Defending scientism: mathematics is a part of science

1+12[Editor’s Note: This essay is part of Scientia Salon’s special “scientism week” and could profitably be read alongside other entries on the same topic on this site, such as this one by John Shook and this one by yours truly. My take on the issue is very different from that of the authors who contributed to this special series, and indeed close to that of Putnam and Popper — as it should be clear from a recent presentation I did at a workshop on scientism I organized. Also, contra the author of the third essay in this series (but, interestingly, not the author of the first two!) I think the notion that mathematics is a part of science is fundamentally indefensible. Then again, part of the point of the SciSal project is to offer a forum for a variety of thoughtful perspectives, not just to serve as an echo chamber for my own opinions…]

by Coel Hellier

While the term “scientism” is often a rebuke to those considered to be overstepping the proper boundaries of science, plenty of scientists will plead guilty to the charge so long as they get a say in how the term is defined. The “scientism” that I defend is the claim that, as far as we can tell, all human knowledge is empirical, deriving from contact with empirical reality. Further, that empirical reality seems to be a unified whole, and thus our knowledge of reality is also unified across different subject areas so that transitions between subjects are seamless.

What we call “science” is the set of methods that we have found, empirically, to be the best for gaining knowledge about the universe, and the same toolkit and the same basic ideas about evidence work in all subject areas. Thus there are no “other ways of knowing,” no demarcation lines across which science cannot tread, no “non-overlapping magisteria.”

A related but different stance is expounded by Pigliucci in his critique of scientism [1]. Pigliucci instead prefers the umbrella term “scientia,” which includes “science, philosophy, mathematics and logic.” This sees mathematics and logic as epistemologically distinct from science. Indeed, Pigliucci has remarked:

“it should be uncontroversial (although it actually isn’t) that the kind of attention to empirical evidence, theory construction, and the relation between the two that characterizes science is ‘distinctive enough’ … to allow us to meaningfully speak of an activity that we call science as sufficiently distinct from … mathematics.”

“… Mathematics is a huge area of knowledge where science has absolutely nothing to say, zip …” [2]

In this piece I argue that mathematics is a part of science. I should clarify that I am taking a broad interpretation of science. Nobody who defends scientism envisages science narrowly, as limited only to what is done in university science departments. Rather, science is conceived broadly as our best attempt to make sense of the empirical evidence we have about the world around us. The “scientific method” is not an axiomatic assumption of science, rather it is itself the product of science, of trying to figure out the world, and is now adopted because it has been found to work.

I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is:

1 + 1 = 2

Do you accept that statement as true? If so (and here I presume that you answered yes), then why?

I argue that we accept that statement as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world [3].

It is a fact about the world in the same way that apples falling downwards are a fact about the world. There is no good reason to place these two different facts (gravity and maths) into two incommensurate domains of knowledge. Our understanding of both derives from empirical reality, and thus both are equally “scientific.”

Having asserted that, let me argue against possible alternative answers to my question of why we accept that 1 + 1 = 2.

Maths is derived from axioms

One answer — perhaps the commonest — asserts that maths is not derived from empirical observation but instead derives from axioms.  You might assert that you accept 1 + 1 = 2 because it is proven so from the basic axioms of maths.  You might point to Peano’s axioms and assert that from those one can logically arrive at 1 + 1 = 2 [4].

My first reply is that I don’t believe you. I don’t believe that there was a time in your life when you were dubious about the assertion 1 + 1 = 2, but then consulted Peano’s axioms, and after some logical thought concluded that, yes, 1 plus 1 really must equal 2. I assert that you accepted 1 + 1 = 2 long before you knew anything about Peano’s axioms, and that you accepted it because it works in the real world: if you had two sweets you could give one to your pal and eat the other yourself [5].

But, even if your belief that 1 + 1 = 2 does derive from axioms, whence your faith on those particular axioms? How and why did Signor Peano arrive at that set of axioms? I assert that they were arrived at with the fact of 1 + 1 equalling 2 being a necessary consequence. Had Peano’s axioms resulted in 1 + 1 equalling anything other than 2 then the axioms would have been rejected as faulty. Signor Peano would have been told to go away and come up with axioms that worked (ones compatible with the non-negotiable truth that 1 + 1 really does equal 2).

Thus, the axioms mathematicians adopt are not arbitrary, chosen by whim or fiat, they are chosen to model the empirical world. Mathematics is thus distilled empiricism. The same can be said about logic and reason. In order to get from Peano’s axioms to derived results you need to use logical reasoning. What validates that logic and that reasoning? Again, I assert that empirical reality validates them. The reason that we adopt logical axioms such as the law of non-contradiction is that they hold in the empirical world [6]. How else would we know which logical axioms to adopt? Thus the whole edifice of mathematics and logic is a distillation of empiricism, created and developed as a model of the basics of how our world works.

Mathematics is arbitrary

Nevertheless, some might assert that no, mathematics is a self-contained logical system entirely distinct from empirical reality, and that any correspondence between mathematics and science is simply a coincidence. Some might even assert this with a straight face. It leads to puzzlement over what Eugene Wigner called “the unreasonable effectiveness of mathematics” when applied to science, but there is no puzzle if mathematics describes deep properties of our empirical universe and is derived from that universe. The idea that mathematics is arbitrary and independent of our universe would be more convincing if mathematicians spent as much time pursuing maths based on 1 + 1 equalling six and a half as they do with 1 + 1 = 2.

A more sophisticated version of this answer accepts that mathematics originally derived from our universe (with, for example, Pythagoras’ theorem resulting from drawing on bits of paper, or from attempts to get a building’s walls square), but points out that nowadays mathematicians experiment with all sorts of axioms that are not first suggested by observation.

As an example, consider the generalization of the “flat” geometry developed by Euclid to the “curved” geometries developed by Carl Gauss, Bernhard Riemann and others. The relaxing of the parallel-line postulate of Euclid to produce non-Euclidean geometries was not motivated by observations but by thinking about the structure of the axiomatic system. Surely this is a non-empirical approach that distinguishes mathematics from science?

Well no. Theoretical physicists do this sort of thing just as much as mathematicians. They take their set of empirically derived axioms (though in physics these tend to be called “laws” rather than “axioms”) and think about them; they experiment with different axioms/laws and work out the consequences. Often they are not immediately motivated by a match to observations but are following their intuition.

They are still, though, working with an axiomatic system that is essentially distilled from the empirical universe, and they are using an intuition that is also very much a product of the empirical universe. Curved geometry — developed by the mathematician Riemann — was later found to be useful in describing the universe when the physicist Einstein — also following a path of logic and intuition — developed the theory of General Relativity. If anyone wants to draw a demarcation line between domains of knowledge, the line would not be between the mathematician Riemann and the physicist Einstein.

Why is it that mathematicians’ intuitions so often produce mathematics that is later found to be useful to physicists? I argue that their experimentations with axioms are productive because their logic and intuitions are also empirical products. Thus a mathematician has a good idea of which changes to axioms are sensible and which are not. Allowing parallel lines to diverge (and thus producing non-Euclidean geometry) is sensible; adopting “one plus one equals six and a half” is not [7]. In both mathematics and physics, if the experimentation produces results that are nonsensical when compared to our universe then they will not be pursued. The empirical universe is in both cases the ultimate arbiter.

At the cutting edge it can, of course, be unclear whether maths and/or physical theories “work.” A current example is string theory, where a generation of theorists is exploring the mathematics of strings. Maybe it’ll lead to new physical theories unifying quantum mechanics and gravity, and maybe not. At the moment, though, one could not really say whether string theory was “mathematics” or “theoretical physics.” This emphasizes the seamless transition between those fields, with string theory straddling the (arbitrary and unmarked) boundary.

An aside before proceeding. Gödel’s incompleteness theorem tells us that even if we have a set of axioms such as Peano’s, which underpin the natural counting numbers and which yield the statement that 1 + 1 = 2, there will be other statements about the natural numbers which are true, but which cannot be shown to be true from the axioms. A further result tells us that the axioms cannot be used to show that the system built from those axioms is consistent. This fundamental limitation of an axiom-based approach shattered hopes of mathematics ever being a complete, consistent, self-validating and self-contained system.

From a scientific point of view, with mathematics being seen as a part of science, such limitations are unsurprising. Science is derived from empirical evidence and our available evidence will always be a small and incomplete sample of the universe, and thus scientific results are always provisional, in principle open to revision given better data.

Our math is the product of pure logic, deriving only from human intuition

Many will disagree with me and assert that human intuition is a primary source of knowledge, distinct and separate from empirical evidence. Indeed this idea is popular with some philosophers, who argue that introspection and thought are the wellsprings of their philosophical knowledge, and thus that philosophy is a domain distinct from the empirical domain of science [8].

However, what basis do we have for supposing that our human intuition produces accurate knowledge about the universe? The first reason is that our intuition has been developed and honed over our lives based on our sense data about the world around us. Thus our intuition is very much an empirical product.

Further, we can ask about instinct, that portion of our intuition that is not the product of life experiences, but is encoded in the genes. Our genetic programming will also be a product of empirical reality. Our brains are the product of evolutionary natural selection, and thus have developed to make real-time decisions that aid survival and reproduction. Obviously, decision-making that bore no relation to the real world would be useless, and thus we can have some confidence that our intuitions are to a large extent programmed to produce decisions well-aligned to empirical reality.

Of course natural selection is not a perfect programmer, and anyhow is not aiming at a perfect and unbiased decision-maker, it is aiming at the one best at survival and reproduction. Thus we would expect our intuition to be reliable only with respect to the everyday world relevant to survival and reproduction, and to be unreliable about aspects of the universe (such as quantum mechanics and general relativity), that are irrelevant for everyday life.

We’d thus expect our intuition to be a folk metaphysics, good enough for many purposes, but full of biases and foibles, particularly so where an inaccurate assessment might actually aid survival and reproduction. An over-active pattern-recognition detector and the Lake Wobegon effect are likely examples of this. Visual illusions such as the checker-shadow illusion [9] show how easily human intuition is fooled, in this case precisely because it is making some assumptions about how the world works, and thus about lighting and shading.

A critic might, though, accept that some of our intuitions are related to empirical reality, but argue that intuition also gives access to knowledge that is not empirical and cannot be arrived at by empirical means.  My response is to ask what basis the critic has for that assertion and what reason he has for supposing that “non-empirical knowledge” has any reliability or validity.

From the evolutionary perspective we have no good reason to suppose that intuition is anything other than an imperfectly and empirically programmed device that models the empirical world — after all, failing to find enough to eat, ending up eaten by a predator, or finding a mate and successfully rearing children, are all aspects of a brute empirical world. Thus we should accept intuition as a useful “quick guide to reality,” but ultimately we should not accept it except where corroborated by empirical evidence. Indeed, the whole point of the scientific method is to use empirical evidence to do much better than just consulting our “quick guide” intuition [10].

Our math is the only possibility

The last alternative answer that a critic might advance is that we accept the claim that 1 + 1 = 2 because it must be true, it is the only logical possibility. Thus, such a critic will say, 1 + 1 equalling six and a half is simply nonsensical. Such a person would not merely be asserting that it is impossible in our world, but that it is impossible in all possible alternative worlds.

Do we know this? And, if so, how? Has anyone given a logical proof of the impossibility of such an alternative scheme? Any such proof could not use any axiom or logic derived from or validated by our empirical world (that would only show that such alternatives did not occur within our world).  But without that, how would one go about showing that the logic of our world is the only one possible?

One could not use our-world logic for such a task, nor could we use human intuition, since our intuition is very much derived from and steeped in the logic of our own empirical world — indeed our brains have evolved precisely to model the logic of our world — and thus we would not expect them to be in any way useful for contemplating radically different alternatives.

But, even if we were to grant the claim that our world’s logic is the only possible system of logic, that would still leave the question of how we came to learn about that logic.  And the only plausible answer is that we learned from observation of the empirical universe and thence deduction about the logic by which it operates.

Conclusion

I have argued that all human knowledge is empirical and that there are no “other ways of knowing.”   Further, our knowledge is a unified and seamless sphere, reflecting (as best we can discern) the unified and seamless nature of reality. I am not, however, asserting that there are no differences at all between different subject areas. Different subjects have their own styles, in a pragmatic response to what is appropriate and practicable in different areas. For example, a lab-based experimental science like chemistry has a very different style to an observational science like astronomy [11]. Further, biochemists studying detailed molecular pathways in a cell will have a very different style from primatologists studying social interactions in a wild chimpanzee troop.

Such differences in style, however, do not mandate that one of those subjects be included within “science” and another excluded. The transitions in style will be gradual and seamless as one moves from one subject area to another, and fundamentally the same basic rules of evidence apply throughout. From that perspective mathematics is a branch of science, in the same way that so is theoretical physics. Indeed, some theoretical physics is closely akin to pure maths, and certainly far closer to it in style and content than to, say, biochemistry. The different subject labels can be useful, but there are no dividing lines marking the borders. No biochemist worries about whether she is doing biology or chemistry, and string theorists don’t worry much whether they are doing maths or physics.

Thus, in arguing that a subject lies within the broad-encompass of “science,” one is not asserting that it is identical in style to some branch of the generally-accepted natural sciences, but that it belongs to a broad grouping that spans from studying molecules in a chemistry laboratory, to studying the social hierarchies of a baboon troop, to theoretical modeling of the origin of the universe, and that it belongs in that group because epistemologically the resulting knowledge has the same empirical source.

I thus see no good reason for the claim that mathematics is a fundamentally different domain to science, with a clear epistemological demarcation between them. This same set of arguments applies to the fields of reason and logic, and indeed anything based on human intuition. All of these seem to me to belong with science, and all derive from our empirical experience of the universe and our attempts to make sense of it.

_____

Coel Hellier is a Professor of Astrophysics at Keele University in the UK. In addition to teaching physics, astrophysics, and maths he searches for exoplanets. He currently runs the WASP-South transit search, finding planets by looking for small dips in the light of stars caused when a planet transits in front of the star. Earlier in his research career Coel studied binary stars that were exchanging material, leading up to his book about Cataclysmic Variable Stars.

[1] Massimo Pigliucci, Midwest Studies in Philosophy 37 (1):142-153 (2013) “New Atheism and the Scientistic Turn in the Atheism Movement.”

[2] See Pigliucci’s article, Staking positions amongst the varieties of scientism.

[3] A pedant might point out that in modular arithmetic, modulo 2, 1 plus 1 would equal 0. I am taking 1 + 1 = 2 to refer to simple counting numbers; one apple plus one apple equals two apples. If we ask further about the basic concepts of “1,” “2,” “+” and “=” I would again base them on patterns discerned in the empirical world, which is of course how humans first came up with those concepts.

[4] Giuseppe Peano, 1889. Arithmetices principia, nova methodo exposita.

[5] A pedant might point out that that equates to 2 – 1 = 1, not to 1 + 1  = 2.

[6] Indeed the great Islamic polymath Avicenna wrote, circa AD 1000, that: “Anyone who denies the law of non-contradiction should be beaten and burned until he admits that to be beaten is not the same as not to be beaten, and to be burned is not the same as not to be burned,” a direct derivation of logic from empirical experience!

[7] The Axiom of Choice is an example of an axiom adopted largely because it feels intuitively right to mathematicians, plus they like the results that it leads to.

[8] There is a vast philosophical literature on this issue, with Kant’s Critique of Pure Reason being influential.

[9] For the Lake Wobegon see here. For the checker shadow illusion see here.

[10] An obvious example being the need for double-blinding in medical trials, which originated from the realization of how unreliable human intuition, based on anecdotes and a partial memory, actually is.

[11] One should ignore commentators who over-interpret overly-simplistic accounts of the “scientific method” and claim that only lab-based experimental science counts as science.

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341 thoughts on “Defending scientism: mathematics is a part of science

  1. > 1 + 1 = 2

    Do you accept that statement as true? If so (and here I presume that you answered yes)

    Of course I don’t accept this statement as “true”. It’s easy to find (mathematical) fields, even infinite ones, in which the statement is not true (fields of characteristic 2). It all depends.

    There’s a reason A why we think that mathematical statements are correct.
    There’s a reason B why we think that mathematically formulated statements about objective reality are correct (as statements about objective reality).
    A and B are different.

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  2. The “scientism” that I defend is the claim that, as far as we can tell, all human knowledge is empirical, deriving from contact with empirical reality.

    What we call “science” is the set of methods that we have found, empirically, to be the best for gaining knowledge about the universe, and the same toolkit and the same basic ideas about evidence work in all subject areas.

    How could we empirically learn, that learning empirically is the “best” or “only” way of attaining knowledge?

    In other words, what is your take on the problem of induction?

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  3. Hi Patrick,

    Of course I don’t accept this statement as “true”. It’s easy to find (mathematical) fields, even infinite ones, in which the statement is not true (fields of characteristic 2). It all depends.

    You might want to read footnote 3.

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  4. Hi Coel,

    My issue with your view is twofold.

    Firstly, I think you are unjustified in seeing continuous variation between fields as license to include them all under one umbrella term when that term as commonly used for only a certain domain. You may as well say that everything is mathematics, or that everything is biology, or that everything is philosophy or reason or common sense.

    Science refers, very specifically, to the formal academic study of the natural world and the human being. I agree that there are no clear demarcations, but it is also true that biomes vary continuously from arctic tundra to equatorial jungle. That does not mean we can sensibly call all environments tundra or jungle.

    “Science” refers to a certain subset of what Massimo calls “Scientia”. It refers to physics, chemistry, biology, medicine etc, and also (more controversially perhaps) to the social sciences. The distinction between theoretical physics and mathematics can indeed get very fuzzy, but that doesn’t mean that we can’t reasonably consider them to be different regions of the continent of Scientia even if the exact borders are disputed territory.

    If you want an umbrella term that will not cause needless confusion and aggravation, I urge you to go with Scientia. It means precisely what you mean by science so you lose nothing important but gain much in the ability to communicate clearly.

    This point also extends to empiricism. You argue that there is no qualitative difference between ‘knowledge’ in the form of instincts primed in the course of evolution over many generations and ‘knowledge’ gleaned from direct observation. This claim is controversial, but even if it is true, the fact that the quantitative difference is so great justifies the usage of different terms. Empirical knowledge is arrived at more or less directly, while a priori or deduced knowledge is arrived at through indirect methods. Even though you can arguably trace a justification of a priori knowledge back to empirical roots, it is just not right to call this knowledge itself empirical. The more indirect a chain of reasoning, the more likely it is that some error has crept in. The great advantage of empiricism is to prefer direct measurement to indirect deductions wherever possible so as to correct for faulty assumptions or reasoning. To call all indirect reasoning empirical is therefore to miss the point of empiricism entirely.

    The second thing I think you get wrong is where you argue that our mathematics is not the only possibility. You think there could be a mathematics, applicable to some other universe but not this, where 1+1=6. This is simply not true, because 1+1=2 is entailed by the definitions of those symbols. Perhaps there is a universe where you could consistently put one apple, then another in a bag and find you have six apples. I doubt it, but let’s suppose that’s true. It would not be the case that in that universe 1+1=6, because the plus symbol is defined to mean addition. Whatever you’re doing in that universe when you put apples in a bag, it is not addition but some other operation (for example, perhaps it is the function f(x,y)=(x+y)^2+x+y). What you propose is like suggesting that there could be some other universe where a bachelor is not an unmarried man. It’s nonsense, because it requires that we ignore what the definitions of the terms mean.

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  5. “all human knowledge is empirical, deriving from contact with empirical reality.”
    I’m a bit worried about this statement; it would be too easy to make it trivially true by simply denying that anything that isn’t knowledge clearly circular.

    I would rather say that all human knowledge derives much of its former and content from the empirical, in combination with human reasoning and collective discourse. The proportions of achievement will vary from case to case, and it’s unlikely that any worthwhile knowledge will be totally free of epically content. But surely your 1+1 example is not typical of the whole of maths. Moreover, at best you’ve shown that the empically contributes to that knowledge, not that it is wholly empirical or that the empirical is necessary and sufficient in every case?

    I wonder where you would place ethics or aesthetics; as science, or not truly knowledge, or what. I’ve say that ethics must be informed by the empirical, but can’t be reduced to it. I’ve seen attempts to reduce aesthetics to the psychology and physiology of evolved perceptions, but they were very unconvincing.

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  6. “I argue that we accept that statement as true because it works in the real world.”

    All that your piece achieves is to showcase your utter lack of acquaintance with both mathematicians (most of whom would laugh about your assertion) and philosophers of logic and mathematics. You write about mathematics (number theory, of all things!) and its Grundlagenproblem like someone who works in applied physics or engineering and is ignorant of even the most basic arguments made in the past 200 years. And while most people from these fields would not rely on their insufficient professional intuitions when publicly engaging a rather sophisticated field of abstract thought, you seem to have no such qualms.
    Disagreeable Me has given you some arguments to think about. Aside from that, go read a book, preferably one by Cantor or Hilbert.

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  7. SciSal: “Gotta celebrate the occasion…”

    Whereas I am in complete agreement with Coel’s whole essay.

    The words host, many, few, are all adjectives expressing quantity in decreasing but imprecise size: one (1) is another such but with useful precision. It is a value like yellow and does not exist in reality unless applied to some other entity. I would defy anyone to show me a ‘yellow’ or a ‘1’: they are each a word with a commonly-understood meaning by which we may communicate an idea (electro-chemical state) between brains.

    IMO ‘A priori’ knowledge, though tempting and ingenious, is only a philosophical ‘illusion’.

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  8. Hi Patrick,
    As DM noted, I was restricting the statement to simple counting arithmetic: “one apple plus one apple equals two apples”.

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  9. Hi mira,

    How could we empirically learn, that learning empirically is the “best” or “only” way of attaining knowledge?

    We learn that from the fact that no other method has been shown to work. If, for example, divination were shown to be a better method of predicting future solar eclipses than science, then I’d have to change my opinion. That stance is then provisional, of course, in that “other ways of knowing” could be shown to work; so far it seems to me that they have not been.

    In other words, what is your take on the problem of induction?

    I think one can make a fair probabilistic justification for using induction (not showing it to absolutely reliable, just probabilistically useful). I attempted that here.

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  10. Hi DM,

    If you want an umbrella term that will not cause needless confusion and aggravation, I urge you to go with Scientia.

    I’m not wedded to the nomenclature and don’t really care whether we call it “science” or “natural philosophy” (the old name for science) or “scientia” (Massimo’s term) or whatever.

    What I do care about is the epistemology, the origins of knowledge. Is our knowledge a unified whole, deriving from our contact with the “real world” (essentially empiricism), or do the different domains of knowledge derive from radically different epistemology? Thus when you remark on:

    … seeing continuous variation between fields as license to include them all under one umbrella term …

    That’s the point, if the different fields do have the same epistemology at root, then it’s fair to use an umbrella term for all of it. Again, I honestly do *not* care what that term is! Why do I use the terms “scientism” and broadly defined “science” for this? First, because it is the most widespread usage for that sort of attitude, and secondly because the word “science” emphasizes the empirical nature of knowledge.

    Why do I not use “scientia”? First because it is much less used (there is no entry for “scientia” in the Oxford online dictionary, whereas there is an entry for “scientism”). Second, because, as Massimo has defined “scientia”, it does *not* convey the idea of a unity of knowledge, because Massimo rejects that idea. Thus it does not convey the essential idea that is the centre of my stance.

    Science refers, very specifically, to the formal academic study of …

    Oxford Dictionaries defines “science” as: “The intellectual and practical activity encompassing the systematic study of the structure and behaviour of the physical and natural world through observation and experiment”. There is no restriction there to “formal academic study”.

    That point is more than pedantry, it is indeed how scientists think about science. In my day job as an astronomer, I have used, in professional academic papers, data collected by amateur astronomers who are hobbiests, not academics. I’ve also used data from ancient Chinese court records, compiled by astrologer-priests. It is all observation of the world, it is all “science”, why would I not use it?

    But again, the central point is that there is no great epistemological divide between simple observation of the world and “formal academic study”. Yes, the latter has refined and honed its methods, but it is still in epistemological essence the same thing. (Will reply to the 2nd half shortly!)

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  11. Hi Coel

    In this piece I argue that mathematics is a part of science. I should clarify that I am taking a broad interpretation of science. Nobody who defends scientism envisages science narrowly, as limited only to what is done in university science departments.

    So it seems to me you are basically saying that mathematics is part of science, just so long as you adopt a definition of science which includes mathematics.

    Yes?

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  12. Hi Coel,

    The reason I’m focusing on the terminology is not because it matters as an end in itself but because the way you (and scientism in general) tends to put forward your view is unhelpful.

    If you are not wedded to the term “science” for the umbrella term, then “scientia” or “reason” or some such will aid communication. If you want to make the point that there is a (remote) empirical basis for all knowledge, that may be legitimate but that doesn’t mean that all knowledge is empirical. You need to emphasise that you are talking about the ultimate foundations of knowledge rather than claiming that the knowledge itself is empirical, which it isn’t.

    Second, because, as Massimo has defined “scientia”, it does *not* convey the idea of a unity of knowledge, because Massimo rejects that idea.

    I’m not sure he does really. We can believe that there is one big continuous “continent” of knowledge, called Scientia, while at the same time believing it to contain within it different “biomes”, such as science and mathematics. The two ideas are not in conflict. There can be both unity and division at the same time.

    But again, the central point is that there is no great epistemological divide between simple observation of the world and “formal academic study”.

    I think the central point you’re making is less controversial than you think. It seems to me that Massimo agrees that there is no sharp epistemological divide, there is just its application to the natural world, where we call it science, and it’s application to the navigation of a public transportation system, where we call it common sense. His objection and mine is not to the idea of epistemic similarity but to the over-broad application of the term ‘science’. Similarly, there is a continuous gradient between domains where different strategies are more productive, e.g. a tradeoff between indirect deduction and direct observation as we move from mathematics to physics. If we can all accept this, which I hope we can, then that’s most of the disagreement resolved.

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  13. Hi DM,

    Reply part 2:

    You argue that there is no qualitative difference between ‘knowledge’ in the form of instincts primed in the course of evolution over many generations and ‘knowledge’ gleaned from direct observation.

    Yes. Both types of “knowledge” are gained by contact with the external world. The word “empirical” comes from a Greek word that gives us “experience” and “experiment”. This is saying that we learn about things by bumping into them. All forms of knowlege are thus derived from that “experience” of the natural world.

    Both the evolutionary-experience (now encoded in our genes) and the developmental-experience (as we grow up and observe the world) are indespensible parts of that: without either of those we’d get nowhere.

    The great advantage of empiricism is to prefer direct measurement to indirect deductions wherever possible so as to correct for faulty assumptions or reasoning.

    But that overstates the extent to which we can get “direct measurements” of anything. The only thing we get “directly” is a list of photon-arrival times impinging on our sense organs. Everything else is a model, and is deduction based on models of the world. If you ask, say, about “direct” evidence of the Higgs’ Boson, you find that the evidence is hugely indirect, coming from a very lengthy train of model-dependent deduction and inference.

    Further, even that “direct measurement” is entirely dependent on our innate (genetic) programming, since if you shone the same stream of sense-data at a rock it would not learn anything. Thus any and all knowlege comes from the entwining of these two things (evolutionary learning and developmental learning). So, in marrying the two in a single word, I am again trying to make a fundamental point about epistemology.

    You think there could be a mathematics, applicable to some other universe but not this, where 1+1=6. This is simply not true, because 1+1=2 is entailed by the definitions of those symbols.

    Sure, but the question is *why* are the symbols (and the whole structure of maths) defined that way? Why is it defined so that it is a good model of the real-world behaviour: “one apple plus one apple equals two apples”? Again, my answer is that maths is a model of the real world that we have adopted through those twin entwined processes of evolutionary-experience of the real world and developmental-experience of the real world.

    Perhaps there is a universe where you could consistently put one apple, then another in a bag and find you have six apples. […] It would not be the case that in that universe 1+1=6, because the plus symbol is defined to mean addition.

    The point is that in *that* universe we would have adopted a set of mathematical definitions in which 1 + 1 = 6 (or whatever symbols they would use to denote “one apple, then another in a bag and find you have six apples”). You’re right that it would not be *our* addition as we have defined it, it would be their addition as they would have defined it, to model the behaviour of their world.

    It’s nonsense, because it requires that we ignore what the definitions of the terms mean.

    It’s not nonsense, the whole point is the suggestion that “the definitions of what the terms mean” is exactly one of the things adopted as a model of our-world reality, and derived from our-world reality.

    If that’s not obvious, consider the claim: “In some other possible reality, gravity might work differently”. Counter: “No, the “gravity” would actually be the same, because what we call “gravity” is defined by the *our* *world* model of gravity, and you were still obeying *our* model in *their* world then you would still be obeying *our* model.

    Response: But the whole point is that *their* world would be obeying a *their*-world model, not an *our*-world model.

    Saying that maths might be different is no different from saying that physics might be different — they are both models of how our world works, derived from experiential contact with that natural world.

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  14. Hi Coel,

    Do we know this? And, if so, how? Ha<s anyone given a logical proof of the impossibility of such an alternative scheme? Any such proof could not use any axiom or logic derived from or validated by our empirical world (that would only show that such alternatives did not occur within our world). But without that, how would one go about showing that the logic of our world is the only one possible?

    So how do you know that logic and mathematics work in this world without using any axiom of logic?

    You might say that you put an orange into a bag and then put another orange into a bag and then get someone else to count and find there are two oranges.

    But why are you assuming that two oranges are two oranges? How do you know that two oranges aren’t twenty six point eight aardvarks?

    If you don’t assume any axiom of logic a priori then naturally you don’t assume the axiom of identity or the axiom of non-contradiction and so you begin with the assumption that two oranges might not be two oranges and that they might well be twenty six point eight aardvarks.

    But you don’t, do you?

    You implicitly trust your prior intuition that two oranges are not twenty six point eight aardvarks. You could not ‘validate’ logic unless you did.

    It is no good saying that you don’t trust your intuition but you test it over and over again because no matter how much you test you are still using those prior intuitions.

    You might say that it is enough that it consistently seems that there are two oranges but again you are using those axioms when you say that if it seems to you that there are two oranges then it seems to you that there are two oranges and it doesn’t seem to you that there are twenty six point eight aardvarks.

    And why do you make the assumption that counting is a valid method for establishing the cardinality of the oranges in the bag?

    By the same token, can you give me an example of empirically validating the axioms of logic which does make a prior assumption about the axioms of logic?

    I guarantee that you can’t.

    And did anybody ever actually observe something implying something else?

    And how do you know there is a universe in the first place? You organised a bunch of sensations using a lot of abstract mathematics is how.

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  15. Hi Graham,

    I would rather say that all human knowledge derives much of its former and content from the empirical, in combination with human reasoning and collective discourse.

    Agreed. But then one can ask where the human capacity to reason comes from, and I argue that that comes from our experiential-contact with the natural world (both the evolutionary experience now encoded in our genes, and the developmental experience of the world around us). See reply to DM for more on that.

    I wonder where you would place ethics or aesthetics; as science, or not truly knowledge, or what.

    I’d say they are science. But, aesthetics is all about brain states in a hugely complex neural network of 10^14 synapses that is far beyond our current capability to analyse and assimilate (and which results from an equally complex set of evolutionary and developmental histories). So I wouldn’t expect “in practice” science to get all that far with it, at present, though “in principle” science can address it.

    As for ethics, well ethics can be divided into two aspects. First, a descriptive account of what human attitudes and emotions on ethical issues are. That I’d regard as science, but with the same provisos as the previous paragraph.

    The second aspect is the supposed normativity of moral realism. Personally I regard moral realism and ethical normativity to be a delusion (beyond human preference and emotion, that is), and subscribe to emotivism. At that point all there is is the descriptive aspect, in which case ethics is within the domain of science. (As one would expect it to be, with ethics being a feature of natural animals that are a part of the natural world.)

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  16. We learn that from the fact that no other method has been shown to work. If, for example, divination were shown to be a better method of predicting future solar eclipses than science, then I’d have to change my opinion.

    This seems to miss the point for you could “only” learn that given that you accept the effectiveness of the empirical method.

    It seems to me that science construed as “empirical reasoning” cannot be fundamentally self-justifying. Whatever reasoning one provides to establish it, it should not be based on empirical reasoning in order to avoid circularity. So whatever one makes of the problem of induction, wrestling with it should not be called science according to your definition.

    I am somewhat sitting on the fence with regard to the whole topic of “scientism”, since I have yet to fully understand what is at stake. But math/philosophy/logic investigating the conceptual space and science investigating our actual world (which seems Massimo’s take) is much more plausible to me.

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  17. Robin, yes. To which you need to add that Coel’s definition of science (and math) is recognized by few mathematicians, philosophers of science, philosophers of math, and possibly not that many scientists.

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  18. Hi Robin,

    So it seems to me you are basically saying that mathematics is part of science, just so long as you adopt a definition of science which includes mathematics.

    I’m saying that the epistemological root of maths is the same as the epistemological root of science. Both are adopted as models of the natural world. What we call “axioms” of maths are descriptions of reality in the same way that “laws” of physics are descriptions of reality.

    I’m thus arguing against the idea that maths and science are separate domains with radically different epistemology.

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  19. Hi Robin,

    So how do you know that logic and mathematics work in this world without using any axiom of logic?

    By a suck-it-and-see approach. In order to, say, build a bridge or design an airplane, you need to use a whole mixture of stuff including logic, maths, physics and engineering. Combine the lot and stir well! Then see if your end product works (carries traffic without falling down; or flies). If it does then you’ve validated the engineering and the physics and the maths and the logic.

    Of course you don’t do this just once, you have a whole array of technology and engineering and experiences of the world, and you’re continually testing your “world model” to see whether it works. This validates the logic and the maths just as much as the physics and the engineering. (The concept is along the lines of Quine’s “web of belief”.)

    But why are you assuming that two oranges are two oranges? How do you know that two oranges aren’t twenty six point eight aardvarks?

    By testing the “two oranges” model against the “twenty six point eight aardvarks” model to see which performs better in terms of explanatory and predictve power and parsimony. In other words you use the standard scientific method for choosing between models.

    It is no good saying that you don’t trust your intuition but you test it over and over again because no matter how much you test you are still using those prior intuitions.

    The analogy is with a floating boat, where you can replace any part of the boat, but not all at once. In the same way you can examine, reject and replace any part of the model, including the prior intuitions, based on whether that makes the overall model work worse or better.

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  20. Hi DM,

    If you are not wedded to the term “science” for the umbrella term, then “scientia” or “reason” or some such will aid communication.

    Scientia is used in that sense here because it is Massimo’s coinage, but so far it’s not used in that sense anywhere else (whereas my usages of “science” and “scientism” are). That might change of course.

    I’m not sure he [Massimo] does really [reject my stance of the unity of knowledge]. […] His objection and mine is not to the idea of epistemic similarity but to the over-broad application of the term ‘science’.

    I’m open to correction from Massimo but it sure seems to me that he disagrees with me over the epistemic similarity and unity of knowlege over the whole of “scientia”.

    e.g. a tradeoff between indirect deduction and direct observation as we move from mathematics to physics.

    OK, but such wordings seem to me to way underestimate the indirectness of “observation” in physics.

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  21. I did not invent the word scientia, it has been in use for a long time to indicate knowledge broadly construed, contributed to by natural and social sciences, math, logic and the humanities. It is similar to, but even broader than, the German Wissenschaft.

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  22. The axiom of identity is a prior intuition, or a mathematical axiom?

    Experiments on infants show that none are born with the intuition of object permanence or axiom of identity. This appears to be why sufficiently young infants do not cry when their mother leaves (they do not realize they are deprived of a still-existing mother) but when they come to realize objects don’t just disappear but are covered up or left, they do. (How much depending upon temperament!) Objection refuted. I must add that ignoring human growth very likely stems from a covert use of a concept of the soul, superficially secularized.

    “You organised a bunch of sensations using a lot of abstract mathematics is how.”

    I must be a moron then, because I don’t recall doing any such thing. In fact, I’m such a deficient human being I don’t remember anything from my infancy. Looking at others I am struck by how they intereact with their environment, and how over a period of time they seem to understand more and more of the world about them, even on a very fundamental level of “intuitions.” I am not altogether certain this term is being used wisely. In particular, the notion that there is a you separate from sensations…is that really an intuition or an ideological concept learned both formally and informally?

    Maybe there’s something wrong with me but my sense of self, my me, somehow incorporates the positions of my limbs and the angle of my torso with respect to the pull of gravity. Even my emotions seem to me to expressed as movements of the viscera! If I somehow locate the presence of a person speaking behind my back, I don’t seem to recall the mathematical axioms used to triangulate their presence to my left or right or directly behind me. Instead, somehow that just seems to be a function of how my hearing works. At least when I’m not suffering yet another ear infection. Somehow the mathematical axioms don’t seem to work any more!

    Things like this are why I can’t help but feel the whole approach above is wrongheaded.

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  23. Bachelors are unmarried, therefore like all unmarried men, they are available for sexual activities as they wish. The logic is irrefutable.

    Valid reasoning and sound reasoning are not the same thing. Sound reasoning depends vitally on empirical conditions. If logical validity is not sufficient to demonstrate truth, I am not at all sure what it means to say that it provides knowledge.

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  24. Aravis: Don’t you mean “All bachelors are unmarried men”? I say this is not a prioi but merely a dictionary definition of an accepted meaning for ‘bachelor’. If you change the accepted meaning of the word ‘bachelor’ and make it to mean a ‘husband’ then it becomes “All husbands are unmarried men”. Just as “All water is wet”.

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  25. Mogguy:

    It doesn’t matter which you use. In either case, the sentence is true by virtue of meaning, rather than some empirical fact.

    All analytic statements are known a priori. Indeed, they are the clearest, most obvious, most basic example of a priori knowledge, acknowledged by Hume and Kant alike.

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  26. Hi Coel,

    I would say emotivism leaves us not only with a descriptive aspect. You could go down the Sam Harris road and argue that science could also be used to assess which alternatives are likely to lead to outcomes people will view as morally desirable.

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  27. Again, the name calling of “scientism” seems rhetorical and journalistic trickery – at best. Just name calling.

    1. No belief system statement can be proven or disproven, e.g. philosophism – or the claim that every statement is a subjective/intuitive/philosophical one or that logic exists, consicousness-language matters,or a god, or spirits…these cannot be verified and no universal/intersubjective claims or evidence can be applied.

    2. By definition, any statement that claims it is evidence-based (so called “science”) must be refutable and disprovable. There is no “-ism” in anything “scientific.

    In fact, there is no uniform set of statements or behaviors that could comprise a “science.” It is a journalistic and political strawman. Which philosophers love to attack with flaming torches.

    There is however a dominant cultural ideology of “philosolipism” which preaches that everything, really just any use of everyday language is “philosophical” or solipsistic and subjective. That is also factually untrue. Airplanes, etc disprove this. But philosophism is just an academic extension of the pop(ular) worship of subjective experiences. A myth increasingly debunked by brain research – in all species.

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  28. I believe that you’re tacitly requiring that induction must be justified a priori as a logical necessity. I think Coel believes it is sufficient that induction be regarded as probabilistic. Let me suggest that if you take the problem of induction seriously you cannot pronounce on the reality of the “external” world. Even worse, when you start subtracting the deceptive sense data, you start erasing your mind. After all, when you pause to remember, the same foods sometimes taste different, the same colors don’t have the same intensity, sneezes and orgasms somehow are different from time to time. Not even the mystical qualia can be relied upon!

    Or to put it in other words, your self when you are least deceived by unreliable sensations is when you are dreaming. Good luck with that.

    If we view ourselves not as disembodied minds (aka souls) or even naked brains, but as whole persons whose very boundaries with the outside world are, literally porous, who ceaselessly gulp in the outside world, the question becomes, How can we interact in such a consistent way with the world? Unless we have a secret team of movie makers projecting a finished, coherent movie in our secret inner theater playing the drama of The World, it’s consistency must be from its own nature. The problem of induction then is the difficulty of correctly generalizing from experience.

    That’s not a deeply principled kind of problem. But the practical difficulties are such that it has taken centuries to make serious progress and we have much, much more to learn. The serious question I think is why it seems to be so urgent to so many to imagine an even more profound problem, that casts all conclusions into doubt? Practically no one really doubts everything, so it must surely be because it is desired to doubt only some conclusions, but with an “argument” that escapes the tedious necessity of producing facts and making arguments, that substitutes a declaration the proponent of those unwelcome conclusions have no standing to make a case.

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  29. Come on, please. The objection I raise is serious. It’s a bit easy to bury it in a footnote and use the word pedant.

    > I argue that we accept that statement (“1+1=2”) as true because it works in the real world.

    It certainly works in the real world. But my point is different. The fact that it works in the real world is not the mathematical reason why 1+1=2 is true (or not true).
    Even if we model our mathematical axioms on the real world, the theorems we derive are mathematically true for mathematical reasons, not because they happen to be useful to describe the real world.

    Mathematical statements and scientific statements about objective reality are made true by fundamentally distinct things. That alone suggests that maths is epistemologically distinct from science.

    .

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  30. Hi mira,

    This seems to miss the point for you could “only” learn that given that you accept the effectiveness of the empirical method.

    You adopt and suck-it-and-see approach. Does it work? Can we predict solar eclipses? Can we build airplanes that fly? Can we create computers and mobile phones? If we succeed then we validate the whole kaboodle of assumptions and premises that have gone into the model.

    It seems to me that science construed as “empirical reasoning” cannot be fundamentally self-justifying. Whatever reasoning one provides to establish it, it should not be based on empirical reasoning in order to avoid circularity.

    It’s more of a bootstrapping method. We simply test out whether, in practice, science does give a good acount of the observable world, in terms of explanatory and predictive power. Since the only claim of science is that it does give a good account of the observable world (or, really, of the stream of experiences that we experience) then that is sufficient to validate it.

    But math/philosophy/logic investigating the conceptual space … is much more plausible to me.

    What is involved in investigating the conceptual space? If it is a matter of adopting axioms derived from the natural world, and then thinking about their consequences, then science does that too (it’s what theoretical physics and other theoretical sciences do). If it is matter of scheming up alternative axioms and thinking about their implications, then again that is what science does also. Of course science does not bother pursuing axioms/laws that are clearly contrary to the real world, but then I suggest that math/philosophy/logic doesn’t do much of that either.

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  31. Hi DM,

    I would say emotivism leaves us not only with a descriptive aspect. You could go down the Sam Harris road and argue that science could also be used to assess which alternatives are likely to lead to outcomes people will view as morally desirable.

    Hold on, the “assessing which alternatives are likely to lead to outcomes people will view as morally desirable” is purely descriptive, and I agree that science can do that.

    What Sam Harris does it try to recover normativity by adopting an axiom saying we “should” maximise well-being. I think this is misconceived (and so do most other people). Indeed I think that any attempt at moral realism is misconceived.

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  32. Hi Aravis,

    The statement “Bachelors are unmarried” is known a priori. Or would you care to identify the empirical conditions that confirm/dis-confirm it?

    As you’re aware, this is really an issue of linguistics, and the issue is to identify exactly what the “knowledge” is in that sentence.

    Asking whether the statement “bachelors are unmarried” is true could — for example — be asking about linguistic usages. It could be asking, “what is the definition of the word “batchelor”?”. For example one could imagine a child asking that when unfamiliar with the word. If that is the “knowledge” being asked about then indeed one knows the answer empirically, since it is an empirical fact that people use the word in that sense.

    But, let’s presume we already have the definition of the term “batchelor” as a “man who is not and has never been married”. We’re then asking whether “a man who is not married is unmarried”.

    If we’re asking that then we’re asking a question about the consistency of the world and of the concept “marriage”, such that one cannot be both married and unmarried. Again, that seems to me an empirical question about the consistency of the world and of the concept of marriage.

    To see that that is not trivial, one can consider a particle being in a quantum superposition of states such that it could be both “spin up” and “spin down”, or both “not spin up” and “spin up”. Again, we learn about that superposition empirically. It so happens — empirically — that such superpositions do not apply to marriage.

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  33. Massimo,

    Have you read Marc Lange’s “What Makes a Scientific Explanation Distinctively Mathematical?” You might find it interesting.

    That being said, there’s certainly a lot I disagree with here, but I’ll try to stick to a few isolated points:

    >>I will take one statement as standing proxy for the whole of mathematics (and indeed logic). That statement is:

    1 + 1 = 2

    I think this is a poor choice of a proxy for all of mathematics. A more representative and philosophically interesting example would be a *theorem* of the sort of that mathematicians are usually in the business of proving, such as the fact that there are infinitely many primes, or the fact that every natural number has a unique prime factorization. These are the sorts of statements that better highlight the difference between mathematical and empirical knowledge. It’s a rather remarkable fact that, with a pencil, a single piece of paper, and a little bit of simple reasoning, you can come to have extremely good (dare I say perfect?) evidence of an interesting fact with huge real-world applications about INFINITELY many numbers (that they all have unique prime factorizations). This mode of discovery – however you wish to call it – on its face seems dramatically different from the kind of activity that takes place in experimental sciences. To be clear, I don’t take this to automatically show that mathematics is in some way fundamentally non-empirical or fundamentally distinct from “science” – I’m just saying that I think these kinds of theorems do a better job of highlighting their *apparent* differences than 1+1=2.

    >>I argue that we accept that statement [1+1=2] as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world

    I think you’re ignoring the fact that the natural numbers and arithmetic are intimately tied to the activity of *counting*. You seem to be thinking of arithmetic like a chemical reaction. Allow me to clarify. You think the model of arithmetic is putting 1 apple in a bag, then putting another apple in a bag, and then looking to see how many apples are in the bag. Of course, in the real world, there are 2 apples in the bag, but we could imagine a world in which, for whatever reason, we regularly found 5 apples instead. In a world where most objects behaved that way, you think we might adopt an arithmetic where 1+1=5.

    However, like I said before, this ignores the fact that our concepts of numbers and arithmetic are intimately tied to COUNTING. Instead of asking “how many apples are in the bag?” to get the answer to 1+1, you should instead ask, “how many apples DID YOU PUT IN the bag?” In ANY imaginary world whatsoever, if you put a single apple in a bag, and then you put another single apple in a bag, the number of apples *you put in* the bag is 2. Likewise, if you put 3 apples in a bag, and then you put 7 apples in a bag, how many apples did you put in the bag? You put in 3+7=10. Even in a world where putting 3 apples in a bag, followed by 7 apples in a bag, results in there being 40 apples in the bag, that doesn’t change the fact that *you put in* 10 apples. People in such a world would still be able to recognize the difference between the number of apples they *put in* the bag and the number of apples that *are* in the bag, and they’d still have use for standard arithmetic (for example, they would like to know that the difference between the number of apples they put in the bag and the number of apples actually in the bag is 40-10 = 30.)

    When arithmetic is thought of this way, I think it’s much harder to make the case that it is empirical or contingent. Suppose you know for certain that you put 3 apples in the bag, and then you put 7 apples in the bag. If you think of 3+7 as representing the answer to the question “how many apples did you put in the bag?”, all you have to do is count three steps up from 7. You don’t need to know any extra empirical fact about what happens when you mix 7 and 3 together. How the answer could possibly be anything besides 10 is beyond me.

    >>An aside before proceeding. Gödel’s incompleteness theorem tells us that even if we have a set of axioms such as Peano’s, which underpin the natural counting numbers and which yield the statement that 1 + 1 = 2, there will be other statements about the natural numbers which are true, but which cannot be shown to be true from the axioms….From a scientific point of view, with mathematics being seen as a part of science, such limitations are unsurprising. Science is derived from empirical evidence and our available evidence will always be a small and incomplete sample of the universe, and thus scientific results are always provisional, in principle open to revision given better data.

    There is no analogy here – arithmetic is not “incomplete” in the same sense that scientific evidence is “incomplete.” As you say, scientific results are always provisional and in principle open to revision given better data. But the incompleteness of arithmetic doesn’t imply that known arithmetic truths are provisional and open to revision. All the incompleteness theorem shows is that there are theorems that can’t be proven or disproven from a (recursively enumerable) set of axioms. That doesn’t mean that theorems which *can* be proven (such as 1+1=2) might later be disproven. That could only happen if arithmetic is inconsistent, but in that case it wouldn’t be incomplete. So to say the incompleteness results are “unsurprising” from a scientific point of view is, frankly, rubbish.

    It’s also relevant to point out that some noteworthy mathematical theories (such as Euclidean geometry) *are* complete. To talk about incompleteness as if it applies to mathematics as a whole is misleading.

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  34. I am very torn on this issue. More or less disconnected thoughts…

    My notion of science is realist, which means I am not quite sure what you mean by mathematical “knowledge” when you don’t use any practical or empirical correspondence to reality. And frequently, potentially always, when you do try to apply mathematics to reality, the problem of tacit or false assumptions comes up, Problems with using Euclidean geometry for plotting transoceanic airline routes or using regular arithmetic when calculating times on a clock or simply using negative numbers should be famous examples. Sometimes are easier than others. It is unlikely any bridges in Konigsberg are Moebius loops nor are we likely to get drunk on wine swigged from a Klein bottle.

    Axiomatization occurred early for geometry according to the surviving historical record, notably with Euclid. But did that have to be the case? Would it be possible to have done geometry with numbers and mathematical induction? Did the emphasis on axioms hold back the development of algebra? Axiomatization and rigor did not play a role in the creation of calculus and analysis. Applied mathematics is the ugly stepsister but it will never be axiomatized so far as I can tell. Doesn’t that mean that the axiomatic version of mathematics is wrong? Or just incomplete? Is the superior prestige of pure mathematics a belated echo of the Pythagorean notion that number holds the secrets of the cosmos?

    Some mathematical objects cannot exist any more than Sherlock Holmes? Can we therefore conclude that knowledge about them is no more science, knowledge about the real world, than our knowledge of Sherlock Holmes’ personality is real? If the difference is supposed to be the deductive method used in axiomatization, then how can mathematics include so much math that wasn’t axiomatized, both in history and in application?

    Overall, I’m inclined to believe that mathematical objects exist in the same kind of way that money or marriage or the game of chess does. Fundamental mathematical notions do appear to be derived from experience and conforms to reality, just as fundamental aspects of marriage and money conform closely to material realities, independent of the human will. That’s why, contra Wigner, the effectiveness of mathematics is entirely reasonable. When playing around, just as the rules of chess or the facts written about Sherlock Holmes permit deductions that are valid, but do not correspond to reality. My understanding at this point is largely influenced by Reuben Hersh, particularly as presented in What Is Mathematics Really?

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  35. Right. I think moral realism is ill conceived, but that doesn’t mean that science can’t help us figure out what is moral, as long as we adopt a convention that what we consider to be moral is for people to be better off in some way.

    My problem with Harris is he egregiously overstates his case. My only point to you is that science is not restricted to a kind of anthropological study of morality in different cultures, it can also (in principle anyway, and in some cases at least) recommend moral prescriptions based on an agreed convention of what kinds of outcomes are morally desirable, in much the same way as medical science can recommend treatments for various ailments.

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  36. The a priori knowledge that bachelors are unmarried means that they have the properties of unmarried men or it doesn’t. I’m not quite sure what the referent for “it” might be. Trying to erase the contradiction by inventing the distinction of linguistic knowledge fails in this case I think, because it assumes a lexical definiteness to words. Words mean how they’re used, not how they’re defined. You can no more definitively capture the meanings in a definition than Popper could identify the falsifiable proposition in science. Which is to say, there is no principle to do so, it has to be done on a case by case basis, i.e., empirically.

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  37. The apparent presumption that being based on counting doesn’t mean arithmetic is fundamentally empirical doesn’t make sense. Sort of concedes the point.

    As to the incompleteness theorems, the usual idea of mathematics is that it indisputably proves things from first principles, unlike the popular view of science which can’t attain real knowledge. I do think the incompleteness theorems do show that this view is wrong (naive if you like.) I don’t think it’s rubbish to notice this.

    Despite the insistence that 1+1=2 instead of 10, the discovery that there are theorems that can’t be proven from a given set of axioms means there are no mathematical forms that can be asserted to describe the universe. Platonism and variations thereof are wrong. I don’t think it’s rubbish to notice that either.

    Also I thought the number line was a part of modern arithmetic and I really had no idea that it’s consistency was proven!

    I disagree with the OP that all discoveries of science are open to correction. I have no idea why an inconsistent universe is somehow deemed to be more possible than an inconsistent arithmetic, save unconscious notions about mathematics being about the true reality.

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  38. I do not accept:

    ” I argue that we accept that statement as true because it works in the real world. All of our experience of the universe tells us that if you have one apple in a bag and add a second apple then you have two apples in the bag. Not three, not six and a half, not zero, but two. 1 + 1 = 2 is thus a very basic empirical fact about the world [3]”

    As surely as science itself has proven nature to be truly immeasurable, to be uncertain or only probable at best, I cannot accept the number of apples in the bag. Furthermore your equation 1 + 1 = 2 is unacceptable as well. 2 = 2 and 1 + 1 is most certainly something else unless of course someone told you they are the same. Are they the same? But then who do you believe, science, religion, mathematics? How many apples are in the bag again? Is science absolutely certain, of anything?

    I do not accept Maxwell’s “C”, the speed of light either, do you? Does not the science of quantum theory tell us that the speed of light as is all of nature is only probable at best? Surely Heisenberg’s uncertainty theory was but a tip of the berg, wasn’t it? Is C truly absolute or uncertain at best? Is quantum theory absolute?

    I cannot accept uncertainty at best, do you?

    I will not accept anything but the absolute.

    Oh and while science buries itself in the quagmire of measuring the immeasurable, I will again write the single absolute right here: =

    A foot note: The old Eastern masters believed the truth could not be spoken. I am certain the truth can be spoken, but sadly for many and most, it cannot be heard.

    MJA
    =

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  39. Hi Patrick,

    Even if we model our mathematical axioms on the real world, the theorems we derive are mathematically true for mathematical reasons, not because they happen to be useful to describe the real world.

    If we model our axioms on the real world, then that’s the same as them being useful in describing the real world.

    If we then derive theorems from those axioms, then you are entirely right that those theorems are mathematically true owing to those axioms. But, it is also true that those theorems are derived from the real world (at one remove, through the axioms being derived from the real world), and it follows that the theorem will apply to the real world, and likely be useful in the real world.

    Mathematical statements and scientific statements about objective reality are made true by fundamentally distinct things.

    Not true, and perhaps people who are disagreeing with me are erroneously regarding science too simplistically as only simple observation, whereas it is hugely model dependent.

    For example, take the scientific statement that the temperature at the centre of the Sun’s core is 1.6 x 10^7 Kelvin. That is a true scientific fact (with the usual provisos about confidence internal and provisionality that always apply in science). But, that is not a direct measurement — we have no way of obtaining a direct measurement.

    What it is is a “theorem”, or a deduction, that is derived from a model using the “axioms” of physics (more commonly known as physical “laws”). So we have:

    Observation => physical/mathematical laws/axioms => deduction => theorem/statement.

    That is epistemologically identical whether we are talking about mathematical theorems deduced from axioms that are modeled on reality, or whether we are talking about scientific facts that are deduced from models that are based on laws that derive from reality.

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  40. Hi DM,

    I think moral realism is ill conceived, but that doesn’t mean that science can’t help us figure out what is moral, as long as we adopt a convention that what we consider to be moral is for people to be better off in some way.

    Sorry, DM, but I think you’re succumbing to Sam Harris-itis! Science can’t help us figure out “what is moral”, but it can help us understand what *we* *think* is moral (which is descriptive). Adopting the convention that “what is moral| is what makes people “better off” is an unwarranted axiom. Further, how one defines “better off” is entirely subjective. Afterall, ISIS think that people are better off under ISIS rule!

    [science] can also (in principle anyway, and in some cases at least) recommend moral prescriptions based on an agreed convention of what kinds of outcomes are morally desirable,

    IF we decide what we want, then science can tell us how to get there. Further, science can inform us of consequences and outcomes, which might affect how we feel about things. But science’s role in both of those things is entirely descriptive. It is still the humans doing the opining and deciding the aims.

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  41. labnut, “when are you going to claim poetry is science?”

    Why should we claim poetry, music and painting are science? Let the poets, musicians and painters be free of science and philosophy, they don’t need it.

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