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 , about which I have already published one essay here at Scientia Salon , 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 .
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 ).
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” , or with an answer to the closely related “no miracles” argument for mathematical realism put forth by Quine and Putnam . 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 .
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).
 The Singular Universe and the Reality of Time: A Proposal in Natural Philosophy, by R.M. Unger and L. Smolin, Cambridge University Press, 2014.
 The Singular Universe and the Reality of Time, by M. Pigliucci, Scientia Salon, 24 March 2015.
 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.
 Nominalism in the Philosophy of Mathematics, by O. Bueno, Stanford Encyclopedia of Philosophy.
 The Unreasonable Effectiveness of Mathematics in the Natural Sciences, by E. Wigner, Communications in Pure and Applied Mathematics, 1960.
 Indispensability Arguments in the Philosophy of Mathematics, by M. Colyvan, Stanford Encyclopedia of Philosophy.
 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!)
75 thoughts on “Smolin on mathematics”
If you had great interest in a subject called “X,” though for thousands of years humanity had been unable to achieve any accepted understandings regarding its associated reality, would you not then look suspiciously at its general conventions? I do suspect that most philosophers would agree with your suspicion — except of course that… OH NO!!! (Surely many have already developed arguments from which to help justify philosophy’s apparent lack of such progress over the years.) While I do seek to change the field quite radically, I also know that Massimo (a true disciple of Socrates!) heartily welcomes worthy speculation in this regard.
So now then imagine a universe which was essentially like ours, though this one contained no “life,” no “computers,” or anything else which might conceivably “think thoughts.” Note that there could be nothing like “English,” or “mathematics,” or “chess” here, because nothing could thus preform these kinds of functions. So to now relate this back to our discussion, wouldn’t electrons still spin in the same essential manner? How about gravity? Or wouldn’t time, space, and other dimensions of existence, occur just as ours do, and even though they held no subjects which could pondered “dimensions of existence”? Yes I’m quite sure that this reality (and by extension our reality) has no mandate for tools like “mathematics,” or any other kinds of thought. Therefore I do find the amazing speculation associated with this wonderful discussion, a bit humorous.
From me you’ll get ideas which are relatively simple, and presumably because I’ve never permitted myself to become sucked into standard practices in the field of philosophy. Nevertheless I do very much seek to have my ideas rationally assessed by qualified people. Thus I will always remain courteous while explaining my positions.
Very nice essay and an interesting discussion. I have a hard time understanding the position of Platonism though as defended by DM and Robin Herbert.
Doesn’t this type of Platonism boil down to the claim that everything that is conceivable exists timelessly? Or is there an example of anything that is conceivable but does not exist on this view?
And if “x exists” just means “x is conceivable” under this view, in what respect is it advantageous to talk of existence?
I would appreciate any comments!
Regarding Gandy’s model and physical computation, there is the contrary view:
Physical Computation: How General are Gandy’s Principles for Mechanisms?
“We will point to interesting examples of (ideal) physical machines that fall outside the class of Gandy machines and compute functions that are not Turing machine computable.”
This impacts what “constructions” are possible, constructively speaking.
The basis of General Relativity is the speed of light in a vacuum(space). The basis of geometry and thus shape, is space. So the question seems to be about time. For GR, time is the measure of duration, but duration doesn’t exist outside of the present, as it is the state of the present, as events form/dissolve and repeat. So what seems to be measured is the rate, or frequency of events being formed by action/dynamics.
“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?”
So what are those procedures and what “generates” them? Doesn’t that necessarily imply/require a dynamic activity? Therefore Smolin’s Time?
If you simply had a static void, what could it determine, or be determined from it?
Doesn’t there have to be cause, in order to have effect? If nothing happens, then the result would be nothing, so without that dynamic, there is no pre-determination of anything beyond zero.
Nice essay. Here is an example of the fictional category: Suppose there is a god which existed before we ever thought of him. It is not difficult to imagine its properties not being rigid. He might choose to be the Christian god to Christians, the Muslim god to Muslims, Kukulcan to Mayans or Thor to ancient Germanic Europeans. However the set called “fictional” is very probably empty.
By rejecting the common understanding of “exists,” you’ve brought us into consideration of problems of epistemology, metaphysics, and communication that extend well beyond the present conversation and, yes, would include knowledge claims by mystics.
The situation you first described still looks like negotiation to me – a process of communication between humans to reach an agreement on future behavior. Computers can be programmed to engage in negotiation-like behavior, but negotiation is a human process. Giving it other names in specific cases can be useful, but can also misguide us into over-abstracting the nature of the process.
If you’re talking about using the rules of a game to generate the rules of another game, or limiting the construction of another game, this could be interesting, but I’m not sure how this applies here. (Presumably, all the rules of the second game would be implicit in the rules of the first.) Mathematicians and scientists are always operating within pre-existent rules and strategies – and they are always negotiating these among themselves. So either I am misreading your point, or the point is unclear.
“If nothing happened differently then, either the facts became demonstrable and true twice (and what does that even mean?)” – It means the facts became demonstrable twice.
I must admit I don’t understand what the epistemic block is to admitting that ideas can be invented twice. Once an idea has been invented, all of its ramifications can then be discovered (within the parameters of the initial invention) as the idea receives further testing and elaboration. Regressions and advances follow the contingent pathways of history. This seems fairly clear to me.
I regret using the word “merely” to describe the verbal construction of Holmes, because actually the construction of fictions is a complex process. There are rules to the game, but they are, as noted in the article, ‘non-rigid.’ Fictional characters are recurrently re-constructed through interpretation – eg., the illustrations accompanying Doyle’s stories in publication, literary criticism, performance in theater and film, etc. An interesting question here is, how many of the signifiers of the original construction must be passed on for the character to be recognizable? For more than a hundred years, Holmes was partly identified with smoking a pipe. However the pipe most people identify with him – the big curved one – appeared in films, not in the original stories (where his favorite pipe is once said to be a small clay pipe, as I remember). And recent revisions have sought to avoid smoking all together, due to cultural pressures against tobacco usage.
The number 2 is also artificial. These cups on my desk do not signify ‘2 cups,’ just by themselves, a number is something we assign to their co-existence. But once the principle of addition is established, then rules of usage become fairly strict.
So the difference in status between the number and the fictional character may arise from these differing usages and differing kinds of rules governing them.
“Me for a start! Can you point out one who has denied it?”
Sorry, doesn’t work that way. I was asking for a published paper by an academic who is both a Platonist and says that chess and other games “exist” out there. Aravis has provided something along those lines, but I’ve never encountered that view anywhere else, and I’ve read a lot about mathematical Platonism. At the very minimum you would have to agree that game Platonism is even more weird than the already pretty weird mathematical variety.
“Platonists often bring up Borges’ Library of Babel for instance”
Not in my readings. And besides, Borges’ Library is actually a pretty good argument *against* Platonism, since it suggests that humans could write all possible books (of a certain format, length and language) and *by chance* get things right in the real world once in a while.
“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.”
I know Andreas, and I’d be surprised if he mentions Platonism, or does he simply mention the Library and you inferred Platonism from that?
“unless I’m really radically misunderstanding my fellow Platonists”
I think you are.
“What you quoted from me does not disagree with this, or indeed with Smolin’s view”
You do, since Smolin vehemently rejects Platonism and you equally eagerly embrace it.
“Platonism *is* trivial. The idea that it is making radical or mystical claims is in my view a misunderstanding”
Again, if you look at the professional literature, *everyone* agrees, Platonists included, that the position is weird and carries unaccountable metaphysical and epistemological implications.
“Modal realism is an implication of the MUH, so that’s fine by me”
I know, but you know what I think of the MUH, so…
“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”
That’s just wordplay, you are defining “creation” to suit your views.
“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”
Why on earth not? Evocation is something that human beings do, so there must be a human epistemic community to do it. There can be more than one such epistemic community, either in time or in space.
“I think the second thesis, the reality of time, has to be considered, to appreciate Smolin’s argument for the third thesis.”
Yes, that’s right, the two theses are interrelated in the book.
“Are all these patterns essentially predetermined by eternal laws, or do the patterns and laws emerge congruent with the physical processes expressing them.”
Unger and Smolin definitely go for the second option.
“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.”
I don’t think it does. However, Smolin and Unger do say that causality is more fundamental than laws of nature: the former explains the latter, not the other way around.
“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.”
Two issues: i) I’d rather not use the word “real” because it has unwelcome connotations, leading people to think along a Platonic track. ii) As I tried to explain in the essay, Holmes and numbers fall under two different entries in Smolin’s table, and I think that’s a valuable insight which would be lost if we treated them both in the same way.
“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?”
No, not really. That’s true for elementary arithmetic and geometry, but as soon as you get to more complex and abstract stuff natural language loses its grip and you really have to use abstract notation.
“I am certainly fascinated by Platonism and neo-Platonism”
neo-Platonism in philosophy has a specific historical meaning, and it has nothing to do, as far as I can see, with the issue at hand. Mathematical Platonism is a fraction of a full Platonist view, to which nobody really ascribes anymore (I mean, I doubt people go around thinking about the ideal chair, or the ideal piece of dung, and so forth).
“Where, for example, does our concept of infinity come from, since nowhere in nature is there an observable infinity?”
That’s actually an excellent question, explicitly raised by Smolin and Unger. They suggest that infinity is a mathematical concept, originating from abstracting away particulars of the real world. Physicists then get into trouble when they forget that and suggest that there may be actual infinities out there. As far as we know, there aren’t, and indeed it would be difficult to even make sense of what that would mean.
“But if the axioms of maths are adopted because they are real-world true, then that tethers maths in the empirical world”
Not all, or even the majority, at these point, of axioms in math are adopted because they are real world-true, which gets to Smolin’s account of the later stages of mathematics, where the field begins to generate new questions internally, not just as a way to deal with the empirical world.
“physics behaves much the same, exploring the wider “space” defined by the axioms/laws, well beyond what is physically instantiated”
Only temporarily. If it does so indefinitely, Smolin would say it’s no loner physics, but metaphysics. And I strongly agree.
“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.?”
I’m not sure I get exactly what you are referring to, but Smolin and Unger do suggest that the “special” sciences, including foremost biology, are a good model for cosmology and physics, because they are historical, and do take the reality of time and the fundamentality of causality seriously.
“Sherlock Holmes may be a fictional character but the Wikipedia entry indicates he was based on four people he knew in real life.”
That doesn’t seem to me to make a difference to the argument.
“the real question of “Platonism” is why mathematical theorems are true”
Yes, for which Platonism doesn’t really have an answer. I mean, simply postulating, with no further evidence, an entire realm of unseen and unseeable objects to “solve” the problem seems like not solving it at all. Smolin, as a counter, offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).
“am I underestimating Smolin, and Smolin is saying that right and wrong in mathematics is just a matter of will?”
You are misunderstanding him.
“How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”
“it does not have to do with transcendence of the will”
Frankly, I’m not even sure what that means.
“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.”
I think you are being uncharitable here. Philosophers and scientists co-opt English words all the time and give it a technical meaning. As long as it is clear what that meaning is, and as long as it is clear from the context that they are referring to the technical, rather than the common language meaning, everything’s fine. And that’s precisely what Smolin is doing.
“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.”
I’ve answered that question already. Mathematical facts, or facts about chess, are not “true” in the sense of corresponding with something out there. A better word is “objective,” and yes, the objectivity of those facts can be evoked multiple times if the same game / set of axioms are invented multiply. I truly don’t understand why you are having such a hard time with this, or why you think this is somehow a problem for Smolin’s scenario. Unless it is the use of the word “truth” that is problematic and causing confusion.
“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”
Others have answered this too: if you change the rules you are not playing chess anymore, you are playing “schmess.” Schmess has its own set of entailed objective facts about it, which can be discovered in the same way as those pertaining to chess, but which are evoked only after schmess is invented.
“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.”
Once again, an answer was given: yes. So what? This is a variant on the original situation, so it only presents a trivial change from that original. I fail to see why these cases would be problematic for Smolin’s concept of evocation.
“It seems to me that if ‘evocation’ has any useful meaning then there should be some reasonably straightforward answers to these straightforward questions”
There are, and they have been given to you.
“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.”
It seems to me that the symmetric (not really circular) question is entirely different: nobody is denying the existence of a physical world, certainly not Smolin.
“most physical processes can support computation. Of course, you will now ask for my definition of computation ;)”
I could, but I won’t because I don’t see it as relevant to the issue at hand.
“why there are so many interpretations of QM…”
Because interpretations are metaphysical in nature, and therefore go beyond the (currently available, possibly foreseeable) empirical evidence. It’s not science, in other words.
“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.”
Yes, in part, and he says so explicitly in the book.
while I actually agree with your thought experiment about an alternative universe without thinking beings, I take exception to this:
“If you had great interest in a subject called “X,” though for thousands of years humanity had been unable to achieve any accepted understandings regarding its associated reality, would you not then look suspiciously at its general conventions?”
You seem to implicitly adopt a concept of progress in philosophy that is modeled on the sciences. That doesn’t work, as I tried to articulate in a number of other discussion threads and posts. I’m finishing a whole book on the subject, so stay tuned…
“Suppose there is a god which existed before we ever thought of him. It is not difficult to imagine its properties not being rigid. He might choose to be the Christian god to Christians, the Muslim god to Muslims, Kukulcan to Mayans or Thor to ancient Germanic Europeans. However the set called “fictional” is very probably empty.”
Interesting, and yes, I tend to agree.
Can you give examples of commonly adopted axioms of maths that are not real-world true (in the sense that they have no utility in modelling the real world)?
Hi Mark English,
I mean things like electrons, protons, quarks, etc (and other stuff made out of those).
Is it untenable? Even if you reduce the above to strings, or whatever, there must be some property that amounts to “physical instaniation” and that we can call “matter”.
DM will like you saying that! :-). Personally I’d say “no”. The difference between “physical stuff” and a “mathematical procedure” is (IMO) that the physical stuff has physical instantiation and causative capabilities.
Thus, at the most basic level, the stuff can surely be described mathematically (the patterns and nature of this basic stuff are likely very simple, so mathematics is entirely the right way to describe them), but I for one am not quite willing to give up the distinction between a mathematical/computational structure that is physically instantiated and one that is not. (I’m willing to admit that this might be a failure of my imagination.)
To me the term “exists” implies properties of physical instantiation and physically causative powers. Mere coherence isn’t sufficient. I can accept the mathematicians’ use of “exist” so long as it’s regarded as metaphorical.
My solution is to say that physically instantiated objects that are causally connected to the universe that we also are causally connected to “exist”. There might, though, be causally disconnected “meta existences”, which have causation internally to themselves but not to our universe. From their point of view, we would “meta exist”.
VP: “Sherlock Holmes may be a fictional character but the Wikipedia entry indicates he was based on four people he (THE AUTHOR) knew in real life.”
MP:That doesn’t seem to me to make a difference to the argument.
Well the person we thought of as a fictional character actually did exist as four real people (to the author) and the characteristics which they had were aggregated into a model that existed in the readers as Sherlock Holmes. Even if he was a fictional character, we were actually discovering the characteristics of four people who existed in the mind of an author. Math theorems are very similar since we believe IN PRE-EXISTENT FORMS.
As far as the entire math debate goes, including time, aren’t we in a similar situation that we were in 200 years ago when we thought colors existed?
“Smolin and Unger do say that causality is more fundamental than laws of nature: the former explains the latter, not the other way around.”
Which opens a Pandora’s Box for conventional physics. For one thing, time cannot be pre-determined block time and the clock can’t run either direction, because effect cannot lead to cause. So the direction of time is not fixed by entropy, but inertia, as cause inexorably leads to effect.
Which then leads to the issue of whether the “fabric of spacetime” is a valid explanation for the effectiveness of General Relativity, given its treatment of time as an adirectional measure of duration, rather than the dynamic processes being measured.
Pretty much the entirety of current cosmology is based on the premise of this “fabric of spacetime” explaining the expansion of space, even though it completely ignores the fact that the speed of light would have to increase proportional to this expanded distance, in order to remain Constant and be relativistic.
Which gets to what I see as the most important missing element in their thesis, that of space. In the Mathematical Universe, space is treated as a consequence of geometry, but rationally it is its basis.
An infinite number of dimensionless points can’t add up to a line, or a plane, or anything else, because anything, even infinity, multiplied by zero is still zero. So unless those points have some prior dimensionality, i.e.. space, they still add up to a dimensionless point.
If we remove all physical features from space, its only remaining non-physical characteristics are infinity and equilibrium, which, not being physical, don’t need cause, yet provide the conceptual parameters for the models used to describe it and the frames in which dynamics occurs.
Not to get too far off topic, but the consequence of positing time as something other than a measure of duration is to remove a foundational premise from many current models.
Hi everyone! There’s been way too many comments to keep track of, so I’ll just weigh in on a couple of recurring topics.
I believe it would help a lot if everyone were to keep in mind that math is a *language*. The Pythagora’s theorem, for example, should be considered as a *sentence* in this language, which may or may not correspond (up to certain accuracy) to something in the real world. It represents a *description* of some regularity that we observe in the real world, and a Platonist should better be careful not to confuse (or worse, equate) the regularity itself with its mathematical description. This is because, well, the Pythagora’s theorem does *not* hold in the real world, if one looks closely enough (i.e. if one measures triangles precisely enough so that curvature becomes observable), and it is in that strict sense *false*.
The proof of Pythagora’s theorem is actually a reformulation of one math-sentence in terms of other math-sentences (axioms of Euclidean geometry, set theory and first-order logic). This reformulation (colloquially called “deduction”) is being done by syntax substitution rules, which are themselves also axioms, i.e. math-sentences. So Pythagora’s theorem is proven to be “true” not in some universal sense of “The Truth”, but rather contingent on the adopted axioms. And the axioms one adopts (for geometry, arithmetic, chess, whatever) may or may not be a suitable description of some real-world regularity we observe. Therefore, the Platonic “world of ideas” is actually a space of all possible sentences that have a certain truth-value in a given axiomatic system. It is exactly analogous to the space of all possible English sentences one can formulate, upholding the agreed-upon grammar and vocabulary.
Also, people often invoke arithmetic rules as something that is invariably true in the real world — things like 1+1=2. Well, guess what — this isn’t always true in the real world! I mean, if you put one apple into a bag, and then put one more, of course you have two apples in the bag. But if you put one bacteria in a Petri dish, and then put one more, the total number of bacteria inside is not simply “two”, but is rather a time-dependent quantity. Another example is the beta-decay: put two neutrons in a bag, wait 10 minutes. How many particles do you have in the bag? 🙂 So the real world doesn’t always follow ordinary arithmetic. Not even our own computers do — a while ago, computers implemented arithmetic in which 65535+1=0, and programmers would have to jump through hoops to work around that. Today they implement 2^64=0.
Finally, Lee’s concept of evocation can (and should!) be understood in the context of math-as-a-language — some notion is “evoked” if math-sentences defining it are spelled out. An important detail is that evocation is contingent on the agent spelling out those sentences. Two agents, on two different sides of a planet, can both independently spell out the same set of sentences (in equivalent languages). Thus the same notion can be evoked multiple times, once per each agent. The evocation is subjective — it represents the change in the state of our knowledge, not the change in the state of some Platonic “world of ideas” which is assumed to exist independently of our knowledge.
I hope this can help clear out some of the confusion. 🙂
Resisting the temptation to involve myself with anything here except the article itself, I had read both books (the Unger ⅔s is hard to read without falling asleep!). The other book is ‘Time Reborn’, Smolin’s more popular-styled one, on the same topic, from two years earlier. I have been unable to find any ‘positive-re-the-ideas’ reviews of either of these books by physicists: e.g. an applauding Guardian review for the latter book is by Ray Monk, a Wittgenstein expert, whose opinion therefore possibly impresses some, but not me and not only because of his Witty-adulation—I did read Monk’s review! He completely mistakes the reason almost all physicists agree with the non-uniqueness of time, the real reason being that relativity is not falsified with 100 years of extensive observation, e.g. agrees to something like 1 part in 100 trillion re pulsars. Monk thinks that Smolin thinks that the reason is because physics is too mathematical.
But a lack of agreement with Smolin by other scientists shouldn’t be overemphasized, as proofs by authority or by democracy are only a little better than proof by incredulity (as below).
Before getting on to this article’s exact topic (Smolin’s 3rd point, that having been narrowed down to the foundations of mathematics and its effect in physics), I have great interest to later hear what Massimo will say about the 2nd point from Smolin. Twice before, but at the time having only been exposed to blurbs, reviews etc., not the books themselves, I’ve mooted here, with no replies, the question of how Smolin/Unger deal with the apparent contradiction between their proposed resurrection of ‘absolute’ time and the well established relativity of simultaneity in special relativity (and its slightly more complicated analogue in GR). The books make it plain that their basic way for this is to deny (relative simultaneity–non-uniqueness of time—ditto for space), as applied to the universe as a whole, but to somehow maintain it as a local phenomenon.
The explanation for this seems to involve
(1) the existence of a quite recent paper, by a few people at Perimeter mostly, which recasts GR in a very different way;
(2) the fact that a symmetry (Lorentz here) of a PDE takes solutions to solutions but not necessarily to the same solution (physicists say ‘a symmetry of the equation may not apply to a solution’); and
(3) the existence of the global time which cosmologists use, for example to give an age for the visible spatial universe.
I’m suspicious of (3) not even theoretically being an exact, but rather a tiny range, of times, because all gross matter of the visible universe seems to be nearly ‘in the same frame’, the latter phrase sloppy since really GR and universe expansion must be taken into account. But I’m far from knowledgeable about (1) and (2) especially (1). It does seem odd that this recasting above is supposed to be “dual” to usual GR (so equivalent?). But no natural way to extract a universal time from GR seems to exist. Anyway I imagine it’s not a contradiction for a reason that physicists (perhaps here) can explain—I’ll read those for sure!
In any case, this business of resurrecting absolute time is clearly more central than anything else in all this work, and Smolin makes very clear that the above apparent contradiction is definitely a very central matter needing justification. That newish paper seems to be the lynchpin for explaining how simultaneity can be observer dependent locally but not globally. When I phrase it that way, it almost sounds like a logical contradiction. (But I’m the one who complains about people misusing the word ‘logic’, confusing man-in-the-street logic with a deep academic subject, so maybe I’d better withdraw that previous sentence!) (con’t below)
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As far as Massimo’s article and this 3rd point goes, some others here possibly recall something of my tentative opinions, which only partially agree with the few here (e.g. Coel, Disagreeable, Marko, Robin) who have done much more than merely rehashing standard positions without any analysis. Massimo of course referred to “ideas” as “crazy”, not to people. We have so far here never had anything other than such ‘proofs by incredulity’ that MUH is wrong, never disputing even the slightest specific in e.g. Tegmark’s paper in the Annals of Physics, never giving even the slightest attempt at a general definition of the purported difference between the abstract and the physical. I know the latter difference is intuitively ‘obvious’, but so is time-absoluteness/space-observer-independence! Major theoretical physicists seem to almost specialize in showing the intuitively obvious to be false (and now with Smolin that the falseness of one intuitive obviosity is itself false!.
A philosopher purporting, with no detail, that someone exhibited discomfort during a discussion with philosophers goes only an infinitesimal way towards refuting that someone’s conjectures.
I doubt there will ever be any definitive resolution concerning the existence of mathematical structures (please, not merely objects, especially not with the “..loose…” words: “…mathematical objects (a term I am using loosely to indicate any sort of mathematical construct, from numbers to theorems, etc.)…”). And please, make a distinction between mathematical structures and mathematical (axiomatic) systems. Of course my Platonic opinion is that these structures exist in reality (but I am always a bit tentative, some might criticize me as mealy-mouthed—in time, humans will learn more about what a structure should be—mathematics is much less settled in its fundamentals than many seem to think). For me, that structure existence is ‘up to’ a very general form of isomorphism, a simple example being not distinguishing between the natural numbers as a structure under [+,x,order] and under [+,x, successor function]. Objects like ‘137’ arise in the obvious way from the structure. And again, don’t confuse the structure with some system like 1st order Peano arithmetic, more-or-less not confusing semantics with syntax. And Tegmark talks semantics, not syntax, to the extent he is explicit there. I do think he has been over-confident about the finality of the logical basics for mathematics.
But this business of a 3rd and 4th sort of existence in general, with so-called “non-rigid” properties, just doesn’t make any sense at all to me so far.
An individual electron—itself emerging from a more basic quantum field (the latter possibly completely basic in some final theory—pace Marko—and neither difficult to describe in mathematical language nor very far from being more like abstract than like physical, but there ain’t no difference to me)—ahem, an electron has a property called ‘charge’. So I guess charge is rigid. Actually just because that electron is physical in the Massimo/Smolin/etc menagerie, apparently that automatically makes all its properties rigid by definition. (con’t below)
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Massimo phrases it: “By “rigid properties” here Smolin means that the objects in question present us with “highly constrained” choices about their properties”. I tend not to be ‘presented’ with choices very often by objects, and can hardly see the properties of a basic object being in any way dependent on me, on any humans, nor on any constraints or otherwise.
He also says: “Sherlock Holmes did not exist until …. wanted with him…” I find it hard to understand why anyone would regard whatever Sherlock is in the brains of different people, or even the same person at different times, as really all being the same existing object, even if it is important for one’s aesthetic enjoyment to pretend they are all the same and really exist outside one’s brain. That is the only purported example of anything populating the bottom line in the table of supposed different modes of things existing. Again, in what sense can an actual object be said to have constrained or unconstrained properties, other than the trivial sense of being completely constrained?
Smolin, unfortunately, doesn’t talk about the bottom-left “fictional” box, nor does Massimo, (presumably non-empty for them??) comprising objects that had prior existence and yet are not possessing rigid properties. So I expect someone will help me to stop saying it doesn’t make sense by offering a decent example. But how can the properties of an object somehow depend on someone or something taking advantage of this non-constraint? Is it a matter of those properties changing as a function of time? Or a matter of Joe saying x has positive barge, but Mo saying x has negative barge, and no one can dare disagree with either, because there are no criteria for judging properties such as ‘barge’?
And, as in 2nd above paragraph, the other, right-hand square, in the bottom line (invented objects with non-rigid properties i.e. by the characterization above, objects with non-constrained properties whatever that could mean: Can Massimo or Smolin or anyone else here give even a single example from this line, other than mistaking all the instances at different times and/or for different humans of something always given the same name for entertainment purposes, mistaking them for being a single existing object known as Jack Frost? Lacking convincing examples, my conclusion would remain that the bottom line is literally fictional, now “fictional” in the sense of being a non-starter. The top line is ‘fine’, we’re back to the age-old conundrums about existence, with the MUH suggested answer shrinking the line to just its left-hand square. So until otherwise convinced, if Massimo’s description is accurate, I doubt that Smolin/Unger have made any progress at all in explaining the age old controversies concerning the existence of mathematical objects.
So far as the unreasonable (or otherwise) effectiveness goes, the famous paper of Wigner is listed, but its main point seems to be ignored by Smolin, and certainly is ignored in the article. That point is the extraordinary fact that many mathematical structures were discovered (not invented IMHO) purely by curiosity/aesthetic considerations, and long before these became central aspects of physics theories—Riemannian geometry long before GR, Hilbert spaces a decade or two before quantum mechanics, and, for a less often mentioned one, Clifford algebras and the spin groups and their representations (Elie Cartan, building on Lie, Clifford, …) several decades before Dirac with his extraordinary equation and his theoretical discovery of antimatter. So the main point there is simply not engaged. And again as yet I see here little contribution to either the solution or dissolution of that question. MUH obviously dissolves it.
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Massimo had the politeness to “answer” me. Let me reciprocate: “Smolin,” says Massimo, “as a counter [to Platonism], offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”
Smolin is apparently unaware of a whole theory of “truth” in mathematical logic, and of the existence of the work of famous logicians such as Tarski. When Smolin was in the physics department of Berkeley, so was the very famous Tarski, in the mathematics department.
What does Smolin say? Nothing recent. Smolin says mathematics is axiomatic, and develops like games. That was at the heart of the effort of mathematical logic, a century ago, to make Axiomatic Systems correspond to sequences of numbers.
Gödel showed that this approach could not work in any system containing arithmetic. Smolin is now trying to reintroduce it, as if Gödel never existed.
Does Mr. Smolin know this? Not necessarily: he is a physicist rather than a mathematician (like Tarski, or yours truly).
Smolin: “Both the records and the mathematical objects are human constructions which are brought into existence by exercises of human will.”
Smolin: Math brought into existence by HUMAN WILL.
So how come the minds of animals follow mathematical laws? Dogs, in particular, behave according to very complicated applications of calculus.
How come ellipses exist? Have ellipses been brought into existence by Smolin’s “human will”? When a planet follows (more or less) an ellipse, is that a “construction which has been brought into existence by exercises of human will”?
Some will perhaps say that the planet “constructs” nothing. That I misunderstood the planet.
Massimo’s quotes me, and asserts there is no value whatsoever to the existence of mathematical objects:
I said: “How come enormously complex and subtle mathematical objects, which are very far from arbitrary, exist out there?”
Massimo replied: “They don’t.”
And Massimo does not explain why he thinks the spiral of a nautilus does not exist. According to Smolin, the spiral is just a “construct of human will”.
I am actually an old enemy of mathematical Platonism. However, I don’t throw the baby with the bath.
I agree that the “Mathematical Universe Hypothesis”, and Platonism in general are erroneous. However that does not mean they are deprived of any value whatsoever.
MUH is: ‘Our external physical reality is a mathematical structure.’
I propose a completely different route: our mind are constructed by (still hidden) laws which rule the universe. Call that the Mathematical Mind Hypothesis (MMD).
Our internal neurological reality constructs real physical structures we call “mathematics”.
This explains why a dog’s brain can construct the neurological structures it needs to find the solutions of complex problems in the calculus of variations.
Dogs did not learn calculus culturally, by reading books. However their brains construct structures which solve very complicated calculus of variations problems. As explained by the MMD, (hidden) physics shows up in neurological constructions we call mathematics.
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>Doesn’t this type of Platonism boil down to the claim that everything that is conceivable exists timelessly? Or is there an example of anything that is conceivable but does not exist on this view? And if “x exists” just means “x is conceivable” under this view, in what respect is it advantageous to talk of existence?
Apparently I squeezed in my own answer at the top of this second page (just thirteen minutes before you asked) and your question does suggest that you see things about the same as I do. We are quite free to define “existence” broadly enough to encompass all that is conceivable, as some quite validly do here. But do we also find this more useful than a purely physical definition? Given some of the noted implications of “nonphysical existence” however, I wouldn’t say so.
>You seem to implicitly adopt a concept of progress in philosophy that is modeled on the sciences. That doesn’t work, as I tried to articulate in a number of other discussion threads and posts. I’m finishing a whole book on the subject, so stay tuned…
Yes you are correct about my implication, though I can’t yet agree that it “doesn’t work,” but merely that it “hasn’t worked so far.” (Note that science really is still quite young.) I will certainly stay tuned, since you must know that I’ve been hooked. Nevertheless I do hope that your point here is simple enough, or that I am sensible enough, so that an entire book isn’t required to convince me!
Still, from my current perspective we might have two separate varieties of philosophy — both a “traditional” one, as well a “science” version of it. The central difference between them, I think, would be that instead of notions of morals/ethics, this new field would be founded upon the biological nature of good/bad for the conscious entity (or perhaps “punishment/reward”).
On the Coel-Disagreeable_Me-Massimo debate on the the philosophy of mathematics, related to the Smolin principle “came into existence at the moment the rules were codified”: Those who say that mathematics is created by homo sapiens are right, and those who say that mathematics is “continuous” with the physical world are right, since the devices (brains of homo sapiens) that created the mathematics are physical devices that produce only physical outputs. But this just shows that there can be physical things that can create new physical things that have never appeared in nature before. Synthetic biology is doing that, too. For mathematics, formulation in the vocabulary of HoTT — Homotopy Type Theory — is useful in showing mathematics to be a synthetic enterprise. Computers (other physical devices) are now creating new mathematics and they (future computers and robots) will do more of this creation of new mathematics in the future, and while what they create is novel, they are not entities that have a nonphysical existence outside of the physical realm.
On Patrice Ayme’s comment about mathematica truth, Tarski, and Gödel: What I think is closer to the Smolin principle is the mathematical pluralism of Joel David Hamkins (professor, CUNY):
“Pluralism in mathematics: the multiverse view in set theory and the question of whether every mathematical statement has a definite truth value”
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Hi Massimo, interesting article and as it turns out quite timely since DM and I are having an off-site conversation on this issue. Basically Smolin describes something very close to my own ideas (I call it “defining” though “evocation” is sexier).
Hi DM, looks like we are directly at your last email to me ☺ I was going to discuss your statement that defining/evoking pops something into existence by force of will. At this point I understand your idea of abstract existence, but I think there is something being missed in the nature of existence for something evoked (I will use that word for consistency with the article). It may be real, but it is a transitory, dependent, imaginary existence which means it requires entities to maintain it. As long as it is held within one’s own mind that is where it exists, and if it is shared with others then it exists with them. It does not exist “out there”. The elements which helped a mind evoke it may exist, but not the item.
As an example, spelunker A is the first person to ever wander into a certain cave and sees three crystals shining on a cave wall. She evokes a triangle by connecting the crystals with 3 imaginary lines. The triangle does not exist anywhere but in A’s mind, which is made explicit as the lines are imaginary and do not exist external to A. Later spelunker B finds the same crystals and muses in the same way, evoking an identical triangle. Spelunker B may have re-discovered the crystals, and re-discovered that they can be used to evoke a triangle, but B did not (re)discover A’s triangle since that went out within A’s head when she left. It was the pattern-recognition software and inventiveness in each of their heads that used the crystals to make a geometric shape, and not the other way around.
Of course A could take C down and show the crystals and explains how a triangle could be imagined, or describe what he saw and imagined at some party to D,E,and F who then go down to the cave to see these crystals (and the triangle C had envisioned). In this way A’s triangle is “shared” between others and becomes real to them. But the existence remains “imaginary” which is to say, dependent on people’s imaginations of it. That is not the same as the crystals stuck in the face of that cave, which can be seen and manipulated and will act upon other bodies whether those bodies can imagine their existence or not. And they will never be affected by some “triangularness” property.
Regarding games, you can re-invent chess, but you don’t re-discover it. Invention requires a creative act of the imagination, including perpetuating its existence, discovery does not. It seems more awkward to believe chess always existed, waiting to be discovered, than that people are able to invent it at different places and times, with no knowledge of the other.
dbholmes: C may not be initially cognizant of the triangle til A says “Look…” or imparts a set of instructions or axioms to convince him. Both parties also share the same language and same biological platforms to see triangles.or “What It’s Like To Be A Mathematician”.
Hi Victorpanzica, I agree with your description of A giving C some details to help. That is part of the sharing process to get C to evoke the triangle.
I also agree (very strongly) with your statement of their sharing the same language and more importantly same biological platform to see triangles. That is a solid bit of evidence that these triangles do not have some real, external existence but are dependent on minds to make them using external (or internal) cues.
BTW I freely admit that I do not have this all sorted out and really I am not sure how these observations impact on the reality of math. I make these comments because here an in the morality discussions it seems many of us might disagree on what is a real property or what is real. So feel free to point out problems with my reasoning and points I miss.
“Smolin, as a counter, offers his model of development of mathematics, which does begin to provide an account for why mathematical theorems are objective (the word he prefers to “true,” in my mind appropriately so).”
This makes some sense given his position on math. Is the problem because truth is often tied to “reality”, which is what is under discussion?
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?”
I think these are good questions. But I think these questions tend to show that even though math is not spatio-temporal it is embedded in reality. If it weren’t embedded in in reality we wouldn’t get the scientific results that are expressed in mathematical terms. It seems that scientific theories themselves have no spatio-temporal qualities. Yet they are embedded in reality.
I admit it is easy to get tied into thinking only material things are real. It’s true material things are the only things that are real in a material sense, but I tend to view reality as having more than that.
Tallness, greenness, and length are qualities that themselves do not have spatio- temporal qualities but they seem embedded in reality to differing degrees. There is no material, spatio-temporial thing we call greenness or tallness or length. That does not mean these concepts are not embedded in reality.
When I look outside and see green grass I realize that light of a certain wavelength is being reflected off the grass while the other wavelengths are absorbed. The greenness is not in the grass so to speak but it is linked to the reality of the grass.
The link below is considered an optical illusion because if you stare directly at the cross in the middle you will see a green dot. But this is called an illusion because the green is not appearing to us due to the normal process where the non- green wavelengths are being absorbed. You can see this by trying to look directly at the green dot instead of the cross in the middle. You will see white which means all of the visible wavelengths are in fact being reflected.
BTW optical illusions like this can also apply to length or size of an object as well.
So we would say the grass is really green where as this green ball in the illusion is not “really” green. And this fact that the grass is reflecting mainly green wavelengths is not just due to a construct we created – like in Sherlock Holmes. It’s true in one sense that Sherlock Holmes is male. But this is just due to our constructing him to be so. It’s a reality we made up, and to that extent it is not reality at all.
Now where does math fit in this? I am not sure. But I tend to think it is real more like grass is green than Sherlock Holmes is male. Chess is also a construct but like Sherlock Holmes (and unlike greenness) it seems to have no bearing on the reality beyond its construct.
Math on the other hand at least seems to have a stronger case for being more like greenness, tallness and length.
You realize, I expect, that your general description of mathematics and logic would take Tarski’s definition of truth as simply worthless within mathematics itself. This is despite it being taught by almost every mathematical logician as central, near the start of any decent course, as the fundamental connection of syntax with semantics in 1st and higher order logic. Perhaps you know better than them that mathematical structures do not exist, there is no such thing as a mathematical model of a set of axioms, only perhaps physical models. A person advocating MUH would agree with the latter phrase, but not the former, since she would not distinguish between mathematical and physical structures. You may possibly be correct about this, to the extent we are even talking about a decidable or at least meaningful question. But you have very little support among 21st century mathematicians, logicians and philosophers in completely removing semantics from within math and logic. People here should realize that this early 1900’s style, of regarding those subjects by some physicists as purely manipulative syntax of formal logic, has many serious doubters.
It is ironic, given your past writings about Godel incompleteness, that most formulations of that need to use the word “truth” and so employ Tarski semantics, though there is of course a syntactic version whose conclusion merely assets the existence of non-deducible propositions whose negations are also non-deducible.
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You say “…General Relativity, given its treatment of time as an adirectional measure…”. This is utterly false, and possibly comes from either or both of
(1) silly statements by pop science book writers along the lines of time not existing in the “block universe”;
(2) the fact that Einstein’s field equation is invariant under the symmetry of time reversal.
The latter fools people into the apparent assumption that GR is no more than that equation. But it also more basically involves a 4-dimensional manifold with EXTRA structure, namely a 3+1-quadratic form (“pseudo-metric”—which is really the entire structure in SR, that I guess would have no physics at all in it for those thinking the field equations say all in GR!), plus one choice of two possibilities for positive cone. That structure induces a partial order, that is what time is, it is perfectly directional in that sense, but it’s not a total order.
So (1) in the context of being said often Smolin’s recent two books is just a catch-phrase to sell books. He is perfectly aware of the directionality of time in both SR and GR, and certainly does not make any mistake along these lines when it comes to really getting down to details.
The “choice” above does say that without it there are two models for ‘reality’ so far been given (or in grandfather murdering acausal models, none) . But it’s not one “adirectional” model (except in the strange cases above mentioned in preliminaries, then excluded axiomatically, in treatments of the Hawking/Penrose black hole existence theorems). It is two possible directional models, though in SR only one up to isomorphism!
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