects are physically present in the universe, there ought be no confusion in holding the second definition to be legitimate.

    You may note that this is hardly a radical position to take. Platonism is pragmatic and sensible and the whole debate about mathematical realism is in many respect based on a pseudoquestion. I endorse the following paper by Carnap, with the caveat that I think Platonism is correct because I think it is useful to define existence as above.

    http://www.ditext.com/carnap/carnap.html

    Like

  20. Coel,

    “Let’s consider the inverse-square law about the force of gravity. Was it “invented” or “discovered”? It was invented, but it was invented as a way of modelling reality, thus it was discovered that the inverse-square law models the world.”

    I believe the example captures exactly what Smolin is trying to say.

    “It is then the case that, given the axioms adopted, one can then reason internally to the system and develop constructs within the system.”

    Again, exactly right. In the chapter Smolin discusses the place at which mathematics begins to develop along internally generated lines, in parallel of course with further development anchored to modeling the natural world.

    “Indeed maths and physics are much the same thing really”

    There I think both Smolin and I would part company. Physics, unlike math, has to be tethered to the empirical world, it can’t be done just o the basis of axioms. Or at the least, it shouldn’t — hence Smolin’s skepticism about notions like the multiverse and such.

    DM,

    “The problem is that this is not uncontroversial at all, but a kind of begging of the question. Actual Platonists will not accept that “there is no reason to think that game existed before we invented the rules.””

    Can you point to a Platonist who actually said as much? I think it would be weird even for a Platonist to argue that chess was discovered rather than invented.

    “The set of possible plays is evoked by the invention of that game, so these plays are real in some sense, even if they have never actually been played”

    No, they are *objective* properties of a real game, once invented. That’s a crucial difference.

    “It is in this sense, that of a space of possibilities, that all possible games exist before they are invented”

    That seems to me to trivialize the notion of Platonism, or, worse, to slide into modal realism.

    “But the same object cannot be created twice”

    Others have already pointed out that this is simply not the case.

    “Regarding whether Tegmark is crazy, well if he is then so am I, so I didn’t particularly appreciate your sideswipe”

    Look at the positive side: you are in excellent company… 😉 (I’m also surprised at your literal understanding of the word “crazy”: I thought it was obvious that I consider the idea to be such, as in outrageous and hard to defend, not the man.)

    Robin,

    “but no evocation of new facts, nothing becomes demonstrable and true because they had already become true with Mehmet’s prior evocation a few moments ago, although they had not exchanged a word on the subject.”

    One could bite the bullet, agree with you, and not much would happen to Smolin’s position as a result. But I don’t need to go that far: I take a social view of science, mathematics, etc. So, for me something becomes “knowledge” only when it is shared. And notice that, again, the proper word here is not “true” but “objective.” (If one takes a correspondence theory of truth, there is nothing to which “truths” about chess correspond.)

    “Would the invention of such a game immediately evoke every mathematical fact associate with chesslike games (under my particular criteria for chesslike)?”

    I don’t see the problem. As soon as players agree on some set of such rules the corresponding objective facts are evoked. If the rules remain undetermined, then so are any facts about them.

    “So did the Ancient Greeks evoke facts about non-Euclidean geometry before Euclid (although they did not see a use for it and did not pursue it)?”

    If they knew about non-Euclidean geometry, they did. Are we hang up on the use of the word “Euclid” here?

    “So what could he mean by ‘demonstrable’ in this case, since it is trivially true that you cannot demonstrate a fact that is unknown and has never been formulated?”

    Forgive me, but why do I get the feeling you are trying to make up one scenario after another to refuse to concede the point? What Smolin was saying is very clear: once we agree on certain rules / axioms, then whatever objective property can be derived from those rules / axioms is evoked. Yes, there will be cases like the ones you formulated well it will be difficult to tell exactly what was knowable at point t1 vs t2, depending on the exact sequence of events. So what? The basic points seems to me to remain unscathed.

    “since the invention of chess is, in effect, a set of such rule formulations, then if the facts about chess were true at the moment when the modern game had become completely formulated, then those facts must have been true before the rules of chess had been formulated”

    Uhm, no. The set of knowable and/or objective facts chess has evolved with the evolution of the game.

    “this suggests that Disagreeable Me’s point about possibility space is more on the right track, and certainly more parsimonious.”

    It doesn’t. The “possibilities” have to be defined by something, and that something has to be put in place by someone. There is no such thing as an unchangeable realm of possibilities, nor are there “possibilities” that exist in a space that hasn’t be formulated.

    “The first mathematical model of the world probably preceded the evolution of man or even our close ancestors”

    Again, no. By mathematical model here we are talking about the result of rational, reflective thinking, not the instinctive actions of pre-human animals. Why on earth would you call that “mathematical”?

    “It would have come the first time that some animal saw three bugs crawl into a hole and two come out and reason that there must be at least one still inside.”

    No, the animal didn’t “reason,” unless you are using that word, again, in a sense that I don’t recognize.

    “were all of these evoked with that creature’s simple arithmetic?”

    No, because the creature wasn’t doing arithmetic, it was simply acting instinctively in a way that mimics our understanding of arithmetics. Would you say that birds of prey are “doing physics” when they fly straight on a victim and manage to catch it without smashing on the ground?

    “What is the upper limit of facts evoked from a particular formulation?”

    Why should there be an upper limit? How is that a problem for the concept of evocation?

    “It seems much more parsimonious and more in tune with the facts if we simply say that mathematical truths are timelessly true”

    If you go that route you have the mystery of an unaccountable realm of non-physical timeless entities, which we are somehow (magically?) capable to access. Are you sure that doesn’t raise more questions than evocation?

    Neil,

    “I wonder, though, whether it matters. As far as I can tell, platonists and fictionalists do their mathematics in about the same way.”

    Well, it may not matter for the way mathematicians do mathematics. But the whole point of this analysis of math by Smolin and Unger is to point out what they regard as the negative effects of mathematics on science: math being about abstractions and atemporal, for instance, has led physicists to deny the empirical reality of time.

    Like

  21. Pete1187,

    […] SU(3) X SU(2) X U(1) […] But those lines on the paper aren’t where the structure comes from, the structure is already out there, instantiated/embedded in reality.

    Really? And you know this how? More precisely, how can you be sure that precisely this group is being instantiated in nature, as opposed to some other? I assume you do know that the SU(2)xU(1) sector is broken down to electromagnetic U(1), i.e. elementary particles explicitly *fail* to uphold that part of the symmetry. I also assume you know that this is not the full symmetry group, but that it should be multiplied by the flavour SU(6), and various U(1) lepton and baryon number groups, not to mention that all that needs to be multiplied by the spacetime Poincare group… or wait, by a broken conformal group… umm, no, by a broken supersymmetry… oh, damn, by an 11-dimensional superconformal group with some compactification to 4D… bummer, make that the infinite Diff(R^4) group of general relativity… grrr… I can’t make up my mind — which of these groups is actually instantiated in nature, and which are just human invention and imagination? Can you enlighten me (and other physicists) how did you manage to figure out the correct one, when the rest of us fail to make up our minds?

    I don’t think so. Your statement is a misunderstanding between a model and reality, just like the example of the Newton’s inverse-square law (that Coel nicely pointed to at the beginning of the thread). It only *appears* to us that it is instantiated in nature, until one finds discrepancies, correction terms, etc. Or just like people were certain that planets travel in circles, until it turned out not to be circles but ellipses. And later on it turned out not to be even ellipses… And every time there is hype about some deep meaning for each of these as being instantiated in the real world. And all the time, the only thing we actually ever do is invent various mathematical structures to approximate that real world by making a model (that later invariably turns out to be not sufficiently correct).

    No mathematical structure is being “instantiated” in nature. It’s just a language we use to describe nature, up to a certain amount of precision.

    Tegmark’s MUH […] We’re talking about a tenured professor of cosmology at MIT

    I’ll see your tenured professor at MIT, and I’ll raise you three tenured professors at Perimeter, Alberta and Tufts (Smolin, Page, Vilenkin, just off the top of my head), that strongly disagree with MUH. I suggest you fold this round. 😉

    Like

  22. George,

    “I may be missing something but Smolin’s argument to me does not seem all that different from e.g. Maddy’s claim that mathematicians are engaged in ‘language games’ and that the rules and methods are for those games are whatever the mathematical community finds convenient.”

    There is more to it. See Smolin’s model of the development of mathematics: since it starts with a solid tether to the empirical world — and it returns periodically to that world, any time that there is an interaction between science and math — then it’s not just a language game. (Chess is, though.)

    “One can prove a great deal about groups from the axioms, but that’s an entirely separate question of what groups exist – not just what physical systems instantiate group properties”

    I fail to see the difference: “exist” according to Smolin doesn’t mean anything unless there is physical instantiation.

    Marko,

    “Perhaps nobody before has summarized these views in a chapter of a book for general public (although I’d be surprised that there are no priors to that too), or more plausibly — perhaps this is the first time for you Massimo to read a nice exposition of these views. But everything Lee says here (at least in the parts you quoted) is very well known, ever since the serious advent of mathematical logic, somewhere at the beginning of 20th century.”

    Could you provide me with some references? Smolin and Unger’s book is definitely not for a general audience, and although they cite plenty of sources, they don’t seem to agree with your assessment of the situation.

    “everyone who has ever invested a moderately serious effort to understand the foundations of mathematics, learned enough about formal systems and mathematical logic, and read at least some history regarding, say, Brouwer-Hilbert controversy, non-Euclidean geometry, Banach-Tarski paradox, or some such — already knows everything Smolin seems to be saying in that chapter”

    If true, that would mean that lots of mathematicians and philosophers of mathematics who consider themselves Platonist “already know everything Smolin” says. Hard to see how this is possible.

    “only people who are ignorant of all the advances of 20th century mathematical logic seem to be bent on the ancient debate of Platonism vs. nominalism”

    Again, I think your claim is at odds with the opinions of actual practicing mathematicians and philosophers of mathematics. A quick Google Scholar search will confirm that they keep engaging in this “ancient debate” (21 hits just from the beginning of this year).

    Why is Life,

    “This argument by Smolin has a hidden assumption – that the reality is not deterministic and as such things like “inventions” make sense”

    Please, let’s stay clear of meta-debates about determinism. This has nothing to do with it.

    Franco,

    “Is there in all this amazing stuff only one, I mean one only, reference to the word ‘prohairesis’?”

    Why would there be? The Stoic faculty involved in giving or withholding assent to impressions is an interesting topic in itself, but I fail to see its relevance here. Or are you suggesting it as a way to support mathematical Platonism? If so, would you care to elaborate?

    pete,

    “That thought really has to sink in, because Smolin (and dare I throw in Massimo and a number of others as well) continues to think of math as a human activity.”

    Guilty as charged!

    “The “understanding” of mathematical content most certainly is. The “content” of mathematics most certainly is not.”

    And you know this how? This is a grand ontological claim, but I like to have my ontology in line with my epistemology. Do you have an epistemological account of the claim?

    “it’s gonna be pretty hard for me to swallow the logic that that structure didn’t actually exist out there in the universe until some pure mathematician wrote some stuff down”

    More difficult to swallow than to posit an entire realm of perennial, non-physical objects? Moreover, Smolin (and others, really) have provided an account for the so-called unreasonable effectiveness of mathematics, and this includes a number of mathematicians and philosophers of mathematics.

    “What the hell then, may I ask, is out there?”

    Stuff, not structures.

    “those lines on the paper aren’t where the structure comes from, the structure is already out there”

    Again, where do you derive this, epistemically?

    “in relation to whether all mathematical structures are instantiated/embedded in reality. I can’t answer this question, we just don’t know.”

    Of course we know. Even Tegmark (!) readily admits that the overwhelming majority of mathematical structures are not instantiated in our physical universe.

    “We’re talking about a tenured professor of cosmology at MIT and arguably one of the most influential philosophers of the last century.”

    Once again: I didn’t call Max crazy, only his ideas. Surely people will see the difference. And I used the word *clearly* in a colloquial, not clinical, fashion!

    That said, I can “raise” you a much larger number of PhD’s from very good institutions who dismiss Max’s ideas, so… Moreover, he most definitely is *not* a philosopher. Indeed, when — a few months ago — he faced a roomful of philosophers at CUNY’s Graduate Center he had a really really awful time defending a coherent view of his MUH.

    “Modal realism especially has plenty of advocates and arguments in favor of it”

    Uhm, no, not really. Very few philosophers take Lewis’ ideas at face value, they mostly consider it an intriguing, far fetched, entirely untestable conjecture.

    Eric,

    “So then how does my “language” explanation seem? Does it not address “…the nature of mathematics and its relationship with the natural sciences”?”

    While I don’t disagree with the gist of what you say, I think mathematics is not a language in the same sense of natural languages like English. It is not only much more formalized, but it is based on abstracting from particulars and focused on discovering patterns. And it does that much better than natural language, which is why it has become the lingua franca in science.

    Liked by 1 person

  23. Massimo wrote:

    “Can you point to a Platonist who actually said as much? I think it would be weird even for a Platonist to argue that chess was discovered rather than invented.”

    ———————

    Jerrold J. Katz, in his book “Realistic Rationalism” argues that the view that games have a temporal location is mistaken. (MIT Press, 1997)

    http://mitpress.mit.edu/books/realistic-rationalism

    He also, however, includes a third category between “abstract” and “concrete,” which he calls the “composite” category. Composite objects are objects that have both an abstract and concrete component that stand in a “creative” relationship to one another, which means that they both contribute to creating the object.

    I published a pretty critical essay on all of this back in 2002.

    Daniel A. Kaufman, “Composite Objects and the Abstract/Concrete Distinction”
    Journal of Philosophical Research
    Volume 27, 2002
    Pages 215-238

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  24. Hi Massimo,

    > Can you point to a Platonist who actually said as much?

    Me for a start! Can you point out one who has denied it?

    I am confident Tegmark would agree (because we tend to think along the same lines). If you’re asking me to provide examples of Platonists specifically opining on the ontological status of board games, I’m afraid I can’t.

    But there are similar examples. Platonists often bring up Borges’ Library of Babel for instance, which posits the existence of a Library containing all possible texts, most of which are gibberish, but contains within it all works of great literature as well as descriptions of all kinds of machines and inventions (and, indeed, board games).

    For instance, biologist Andreas Wagner in his book Arrival of the Fittest discusses how Platonism and the idea of a Borgesian Library of Genotypes helps to understand evolution.

    So, unless I’m really radically misunderstanding my fellow Platonists (and I don’t think I am), many of them would agree with me that the rules of chess (which constitute chess itself, in my view), can be thought of as a mathematical object, and so existed as a mathematical object before chess was invented.

    > No, they are *objective* properties of a real game, once invented. That’s a crucial difference.

    What you quoted from me does not disagree with this, or indeed with Smolin’s view. You’re disagreeing with me here where I’m just describing your own views back to you and not saying anything specifically in support of Platonism. I am saying that while Smolin believes in the possibility space for moves in a game, I also believe in the possibility space for moves in the game of designing games!

    > That seems to me to trivialize the notion of Platonism

    Platonism *is* trivial. The idea that it is making radical or mystical claims is in my view a misunderstanding (again, I would reference Carnap here).

    > or, worse, to slide into modal realism

    Modal realism is an implication of the MUH, so that’s fine by me.

    >> “But the same object cannot be created twice”

    > Others have already pointed out that this is simply not the case.

    Unfortunate that I clarified this just before you posted. I hope you read and respond to that clarification and evaluate the argument again. Again, the point is that an object can only really be *created* once if we define creation as the act of bringing something into existence. The second time it is claimed to be “created”, it already had a prior existence so true creation is impossible. Instead we have rediscovery of some kind.

    > So, for me something becomes “knowledge” only when it is shared.

    So if something is shared between two mutually exclusive groups, then Robin’s point is raised again. Something is evoked twice. Which doesn’t make sense, because once something is evoked once it cannot be evoked again, because it is already real.

    Liked by 1 person

  25. I think the second thesis, the reality of time, has to be considered, to appreciate Smolin’s argument for the third thesis.

    Are all these patterns essentially predetermined by eternal laws, or do the patterns and laws emerge congruent with the physical processes expressing them.

    To clarify my point of view, I don’t necessarily agree with this line of reasoning, so I’m not defending it, just making the point that his “evoked math” would seem premised on the primacy of time.

    Like

  26. On fictionalism about numbers, I do believe numbers and fictional characters have the same ontological status–both being (mind-dependent) mental constructs–but I think it’s questionable whether it’s appropriate to call numbers, even in this case, “fictions.”

    What makes a mental construct a fiction is (roughly) falsely representing something beyond itself. ‘Sherlock Holmes’ does this, ‘2’ does not. ‘Sherlock Holmes’ carries an empty existential claim, in a manner of speaking, regarding a certain detective, while ‘2’ represents nothing beyond itself. For ‘2’ to make sense as a fiction, it would have to be an empty representation of some non-existent other entity “2”.

    When we say that we speak of Sherlock Holmes as if he were real while knowing that he is not real, that which is not real is the detective that the representation ‘Sherlock Holmes’ falsely represents. Since ‘2’ does not represent anything beyond itself, it doesn’t make sense to say that we speak of 2 as if it exists while knowing that it does not.

    As on the level of mental constructs both fiction characters and numbers are real, it seems we should say that numbers are real even though they have the same ontological status as fictional characters.

    To make sense of this, I would propose calling numbers (and fictional characters, as representations) “artificial entities.” On this view, we would say that numbers are real, though artificial, and that generally, artificial entities are such that some are fictions and some are not.

    Like

  27. Hi Massimo,

    Forgive me, but why do I get the feeling you are trying to make up one scenario after another to refuse to concede the point?

    I don’t refuse to concede the point, I would be happy to concede the point were it coherent and backed by evidence or reason rather than being full of holes and depending on ambiguous language.

    My point at this stage was that, in any sense that F is demonstrable at t2, it is also demonstrable at t1. I am not sure how your confident declaration of victory answers this.

    Smolin’s point here depends very much on not thinking too hard about how he is using the word ‘demonstrable’.

    And notice that, again, the proper word here is not “true” but “objective.”

    If that were the proper word then Smolin should have used it, but ‘true’ was the word he used at this point and that is what I am addressing.

    Yes, there will be cases like the ones you formulated well it will be difficult to tell exactly what was knowable at point t1 vs t2, depending on the exact sequence of events. So what? The basic points seems to me to remain unscathed.

    Um, really? Smolin’s point remains unscathed even if you cannot say whether the facts about a set of chess rules became demonstrable before (t1) or after (t2) the invention of those chess rules?
    Hi ejwinner,

    Until rules are set and play begins, there isn’t any game; but once the rules are set, the matrix of possible moves can be fairly well determined.

    In metachess game play begins when the players begin negotiating a set of rules in accordance to the rules of metachess (which were formulated before the game began). In the first phase of game play they can negotiate any rules which meet the ‘chesslike’ criteria in the rules of metachess. Once they have agreed on some specific set of rules, the second phase of the game begins where they start applying the negotiated rules.

    Massimo asks “Why should there be an upper limit?” well that is just what I am asking – if there is no upper limit then the first phase of metachess game play ‘evokes’ all the facts about every possible game which meets the ‘chesslike’ criteria defined in the rules of metachess. We could then ask about the game of metametachess where players can negotiate ‘chesslike’ according to another criterion set in the rules of metametachess.

    I don’t think that either of these are minor quibbles, they both go to the question of whether Smolin’s formulation of ‘evocation’ is meaningful, coherent or useful in any way.

    Liked by 1 person

  28. Rules of the sort one finds in games — and for that matter, natural language — are descriptions of established practices that have become entrenched. They are part of — you know what’s coming — a language game. They do *not* exist prior to practices, exerting some sort of regulatory control over them. To think that they do is to run afoul of one of the most potent skeptical critiques ever conceived, namely Wittgenstein’s Rule-Following argument. If you have trouble reading The Investigations, Kripke’s “Wittgenstein on Rules and Private Language” is an acceptable proxy, *in this context*. (It is not a suitable replacement, generally.)

    Also relevant are Ryle’s remarks in “Knowing How and Knowing That,” in which he talks about chess,explicitly.

    Like

  29. DM,

    Your point has a charitable interpretation with a trivial result; or an uncharitable interpretation with a groundless result with unfortunate implications.

    Charitably, you might be arguing that the words ‘invention’ and ‘discovery’ cannot be distributed across cultures or across time. Baudhayana ‘invented’ the Pythagorean Theorem which Pythagoras later ‘discovered.’ As this is an empty difference, it would seem to make sense to use one term for both events. But this would be a trivial discussion.

    Uncharitably (towards your argument, not towards you), the argument seems to conflate epistemology with ontology. If I understand it correctly, once an an idea is brought into being, this being is absolute and universal – Baudhayana cannot invent what Pythagoras discovers. Therefore they are both engaged in either discovery or invention; however the nature of the entity having being must have been available to both of them, therefore it had being prior to either, therefore we can only say that both discovered what was already there in principle existing “as a possibility in the space of possible” theorems.

    This teeters on the cliff of mysticism. At any rate, since there is no demonstrable “space of possible” theorems (or games), I don’t see how this can be accepted as anything other than an article of faith. Further, since this so clearly violates Occam’s rule against multiplying terms unnecessarily, I don’t see how Robin could say this was more parsimonious than simply saying ‘two people invented this theorem in equivalent cultures within two centuries of each other’ – no ‘space of possibles’ need apply.

    What would constitute a demonstration that the “mathematical object” that *is* chess existed prior to the human invention of the rules? What is even the standard of demonstrability here? Right now, I don’t see how it can be demonstrated that chess-rules or mathematics exist prior to human invention, so I admit I have no idea how anyone might supply counter-demonstration that they do not. But, alas! the same applies to many mystical claims as well….

    Robin Herbert,

    What you describe is not “metachess,” it is a wholly separate language game called ‘negotiation’ over rules of the game to be agreed to. It actually has its own quite specific rules within a culture, socially determined and understood by all the players. (Misunderstanding leads to disagreements, and possibly even to conflict – say, fisticuffs or lawsuits.) I’m afraid Wittgenstein is still ahead of this curve.

    Further, chess with rule-change X would not be chess as we know it, but a variant game (let’s say. chess*). Yet if the rules of chess* are established at t1, F would be a knowable truth concerning chess*, implicit in its structure. A lot of human inventions carry with them mathematical facts implicit in their structure, the principle of these inventions being music. That is because humans approach various problems in a pattern forming, organizational way – structures are introduced, and structures are measurable. I really don’t see the mystery in this.

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  30. Great to see we’ve gotten the discussion going.

    Aravis –

    I don’t really understand the whole metaphor/not saying anything. Last time I checked I wasn’t a postmodernist, and I hate obscurity. You raise a good point about abstract objects and their relation to spatio-temporal “physical” reality. I’ve tried to answer this before by asserting that modern science leads us to the conclusion that “physical” objects in nature that the nominalist would try to fall back on aren’t actually physical. It’s mostly empty space, and the subatomic particles have no known substructure down to absurdly small scales. Massimo believes this to be a limitation of modern science, which is fair. But what does the most fundamental limit consist of? “Vibrating strands of energy” or “loops of spacetime”? What the hell would that even be? And where is their spin or charge located?

    “Oh but Pete, its all fields!” Well what the hell is a field? No one can even tell me that, other than it being the most fundamental component of the “physical” world we know (by the way, when this question is asked and a PhD physicist answers, it is almost always in terms of being the “fundamental mathematical description” that we currently have). It turns out that your abstract/concrete distinction is a whole lot thinner (nonexistent?) than you may have thought.

    Marko –

    Thanks for advising me, but I don’t think I’m gonna sit this one out bud. It’s completely true that things like symmetry breaking are taken into account when describing the Standard Model. I’ll inform you from the outset that my undergrad majors were philosophy and economics, and there’s been no formal schooling since, so my knowledge of theoretical physics won’t be anywhere near yours. I won’t know all of the intricacies involved, but we’re not really asking for that here. I’ve read a boatload on the topic (Green, Hawking, Carroll, Penrose, Susskind, Feynman) and try to learn even more whenever I have the time, but I think I have a good understanding of the basics. What I will say is that we obviously don’t know which mathematical structures/patterns are at the foundation of our universe, but we’ve been getting closer and closer over time. That goal is far off but forms the basis for trying to get to a TOE. And its not you physicists “collectively making up your minds” that’s going to lead us to an answer (assuming, which I do, it exists); its going to be a symmetry group that leads to all of the generations of particles and helps us understand and predict the actions between them.

    The SEP article on Symmetry and Symmetry Breaking gives a great rundown of things:

    “The study of symmetry breaking also goes back to Pierre Curie. According to Curie, symmetry breaking has the following role: for the occurrence of a phenomenon in a medium, the original symmetry group of the medium must be lowered (broken, in today’s terminology) to the symmetry group of the phenomenon (or to a subgroup of the phenomenon’s symmetry group) by the action of some cause. In this sense symmetry breaking is what “creates the phenomenon”. Generally, the breaking of a certain symmetry does not imply that no symmetry is present, but rather that the situation where this symmetry is broken is characterized by a lower symmetry than the original one.”

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  31. Hi Massimo,

    >While I don’t disagree with the gist of what you say, I think mathematics is not a language in the same sense of natural languages like English. It is not only much more formalized, but it is based on abstracting from particulars and focused on discovering patterns. And it does that much better than natural language, which is why it has become the lingua franca in science.

    Thanks for the support! I didn’t really expect to convince anyone that I had indeed discovered “the nature of mathematics and its relationship with the natural sciences,” by means of my “language” explanation here (though I do personally think that I’ve done so). This business is actually just a fun diversion from the important stuff to me however. But if it is important to you I’d also hope for you to further consider my “language” solution from which to dismiss “platonism” (or at least once given pure physicalism).

    Have you ever noticed that everything which is expressed in mathematics, such as “Two plus two equals four,” should also be possible to express in a language such as English? (I’m no mathematician, but I’d be quite impressed if we could express something like, “I’m thirsty” through mathematics!) Languages such as English are of course many order more complex than mathematics is, though this can also seem strange given that we were specially designed to understand these advanced languages (or “natural” as you’ve pointed out), though not mathematics.

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  32. Paul Paolini,

    “What makes a mental construct a fiction is (roughly) falsely representing something beyond itself. ‘Sherlock Holmes’ does this,” – Ah, now this is *really* interesting!

    ‘Sherlock Holmes’ is not a false representation, he’s actually merely a verbal construct. The author and his readers know this; they are therefore engaged in a contract-style agreement to play a certain kind of game *, wherein the reader will respond to Holmes *as if* he existed.

    There are violations to the rules of fiction on the part of authors (who present evident fictions as though they were reports on reality), and on the part of readers (who decide that the fiction is real), but such violations are pretty easy to discern once all the evidence is in – although this can get complicated by cultural bias: eg., Salman Rushdie wrote a book in which a fictional character named Mohammed plays a part; but some Muslims assume that any remarks concerning a fictional construct bearing that name must be an ontological claim on the historical entity known by this name. There are evident similarities between this and Mathematical Platonism.

    You are aware of this, I think; and I like the notion of ‘artificial entities’ rather than ‘fictional’ (if we conceive of ‘fiction’ as simply ‘false representation’). In this regard, of course the number 2 does not have the same status as a fictional character, but it is indeed artificial. Yet it is also a marker in a number of social interactions we can think of as similar to implicitly contractual games: I can use the number 2 (however called) in the process of counting; once we begin articulating formulas incorporating √ 2, it is then a construct in a special language that other practitioners need to agree to and use in practice for further formula construction.

    This notion of verbal construction within social agreement has considerable epistemological value. C.S. Peirce once argued that unicorns do have a ‘reality.’ since they exist as constructs in the mind – they are ‘really’ there, but only within the mind. One can then make truth-claims about them (they are equine, not bovine), and use them in certain further constructs (such as story-telling). If we marry Peirce to Wittgenstein, we get a rich epistemological field for further exploration and discussion.

    Or we get Ludwiga Peirce, the daughter notorious for her mathematical description of Mohammed’s sex life.

    Even jokes have their place in philosophy.

    While I’m near the topic –

    Massimo,

    I like both Downey and Cumberbatch as actors. Unfortunately, their interpretations of Holmes violate so much of what we previously understood about the character, that I can not accept them as adequate representations.

    (BTW, have you caught the Russian version produced last year available on Youtube? ** Also a re-invention, and unnecessarily violent, but closer to the spirit of Doyle’s original invention than the American and British revisions we’ve gotten recently.)

    —–
    *Wittgenstein gains greater relevance the further we go on with this conversation!

    ** https://www.youtube.com/watch?v=VzwBIvVwuOg

    Liked by 1 person

  33. Hi EJ,

    > Your point has a charitable interpretation with a trivial result; or an uncharitable interpretation with a groundless result with unfortunate implications.

    I agree with both of your interpretations. Let’s take the charitable interpretation first.

    > Baudhayana ‘invented’ the Pythagorean Theorem which Pythagoras later ‘discovered.’

    Yup, but not in my language, in Smolin’s language. I would say both discovered it.

    > As this is an empty difference, it would seem to make sense to use one term for both events.

    I agree wholeheartedly, which is why I say they both discovered it. I am even happy to say they both invented it independently as long as we are talking in a context that draws no sharp distinctions between invention and discovery. Unfortunately, Smolin’s approach means we do need to draw a distinction, because Smolin would have us believe that the facts about the Pythagorean theorem only become real for the first time when Baudhayana evokes them and not when Pythagoras discovers them. I reject Smolin’s approach for precisely this reason. Really, we should be in agreement.

    Uncharitable:

    > If I understand it correctly, once an an idea is brought into being, this being is absolute and universal

    Well, the Platonist view is that mathematical objects just exist necessarily and timelessly. They are not brought into being from a state of nonbeing. But otherwise yes.

    > This teeters on the cliff of mysticism.

    Well, I beg to differ. What I said to Marko upthread is that the Platonist just has a broader definition of the verb “to exist” as well as associated words such as “real” and so on. There is nothing mystical there. It’s just using the language of existence more broadly than you do. Think of it as a mental tool or framework rather than a vision of mystical immaterial objects floating in a Platonic void.

    > At any rate, since there is no demonstrable “space of possible” theorems (or games)

    If you define the words as a Platonist does, then this space exists necessarily, by definition. To ask me to demonstrate, perhaps by pointing it out or giving you a photograph of it, is to misunderstand what I mean when I say it exists. The question is not whether this or that abstract object actually really exists, but whether it is appropriate to talk of all well-defined abstract objects as existing.

    > I don’t see [..] this was more parsimonious than simply saying ‘two people invented this theorem…’

    You can say that, but you can’t say two people invented this theorem for the first time. Smolin’s schema depends on something significant happening when something is invented for the first time. Parsimony is great, but it doesn’t do to make things simpler than coherency allows.

    > What would constitute a demonstration that the “mathematical object” that *is* chess existed prior to the human invention of the rules?

    To ask for a demonstration shows you don’t understand how Platonists think of existence. It existed by definition, according to how a Platonist defines and thinks of existence. Chess presumably didn’t exist prior to its invention according to how you think of existence.

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  34. I’m certainly enjoying the exchange here. I believe that the penultimate issue in philosophy (and in science and in theology and in psychology and in . . .) is the relationship between the human mind’s conception of reality and whatever-is-out-there. On the one hand there is solipsism, on the other naive realism. If one rejects both of these (which I surely do) one is left to find a middle ground. The greatest philosophers (IMO) have address this question: Plato and Kant come to mind. And the quantum mechanical question of “measurement” and the wave-function collapse is closely related – may be identical, in fact.

    I am certainly fascinated by Platonism and neo-Platonism, but mainly because it stimulates thought. I don’t think it’s the entire answer, but there seems to me some truth there. (Where, for example, does our concept of infinity come from, since nowhere in nature is there an observable infinity?) In any case, I am cheered that serious philosophers/scientists are grappling with the problem.

    Liked by 2 people

  35. No one seems to have mentioned Wolfram and cellular automata as an expression of how seemingly deterministic consequences necessarily still need causal properties to produce them. Possibly this might tie into Smolin’s specific argument for evoked structure and the essence of the dynamic process of time to produce it.

    For a very simple example, it would seem that 1+1=2 would be about as set a physical principle as there is, yet it is still a dynamic process. You still have to commit the act of addition to reach the conclusion, otherwise there is no result. So it would seem in this instance, the process of time is required to “evoke” the 2.

    Yet the question then is, are there static principles and axioms which do not require a dynamic process and it would seem that much of geometry would fall in that category. Which might then ask as to what other than time is required as foundational, to support these principles and these static factors would seem to require space as an equally necessary foundation for the mathematical universe.

    Are three dimensions the basis of space, or simply a mapping device for particular frames, such as longitude, latitude and altitude and there is no universal frame?

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  36. Hi Massimo,

    Physics, unlike math, has to be tethered to the empirical world, it can’t be done just on the basis of axioms.

    But if the axioms of maths are adopted because they are real-world true, then that tethers maths in the empirical world. [At this point someone is bound to say “Axiom of Choice and Banach-Tarski”, but the reason the AoC is adopted is that it is clearly real-world true about finite sets (it’s only when it’s applied to infinite sets that things go a bit haywire); anyhow, AoC is debated by mathematicians for exactly this reason.]

    Further, physics behaves much the same, exploring the wider “space” defined by the axioms/laws, well beyond what is physically instantiated. This is because properly understanding a particular instantiation requires understanding the wider set of possibilities. Thus it is natural for cosmologists to consider all the possible universes that are compatible with the basic laws. So one asks questions such as “what would happen if the cosmological constant were negative instead of positive?”. This “what if?” exploration based on laws/axioms is much like mathematics.

    As another example, if we want to understand the layout of planets in our own solar system, we consider the ensemble of all possible planetary systems, and figure out which subset of initial conditions leads to ones like ours. Thus we are very much considering “possible planetary systems” that we don’t have empirical knowledge of since they are not instantiated and don’t exist.

    Hi DM,

    Platonists [define] Exist: “2) To entail no contradictions with respect to a given mathematical framework”.

    My problem is that Neanderthals and indeed unicorns “exist” in that sense, and it would be rather confusing to use the term “exist” for things that don’t actually exist.

    Mathematicians get away with it in the sense defined by Marko, because the metaphorical use of the word is understood (or should be!).

    Platonism *is* trivial.

    It is so trivial that it amounts to applying a label (“exists”) that is literally meaningless, since you assign zero properties to this “existence”.

    … the point is that an object can only really be *created* once if we define creation as the act of bringing something into existence.

    The thing that is “brought into existence” is the physical instantiation of the thing. I can “create” a wheel, even though others have made wheels before. I can “invent” a wheel independently of a prior invention. I can “discover” that a wheel is useful, subsequent to others doing so. Further, the *concept* of a wheel can also be created and re-created, since that concept is physically instantiated in our brains, and new physical instantiations can be created.

    Hi Harry Ellis,

    Where, for example, does our concept of infinity come from, since nowhere in nature is there an observable infinity?

    It’s a model, a concept invented because it is useful in modelling reality.

    Hi Marko,

    I’m quoting this to agree with it:

    And all the time, the only thing we actually ever do is invent various mathematical structures to approximate that real world by making a model (that later invariably turns out to be not sufficiently correct). No mathematical structure is being “instantiated” in nature.

    My view in essence:

    The real world consists of stuff in patterns. The patterns are already there in nature. We discover the patterns. The concepts (maths) that describe these patterns are invented. They are invented because they are useful, and they are useful because they model pre-existing patterns.

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  37. “But in what sense is mathematics analogous to chess? Here is Smolin again: “There is a potential infinity of formal axiomatic systems (FASs). Once one is evoked it can be explored and there are many discoveries to be made about it. But that statement does not imply that it, or all the infinite number of possible formal axiomatic systems, existed before they were evoked. Indeed, it’s hard to think what belief in the prior existence of an FAS would add. Once evoked, an FAS has many properties which can be proved about which there is no choice — that itself is a property that can be established. This implies there are many discoveries to be made about it. In fact, many FASs once evoked imply a countably infinite number of true properties, which can be proved” (p. 425).”

    What comes to mind is biological cells and biological systems, or aren’t we ourselves games and FAS’s that follow a set of rules for environmental conditions, nourishment, mating etc.? What comes to mind is that our brains are not just organs of individuality as much as organs of social adaption and rule making / rule following systems themselves or we are actually games. Like the seagull that lands on the field during the World Cup Soccer Match, it has no idea what the rules of the human game are. Language itself may be previously invented but also evokes thoughts and emotions? Just may be more of a confusion.

    BTW I am enjoying Dr Devlin’s series on mathematics, he gives an interesting overview of history.

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  38. Smolin:
    “the main effectiveness of mathematics in physics consists of these kinds of correspondences between records of past observations or, more precisely, patterns inherent in such records, and properties of mathematical objects that are constructed as representations of models of the evolution of such systems … Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence. Both are static, not in the sense of existing outside of time, but in the weak sense that, once they come to exist, they don’t change”

    Patrice Ayme: Smolin implies that “records and mathematical objects are human constructions which are brought into existence by exercises of human will; neither has any transcendental existence”. That may be true, to some extent: after all anything human has to do with human will.

    Seen that way, Smolin’s point is trivial. However, the real question of “Platonism” is why mathematical theorems are true.

    Or am I underestimating Smolin, and Smolin is saying that right and wrong in mathematics is just a matter of will?

    I have a completely different perspective. “Human will” cannot possibly determine mathematical right and wrong. (As many students who are poor at mathematics find out, to their dismay!)

    So what determines right and wrong in mathematics? How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?

    I sketched an answer on my site:
    https://patriceayme.wordpress.com/2015/04/21/why-mathematics-is-natural/

    Hint: it does not have to do with transcendence of the will.

    Liked by 1 person

  39. ejwinner,

    Thanks for the response.

    >”‘Sherlock Holmes’ is not a false representation, he’s actually merely a verbal construct.”

    I would say that the verbal construct ‘Sherlock Holmes’ expresses the mental construct (representation) ‘Sherlock Holmes’, which Doyle may have constructed to a great degree (through stipulative attributions) before he ever wrote or uttered the linguistic expression ‘Sherlock Holmes’. Btw, note that with fictional characters, the stories themselves are modes of constructing the characters;

    I would agree that my characterization of fictional characters as representations was a bit rough. “Falsely represent” could mean misrepresenting an actual being. It would be better to say that fictional characters carry false existential representations. What I mean by this might best be conveyed through an illustration: suppose someone is reading a Sherlock Holmes story aloud and someone comes in who doesn’t know the story being related is fiction. Suppose the person asks a question that implies he thinks Holmes is real. What would others tell him? That Holmes isn’t real, just a fiction, of course. The potential to be understood as a true representing, in the sense of representing an actual being, suggest the way it is a false representation.

    But I agree that a fictional character is more than just a false existential representation. When a con artist invents a “fictitious” person for nefarious purposes, that fiction differs from an artistic fiction in that it is not invented or put forth with the intent or expectation that others will understand that the person is a fiction. This relates to the “social contract” context of artistic fiction that you mention, which I agree is essential to fictional characters in the sense of Holmes. I might use different terms, however, such as ‘the institution of the art of fiction’, which is indeed game-like in some ways.

    >”I like the notion of ‘artificial entities’ rather than ‘fictional’ (if we conceive of ‘fiction’ as simply ‘false representation’). In this regard, of course the number 2 does not have the same status as a fictional character, but it is indeed artificial. ”

    This is an interesting point. Given that both fictional characters and numbers are mental constructs, would the fact that fictional characters are false existential representations while numbers are not mean they differ in ontological status? My sense was that they would still have the same status, but I would be open to hearing reasons why they wouldn’t.

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  40. I think that the use of a familiar term like ‘evoke’ for this concept is misleading – it gives the impression that Smolin is saying something clear and straightforward. And yet when I try to establish just what it means I make no headway (and neither, it seems, does anyone else)..

    I have asked a set of fairly straightforward and specific questions about it and can’t see that I have received any answer to them.

    Firstly I wanted to know, if two people formulated a new game independently of each other, then do the objective facts about it become demonstrable and true both times, or just the first time it is formulated. In the latter case, I asked, what happened differently the second time the game was formulated, that this ‘evocation’ does not happen. If someone could explain what happened differently, then I could immediately understand what is meant by ‘evocation’. If nothing happened differently then, either the facts became demonstrable and true twice (and what does that even mean?), or they didn’t become demonstrable and true either time.

    Secondly I asked about the sense in which Smolin is using ‘demonstrable’, in my example where X is a rule change to chess and F is some mathematical fact that pertains only the new version of chess, in what sense is F ‘demonstrable’ – after X is formulated, but before F is known or formulated – that it was not ‘demonstrable’ before X was formulated.

    I suggested that he could only mean ‘It may be possible to demonstrate F in the future’ and pointed out that if this statement was true after X was formulated then it was also true before X was formulated.

    Thirdly I asked about another game which also has rigid properties, but in this case the rigid properties are a set of criteria for ‘chesslike’ games and the players must negotiate a rule set within these criteria before commencing the game that they have defined. If anyone doubts that this game has rigid properties, note that you could program two computers to play this game against each other. (Note to ejwinner ‘metachess’ is just the name of the game. You could call it ‘SuperMonkeyFunGame’ and the point would remain the same).

    The question I asked about ‘metachess’ is: do all the mathematical facts about metachess become demonstrable (in whatever sense Smolin has in mind) and true when the rules of metachess are formulated? That would mean that all the mathematical facts about any ‘chesslike’ game, as defined in the rules of metachess, would become demonstrable and true.

    It seems to me that if ‘evocation’ has any useful meaning then there should be some reasonably straightforward answers to these straighforward questions. I can’t see that any have been answered.

    But if this concept concept cannot be coherently applied to what ought to be a completely uncontroversial case, then there is no hope of it telling us anything useful about the nature of mathematics.

    Liked by 5 people

  41. Hi Coel,

    > My problem is that Neanderthals and indeed unicorns “exist” in that sense, and it would be rather confusing to use the term “exist” for things that don’t actually exist.

    Mathematicians use the term to describe well-defined mathematical objects with objective properties, so I ought to have made this clearer in my definition (I didn’t because I was deliberately copying the form of Marko’s described usage). Unicorns don’t fit this category. You might be able to do something with Neanderthal, for example by codifying the Neanderthal genome as a mathematical object, and if so, then that mathematical object exists according to Platonism.

    Also, you appear to be confusing physical objects and abstracta. The concepts of Neanderthals and Unicorns might be said to exist if they could be sufficiently well-defined, but that is not to say that the physical creatures exist.

    > It is so trivial that it amounts to applying a label (“exists”) that is literally meaningless, since you assign zero properties to this “existence”.

    We agree that it is useful for mathematicians to use the word in this way. We only disagree on whether this ought to be considered a metaphor or an alternative definition of existence. The debate is as silly to me as arguing over whether a wireless computer network is an actual network or a network only in a metaphorical sense by analogy to physical nets.

    The term is hardly meaningless. The properties we assign to entities which exist are that they are well-defined and coherent with respect to a particular framework. Entities which do not exist (square circles) do not have these properties.

    On the other hand, I regard the idea of objective physical existence to be meaningless and without content. I have never come across a definition adequate to discuss the concept of hypothetical causally disconnected universes, whether there might be such things or not. To you, they don’t physically exist. To observers in those universes, you don’t physically exist. Attempts to define physical existence so as to resolve this problem and come to an objective description of physical existence inevitably lead to a carousel of circular definitions of the sort we got when trying to get Aravis to define semantic reference.

    My view is that existence in general really is pretty meaningless (since pretty much everything exists on the MUH) and only makes sense when used relatively to describe what is real to a particular observer.

    > I can “create” a wheel, even though others have made wheels before.

    Yes. You can create new wheels. You can’t create The Wheel, as in the concept, because it already exists. If you didn’t already know of it, you could discover it. You might say invent. What you can’t do is invent it for the first time and so evoke everything that the invention of the wheel is supposed to evoke.

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  42. The question is about the nature of thinking and knowing. In a circular fashion, what is the physics of contemplating a mathematical theorem? Our best guess is that it involves computation performed on a biological substrate – a sequence of real physical events. If Aisha and Mehmet both think of their novel (to them) game rules, I suggest that there will be an isomorphism between the events in each person’s brains. The isomorphism will be at a “high” level, in that low level neural encodings etc will likely be quite different for this type of task, but it will embody the commonsensical idea that the two individuals have had the same idea (an abstract object is the pattern that remains invariant under different representations). So does their idea exist? Yes, it can be measured using appropriate methods, usually just by asking them, but also indirectly using some yet-to-be-invented imaging technology. How many instances of it have there been: 0, 1, and now 2. If Aisha thinks further about it, there will be another physical instance. To use zero in a principled fashion, I guess we have to be kind of Platonic: “zero previous instances identical to this brain pattern”, but only in a retrospective fashion. Have I begged the question somewhere?

    About physics of computation, Gandy’s model is that “all” you need is: homogeneity of space, homogeneity of time, finite velocity of propagation, and that quiescence (a canonical state) exists for any region of space. That is, most physical processes can support computation. Of course, you will now ask for my definition of computation 😉

    About mathematical models in physics, Bunge has it that “a correct axiomatization of a physical theory must include one interpretation postulate along[side] every mathematical postulate”, why there are so many interpretations of QM…

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  43. Coel mentioned the axiom of choice above. The interesting development of homotopy type theory

    http://homotopytypetheory.org/

    develops constructive mathematics, ie without the Law of the Excluded Middle and the Axiom of Choice:

    A[n] aspect of constructivity is that type theory has a computational interpretation as a programming language, where functions are interpreted as computable functions from their domain to their range. Crucially, type theory provides a true higher-order notion of computability, where computable functions are treated as extensional mathematical entities — two functions that behave the same on all arguments are indistinguishable — which is not true of encodings as Turing machines in ZFC. Strikingly, the absence of excluded middle and related principles is key not only to synthetic mathematics but also to computation: as a computable function, the law of the excluded middle would require that every computational problem can be solved.
    [Another] aspect of constructivity is that type theory unifies the notions of “proposition” and “set” by the propositions-as-types principle, according to which proofs are first-class mathematical constructions, on the same footing as numbers or groups or mappings. This enables a proof-relevant conception of mathematics, where proofs are themselves objects of study.

    This formalizes an intuitionist logic.

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  44. Sherlock Holmes may be a fictional character but the Wikipedia entry indicates he was based on four people he knew in real life.

    Geometry and mathematics may not pre-exist but the four forces of nature pre-exist our biology. There are no such things as a point or straight line but drop anything and the gravitational force pulls it in a straight line towards the earth. Orbital ellipses and Gaussian curves exist due to natural forces.

    Think about a walk throught the garden and you may know how many trees you own but try to recall unique leaves and other fractal structures from memory. What comes to mind is Hubel’s discovery of edge and straight line detection neurons in the visual system or man’s ability to exapt geometric structure from reality.

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  45. Smolin has the right scent in the maze for truth…..but please note that all the big ideas are supplied by the genius Julian Barber whose lifework seeking discovery and correction of general relativity has delivered Shape Dynamics. Barber says there’s no time, smolin says time is fundamental. On that count I would certainly agree Smolin will prove correct. Otherwise Smolin wholly draws on Shape Dynamics. Why can this be so if the founder says one thing about time (which is core0 and the top table profile pioneering the shape dynamical world view says the opposition. The answer to that is in one of the themes in shape dynamics which is that of *pure* relativistic natural law. However, both of them appear unable or unwilling to maintain fidelity to pure relativism beyond a certain .
    the same paint in fact. Smolin’s notion of absolute/ fundamental time is where he is unfaithful or loses faith. Barbour, by suggesting time does not exist if in fact it is a constructed from other more fundamental effects. Clearly, a pure relativistic situation, would have all laws essentially as effects brought about by underlying more or less fundamental forces. Implying….if it’s real and significant and influential and stable and enduring.. It’s REAL. That’s the purpose of the word.

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  46. Hi Coel

    “The real world consists of stuff in patterns. The patterns are already there in nature. We discover the patterns. The concepts (maths) that describe these patterns are invented. They are invented because they are useful, and they are useful because they model pre-existing patterns.”

    I am generally sympathetic to what you have been saying but I have a few doubts about this. A couple of questions…

    What do you mean by ‘stuff’? Does it depend on the commonsense (but apparently now untenable) notion of matter?

    Our (mathematical) concepts model preexisting patterns. But doesn’t this suggest that perhaps the patterns themselves were generated by mathematical (or, more specifically, computational) procedures?

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  47. Hi DM

    “… I regard the idea of objective physical existence to be meaningless and without content. I have never come across a definition adequate to discuss the concept of hypothetical causally disconnected universes, whether there might be such things or not. To you, they don’t physically exist. To observers in those universes, you don’t physically exist. Attempts to define physical existence so as to resolve this problem and come to an agreement inevitably lead to a carousel of circular definitions…”

    Interesting. I seem to remember Seth Lloyd saying something along these lines in respect of the many-worlds interpretation of QM.

    Also, prompted by your link, I’ve been looking again at Carnap. The distinction between internal and external questions is good and useful in respect of formal systems, but I don’t know about its applicability to natural language contexts.

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