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.

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

  1. I have to admit to pretty much agreeing with Coel’s main points in this post. The idea of math and logic resting on empirical foundations seems to be ferociously resisted, I think because those foundations don’t feel empirical, mainly because we don’t learn them empirically. The human brain is not a blank slate. It comes with pre-wiring for a number of capacities, including logic and some math. We don’t always use it, but we evolved it, probably due to its survival advantages.

    However, unlike Coel, I’m not insistent on mathematics being a part of science. I’m content to leave science to endeavors that involve a heavy amount of empirical investigation and the logical and mathematical consequences of that investigation. Mathematics may have empirical foundations, but I think it’s pretty obvious that mathematicians aren’t doing empirical work, but finding interesting and (sometimes) useful tautologies.

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  2. I just think you’re not really following me, Coel. We’re really saying the same thing. We can arbitrarily decide that what morality is about is well-being, which for the sake of argument we can define according to criteria such as having a society of people who are safe, free of worry and having a high human development index and so on. We can then ask whether these goals are better achieved by allowing or banning abortion, gay marriage, divorce etc or by raising or lowering the drinking age, by encouraging safe sex or abstinence in teenagers and so on and so forth. If you adopt a definition of morality by convention, it is therefore at least plausible that science can identify which values are compatible with that convention. I’m not arguing that science can determine what that convention ought to be.

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  3. Coel: I think the same argument you make about mathematics applies to normativity. Mathematical knowledge comes to us empirically, through our connection to the world. The structure of the world is such that 1 apple and 1 apple will make 2 apples. We might perceive it incorrectly in some situations, but the structure of the world will continue to produce the 2 apples. In the same way, the structure of the world is such that certain outcomes will be better or worse in terms of the teleology that self-organizing systems will have as a result of their structure (they’re part of the world too). We come to knowledge of the better and worse empirically. We may judge it incorrectly for various reasons, but the structure of the world will keep producing the same better and worse. The only difference is that because better and worse are more complicated than counts of apples, the normative aspects of the world will manifest as strong biases rather than immutable laws (there are more ways for something to be better or worse than there are ways for something to be two).

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  4. I think labnut was being sarcastic, Mario. He’s asking the author to explain what’s going on in his “world model” beyond statements like this: “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 . . . . So I wouldn’t expect “in practice” science to get all that far with it, at present, though “in principle” science can address it.”

    And, still, despite this practical limitation, in the meantime, there are theories of aesthetics for those of us who lack the rigor and patience to await the prospect of science addressing such matters “in principle.” Similarly, I find myself hard-pressed to control my laughter upon hearing a good joke while awaiting a proper scientific explanation for my laughter. Any empiric explanation of humor would seem itself the butt of a joke.

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  5. The idea of math and logic resting on empirical foundations seems to be ferociously resisted, I think because those foundations don’t feel empirical.

    ——–

    You may not be aware of this, but there are substantial arguments *against* Empiricist mathematics, most notably in Frege. So, the idea that people are against it just because of how they “feel” is just false.

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  6. Hi C,

    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 [. . . ] 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.

    But experiment in science is only half of it, and that half is entwined with theorizing. Scientific theorizing is essentially the same as mathematics — you take axioms/laws and apply reasoning and make deductions. That is an essential part of all science, even “descriptive” and “experimental” science, since those things don’t get anywhere without theorizing and model-building.

    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, …

    Perhaps they would, yes. Perhaps they would develop more than one concept of “counting”. Thinking of fundamentally different realities is counter-intuitive for us, so hard to imagine. Suppose when they put something into a bag the first time, one object went in; when they did a second time a second object materialised along-side the one they were putting in, and also went in; and when they did it a third time, two extra objects materialised and went in. Would they then develop “counting” that went 1, 3, 6 … et cetera?

    Anyway, counting is surely a real-world empirical activity, and would have originated with a farmer counting his sheep or something like that. So this is in line with my claim that maths fundamentally derives from real-world behaviour.

    It’s also relevant to point out that some noteworthy mathematical theories (such as Euclidean geometry) *are* complete.

    But doesn’t Godel tell us that such a system cannot be shown, using its own axioms, to be consistent? That does seem similar to science not being able to show something to be ultimately true.

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  7. * error corrected *

    Coel, “all human knowledge is empirical, deriving from contact with empirical reality”

    Can you please explain how this applies to the knowledge that the square root of 2 is irrational?

    What was the empirical test that led to this result? When was the experiment performed, what was the apparatus used, and what did the data points look like? Was it a 2-sigma or 5-sigma result?

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  8. I have to admit to not having read Frege, but when doing research for a recent post on logic, I came across, and was impressed with, his definition of logic; “The science of the most general laws of truth”

    Since mathematics is often referred to as quantitative logic, Frege doesn’t seem that far off from this concept.

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

    It’s pretty clear that this kind of thing is a difference in how terms are used. Knowledge resides in some kind of thinking being, and whatever it is in that thinking being that thinks requires “experience” to even engage in thought. And there’s no difference in kind between sensory experiences and introspection. It’s all brains doing basically the same sorts of things. Blah blah blah.

    But I wouldn’t say that “a priori”, even in the sense that you’re using it, is uncontroversial. It seems weird to say, “you need all this experience to know how the rules work and how to apply them and what the game is, but then the things you discover with those rules are not grounded in experience”. I think one of the things Wittgenstein was trying to get across is that this sort of thing indicates that we’re thinking about it wrong.

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  10. Hi Self,

    I think it’s pretty obvious that mathematicians aren’t doing empirical work, but finding interesting and (sometimes) useful tautologies.

    Sure, and so are a lot of theoretical physicists! (And theorists in other areas of science.)

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  11. Hi gfrellis,

    Can you please explain how this applies to the knowledge that the square root of 2 is irrational? What was the empirical test that led to this result?

    The process is empirically verifying some axioms and then reasoning from those axioms to show that root-2 is irrational. That is how *all* empiricism is done — by model building. If we ask what we can observe directly then the only such thing is a stream of photon-arrival times as they impinge on our senses. Everything in science is a model constructed out of that, which means that deduction from axioms/laws is a core part of all empirical enquiry.

    See an early comment up-thread about the temperature of the core of the sun for an example of this.

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  12. Let me try to put that a different way. The distinction that Kant and Hume were recognizing is a real distinction. There’s a real difference in the kind of thing that’s happening between “bachelors are unmarried” and “the sky is blue”. But the distinction is only important within the framework of an inquiry into what we can know about things. And specifically, the concern is whether there’s a way to sort of “traverse” those realms. And even Kant, who was hella smart, didn’t see a way across. So – seemingly fundamentally different things.

    And Coel would seem to be saying that the reason there’s no way is that bachelor realm is really just a housing development in sky realm. It’s abstracted, sublimated, etc., but there’s no other realm really. And many would argue that this just amounts to conceptual redistricting and the only consequence of it is that we’ve lost a useful distinction. And others would argue that if we don’t see our access to knowledge in a unified way, we’re making a distinction that misleads us.

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  13. Hi Self,

    I came across, and was impressed with, his definition of logic; “The science of the most general laws of truth”

    You’re right. What Frege (and other mathematicians and logicians such as Peano and Russell) tried to do was to distill maths down into axioms of logic. Thus, overall, basic “real world truths” were distilled into axioms of maths and thence to axioms of logic.

    This is exactly what physicists try to do, they observe real-world behaviour and try to distill it down into more-and-more fundamental “laws” of how things work. This corresponds exactly into distilling things down into more-and-more fundamental axioms of logic.

    The idea that maths/logic is “based on” or “derived from” axioms is roughly the opposite of what happens, rather mathematicians/logicians proceed towards axioms, in the same way that physics is not “derived from” laws of physics, rather physicists proceed to laws of physics.

    Of course, in addition to deriving those fundamental laws/axioms, another aspect of maths/physics is mathematicians/physicists reasoning from those axioms/laws to build up edifices of theories/models.

    tried to do was to distill “

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

    I wasn’t talking about particular acts of counting. If you want to accurately count the number of apples in a bag, for example, you need to be able to perceive all the apples, so that’s obviously an empirical process. But the general “laws” of counting (if you will) which correspond to arithmetic don’t need any empirical support. To put a variation on my previous example, suppose Mary puts puts 7 apples in the bag (one at a time), and then she puts 3 apples in the bag (one at a time). Now suppose Bob puts 3 apples in a bag (one at a time) and then 7 apples in the bag (one at a time). It should be perfectly clear to you that the number of times Mary put an apple in the bag is the same as the number of times Bob put an apple in the bag (because 7+3=3+7). The author would have us believe that we are only justified in believing this because it happens to work in our experience. But I think you’re fooling yourself (or you just don’t understand the nature of counting) if you really think you need to appeal to experience to know that counting to 7, and then going 3 steps further, lands you at a different number than counting to 3, and then going 7 steps further.

    >>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.

    The incompleteness theorems don’t refute that view at all. Mathematicians prove things indisputably from first principles all the time (for example, the fundamental theorem of arithmetic.) The incompleteness theorem is no threat to that. At best, the incompleteness theorems show that there are some statements that *cannot* be proven from first principles (or, alternatively, that one needs to beef-up one’s first principles.)

    >>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.

    What? I don’t understand what you’re saying. I will point out, however, that the fact that there are theorems that can’t be proven from a given set of axioms is completely trivial. For example, here is a set of axioms:

    A1. For any n, 0+n = n
    A2. For any n, n+1 = 1+n

    There are lots of statements that can’t be proven from these two axioms. You can’t prove that there are infinitely many numbers. You can’t prove that addition is commutative. You really can’t prove anything interesting at all. But that’s just a trivial consequence of the fact that I didn’t give you much to work with. “The discovery that there are theorems that can’t be proven from a given set of axioms” doesn’t warrant any profound philosophical conclusions.

    >>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 assume this is a response to my remark about Euclidean geometry. The natural numbers are not definable in the theory of real-closed fields, which is why the theory of real numbers can be complete even though arithmetic isn’t.

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  15. I agree with Coel on a lot of things, but this is just odd. 1 + 1 = 2 is a cleverly chosen example because yes, we can empirically observe that one apple plus one apples makes two apples (for certain definitions of 1 and 2). But where is there a straight line in nature, especially one that has zero width and extends eternally into both directions? Where is there a circle for which it is really true that area = 2 x Pi^2? They exist in the realm of math but real life structures are wonky and thus mere approximations of these shapes. Funnily, in science it is exactly the other way around: our models are approximations of reality.

    What is more, in school I had math and science classes. It was some time ago but I distinctly remember that science was taught by showing plants to us, by conducting chemical experiments, by building electric circuits. But math was taught by sitting in a room deriving formulas from other formulas; never did the teacher demonstrate how to work with integrals and derivations (or complex numbers, for that matter) by showing us empirical evidence.

    Surely there is a reason for that?

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  16. Indeed, my point about arithmetic and counting was meant to echo Frege’s point about how numbers apply to anything – even non-spatiotemporal things (e.g., thoughts, spirits, ways of counting, factorizations, etc.)

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  17. After posting this it immediately occurred to me what the reply will be: Math is also an approximation of reality, so the situation is the same. But really here we will have to agree to disagree.

    As far as I can tell, math starts out developing a system of relationships and then accidentally some of them are found to be useful in the real world while many others aren’t. It could really be done in a cell by somebody who has never in their life so much as seen a tree or a rock.

    Science, on the other hand, starts out with the real world observations and only then has any incentive to propose a model. There is no theory or model of non-existing but logically possible things in science waiting idly for that thing to pop up later.

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  18. Coel, thanks for your defense of the empirical nature of at least the axioms of mathematics used in science. One quibble – any fundamental axioms of mathematics that cannot be empirically justified should not be considered part of objective science. Right?

    In appreciation of your post and all the interesting comments, I donated to Scientia Salon and want to thank Massimo for all he has done to make this site possible and, in particular, his willingness to provide a forum for positions that contradict his own.

    Consistent with your arguments about empirically justifiable axioms, perhaps something like the following is true: The fundamental axioms of mathematics relevant to physics are necessarily true due to our universe’s innate symmetries (which are responsible for our conservation laws).

    Consider the premise (or hypothesis) that these physics applicable fundamental mathematical axioms of our universe apply to all possible universes. (Perhaps mathematicians who believe mathematics is independent of science might provide justification for such an idea.) Then all possible universes would share our universe’s innate symmetries.

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  19. Hi Coel,

    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.

    No you haven’t, you have only validated logic and mathematics using the prior assumption that logic is valid.

    How can you validate that something can fly if you are assuming that the fact that it flies does not imply that it is capable of flight?

    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.

    I don’t think you even read the question. You cannot have explanatory power if your prior assumption is that the fact that you observe two oranges does not imply that you have observed two orange.

    Under your prior assumption that an observation of two oranges might be an observation, not of two oranges, but of twenty six point eight aardvarks, then either model has equal explanatory power and parsimony.

    Under that prior assumption either model satisfies the observation equally and so there is nothing to choose between them, they both perform equally well.

    Also, don’t forget that under your prior assumption the fact that something is more parsimonious does not imply that it is more parsimonious.

    You can’t even have a scientific method without the prior assumption of the axioms of identity and non-contradiction.

    If you observe that X is the case and conclude therefrom that X is the case then you have used the prior assumption of the axioms of identity and non-contradiction – in other words you have used a prior intuition about logic.

    Can you answer the question properly and show me how you can validate maths or logic without first assuming the axioms of identity and non-contradiction?

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  20. Hi stephen,

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

    False dichotomy, it can be both – an axiom can be based on an intuition.

    But that is simply besides the point, if you are not using it as a prior intuition then all of your knowledge about the world depends entirely on a mathematical axiom.

    So you can hardly say that mathematical axioms are derived from our knowledge about the world if our knowledge about the world is derived from mathematical axioms.

    On the other hand if you observe X and conclude that you have observed X then you are using this axiom as an intuition.

    So if you have an orange and put it in an empty bag and then a second person has an orange and puts it in the same bag then we have two hypotheses – 1) There are two oranges in the bag and 2) There are no oranges, only twenty six point eight aardvarks in the bag.

    Open the bag and count them – you count two oranges.

    Now if you claim that you are not using the axioms of identity and non-contradiction as prior intuitions then you fully accept that your observation of two oranges might be an observation of no oranges, but of twenty six point eight aardvarks.

    So, without this prior assumption either 1) or 2) satisfies the observation equally.

    1) is only a better model under the assumptions of the axioms of identity and non-contradiction.

    So if you are proposing to validate mathematics by the method, you could only conclude that maths and logic are valid just so long as maths and logic are valid.

    So your “refutation” is simply missing the point.

    I must be a moron then, because I don’t recall doing any such thing.

    I don’t recall pooing when I was an infant. Does that imply that I didn’t poo?

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  21. Hi Coel,

    I think that Sean Carroll would disagree that there cannot be a superposition of married and never married.

    But the quantum argument is a furphy, a category mistake.

    If I am in a superposition of “married/never married” does that imply that I am a bachelor – since I am “never married”? Or does it imply that I am a not a bachelor because I am married?

    No, in fact I would be in a superposition of being “married, non-bachelor/never married bachelor”

    We could not even speak of superpositions if we did not have definitions. “Bachelors are unmarried” follows from the definition of the words used.

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  22. >>But doesn’t Godel tell us that such a system cannot be shown, using its own axioms, to be consistent? That does seem similar to science not being able to show something to be ultimately true.

    Whether a system can prove its own consistency has nothing to do with “showing something to be ultimately true.” After all, suppose you’re not 100% confident that PA is consistent. Then suppose PA could prove its own consistency. Would this be a definitive proof that PA is consistent? Not really – after all, PA would be able to prove its own consistency even if it were *inconsistent*. Using a theory to prove its own consistency is kind of like asking somebody if you can trust them. If you already *do* trust them, then there’s no reason to ask in the first place. But if you *don’t* already trust them, then hearing them say, “Sure, you can trust me!” is not exactly a good means of vetting them.

    The significance of PA proving its own consistency had more to do with Hilberts formalist/finitist program. It wasn’t about proving things to be “ultimately true.”

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  23. The big problem I see is that if mathematics were empirically derived it would be also be empirically falsifiable. But it is not.

    In fact we have no idea of what a falsification of any mathematical truth would look like. The debate between Descartes and Marsenne hinged on whether God could have decided different mathematical laws. Marsenne’s objection was that he had no idea what Descartes meant by this, because mathematical laws seemed to him self-evident. On the other hand Descartes insisted that had God wanted to, he could have made any mathematical truth different. The reason why Descartes was arguing this was twofold: first to hold onto God’s autonomy (he must not be bound by an external necessity) and secondly to put God above the paradoxes of omnipotence. Descartes admits an insuperable distance between the finite and the infinite positing this beyond the rule of logic. Thus Coel finds himself in a double bind: he either opens the possibility for domains to which our empirically determined rationality does not apply or he has to admit that falsifiability is not a demarcation between what is science and what is not science. Neither of these options is desirable in my opinion.

    Another big problem with incorporating math into science is that while empirical sciences tend to produce a lot of theories (most of them wrong) math tends to produce one theory which is consistently right.
    Not only that, but when a different mathematics is needed to describe the world, we don’t lower the status of that mathematics but we actually elevate it.

    The key example here is the fifth Euclidean postulate which, since it was not empirically verifiable (making claims on infinity), was presumed to be logically deducible from the other four. When we found out that our world quite often does not respect the fifth postulate we didn’t throw it out into the dustbin of failed theories, but we actually recognized its status as an axiom that has to be assumed as true in itself.

    That is why I don’t agree when Coel claims that mathematicians don’t explore 1+1=6.5 for empirical reasons. The reason they don’t explore 1+1=6.5 is that it would produce a system that is isomorphic to our 1+1=2; i.e. a system that is reducible to the same set of axioms.

    Contra Coel I believe that if we want to understand how mathematics works we should still look at formalism. For example Peano’s axioms, as most axiomatic systems, start with the posing of self-identity (x=x, law of noncontradiction), poses definitions of objects (0 is a natural number) and allows the transformation of the axioms via truth-conserving functions.
    I think that looking at mathematics this way shows that the backbone of math is not the self-evident truths that we think of in our everyday life (1+1=2) but rather that math is a series of ways to talk about objects when defined in a certain way.
    But the objects it defines are abstract, they have no real reference in the world, but only what is imposed on them by our interpretation.
    With Wittgenstein I believe that mathematical propositions are neither true nor false, in the sense that our ordinary language is, but only that when taken together they are consistent. It is only when we interpret the propositions of math as applying to certain objects in the world we manage to create sentences that are passable of being judged true or false. A correct interpretation will produce true sentences while an incorrect interpretation will produce false sentences.

    If I want to count how many oranges are in the bag but I can’t distinguish between oranges and grapefruits I will produce false statements. If I treat my trapezoidal table as a square I will produce many false statements about it. All these statements would be empirically falsifiable but this would not affect in any way the status of mathematics.

    That is why, while the empirical world may be a source of inspiration for mathematicians, mathematics is certainly not reducible to empirical knowledge.

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  24. Consider modus ponens, is it validated empirically?

    No, in fact validating modus ponens empirically is a meaningless concept – it can’t be done. It is like empirically validating that there can be no married bachelors. Sure I can empirically verify that I am married but how do I then empirically verify that I am not a bachelor so I know that know I am not a counter-example? I can’t because the statements truth follows from the definition of the terms.

    Modus ponens is the same. That ((a->b)&a)->b is a tautology follows from the definitions of the terms.

    So we can easily empirically verify a case in which a & ~ b, but we could not possibly verify (a->b) because, for the purposes of logic, implications is not a thing, it is a concept.

    So could there be a possible kind of reality in which modus ponens was not a tautology? Maybe in the sense that there might be a reality in which the concepts of implication and conjunction could not have any possible meaning and so modus ponens could not have any possible meaning.

    But you could not have a reality in which (a & ~b) & (a->b) was a tautology because that would simply mean that there was a different meaning to the terms.

    So could there be a reality in which there was a different logic? Well, no, because if there was a reality in which even the concept of logic existed, as we use the term, then all of those concepts mentioned earlier could exist, no matter what the physical reality. And if those concepts existed then modus ponens would be a tautology.

    There might be a reality in which something that I arbitrarily label “logic” existed but was not logic as we mean the term.

    In that case you would simply be saying that there was a possible reality in which needle-nardle-noo. Maybe there is, but that would entirely depend upon what needle-nardle-noo means.

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  25. 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.

    If that is right then it represents a big problem with string theory, doesn’t it?

    Because if string theorists genuinely do not care if they are doing physics or maths then they genuinely don’t care if any of their results are ever experimentally confirmed. I would contend that if they really are completely unconcerned about experimental confirmation then they are not doing science any more.

    For a mathematician experimental confirmation is not the point. Their results are correct if they are proved from the axioms – end of story. Certain results can never be proved and mathematicians use inductive methods to get a second best result but it will still be the case that these truths stem from the axioms and rules used rather than being descriptive of an external reality.

    The Banach-Tarski theorem is correct because it has been proven. Anybody might claim that it makes no physical sense – but so what? That would only be a problem if mathematics was a branch of science, or if it was an empirical epistemology.

    Some complain that it can only be proved by accepting the axiom of choice and that the axiom of choice is not a True Scotsman because it can never lead to physically meaningful results.

    But again, that is just to assume the conclusion that mathematics is an empirical science.

    From a mathematical perspective there is no problem with the axiom of choice.

    The knowledge of mathematics is not, and has never been that such-and-such axiom is true. The knowledge of mathematics is that such-and-such a result is true of such-and-such axioms.

    As such it is a fundamentally different exercise from physics.

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  26. Coel, the flaw in your argument is in the triviality of your math examples. “1+1=2” is not much of a theorem, and is more accurately the definition of “2”. Try applying your argument to a real theorem, such as the infinity of primes, as someone suggested.

    There certainly is a sharp qualitative difference between the work of Riemann and Einstein. The mathematical theory of general relativity was worked out by Minkowski, Grossmann, Levi-Civita, and Hilbert — all mathematicians. Einstein did not prove any theorems and rarely even made any mathematically precise statements.

    Your comments about Godel’s theorem are about like saying that the irrationality of the square root of 2 shattered hopes for geometry, or that comets shattered hope for astronomy. And it surely does not help your argument, unless you can explain how the theorem can be empirically understood or validated.

    Your lesson from this is that “scientific results are always provisional”. Maybe so, but mathematical results are not. Godel’s theorem is not provisional.

    You can, of course, define “science” any way you please, but you have failed to give a definition that includes mathematics. To you, science is empirical and provisional, but you do not give a single mathematical result with these properties. Do you really want to argue that “1+1=2” is a provisional result subject to empirical acceptance or rejection? Will you please tell us how this equation might be rejected?

    You deny a “clear epistemological demarcation”, but you do not give an example on the boundary of math and science. Your closest example is string theory, but you must realize that most of that subject is viewed by outsiders as neither science nor mathematics.

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  27. Coel:
    “all human knowledge is empirical, deriving from contact with empirical reality.”

    You suggest that this will be one of the conclusions to your argument; in fact this is itself an argument:
    All human knowledge is empirical, [because] it derives from contact with empirical reality.
    Or:
    All knowledge derives from contact with empirical reality;
    what derives from contact with empirical reality is itself empirical;
    All knowledge is empirical.
    I’m not buying this.
    The first problem is terminological; the “empirical” describing reality is not the same “empirical” describing knowledge. Reality, just as such, is not empirical (or anything else), it is just what is.
    The proper phrase here is “empirical contact with reality” (where “empirical” means ‘sensory experience’ of reality). But once we rephrase this, the problem becomes clear – whatever could we mean by “all knowledge is ‘sensory experience'”? Obviously what you want to say is that “all knowledge is derived from sensory experience,” but then your argument is a tautology:
    All knowledge derives from sensory experience; whatever derives from sensory experience is (derived from) sensory experience;
    therefore all knowledge is (derived from) sensory experience.
    One way to see this problem is that the “empirical” of “all knowledge is empirical” is part of an assertion of empiricist epistemology, while the “empirical” of “contact-with-reality” is a common language description of experience with reality.
    So what you really want to say is, “all knowledge derives (originates from, is developed out of) sensory experience.” After this you can then make a case for empiricism; but that won’t get you a case for ‘scientism.’
    Just as I can grant that mathematics derived (originated) from counting, I can grant that all knowledge derives from (originates from, in the sense that the questions developing knowledge begin with questions about) sensory experience. But as I’ve just made clear, this is not at all the same as saying that “all knowledge *is* [sensory experience].” That’s just silly. That knowledge originates in sensory experience (via questions about it) makes no claim at all on its later development.
    You spend a lot of time hashing over the origin of number as derived from counting, e.g. “1+1=2.” I know some philosophers do go back to that – but as a problem. The problem is, how do we get from there to complex deductive mathematics? That answer is not at all clear, and you offer no view on the issue. Algebra is your problem (let alone, for now, calculus, and higher order mathematics dealing with large numbers, or say, n-dimensional geometry), not simple addition. Talking about adding apples as a basis of higher math is like discussing Newton’s law of gravity in terms of an apple falling on his head.
    Newton himself said that he *deduced* the law of gravity; while the physical issues involved, e.g., the moon and the earth, and while, had the movements of these spheres not tallied with the law, the hypothesis would have been falsified; the mathematical deduction itself involves abstract conceptual entities, not empirical or physical spheres: “Every point mass attracts every single other point mass by a force pointing along the line intersecting both points” (Newton). Now, physics seems to me to be a complex interweaving of mathematical deductions and empirical testing; that’s exactly one of the things that makes epistemology of science, and the philosophy of science in general, so fascinating. You want deduction, empirical testing, knowledge about abstract entities, knowledge about physical entities, and the sometimes difficult trail that somehow connects them all, to reduce to the same basic stuff. I think you need, at the very least, to engage in the epistemological problems directly before making such a claim.
    I should remark, that what you seem to be arguing for is a vigorous, but, frankly, primitive, empiricism, which you then turn around and claim to be “science” by nature or per se. At one point you say you don’t really care what label gets used for it.
    But labels are important as markers for positions in a field of competing claims. They help us understand what claims relate to others, and what their issues are. John Stuart Mill once claimed that all mathematics were reducible to counting (he was quite a sophisticated empiricist, but he didn’t understand mathematics). And Hegel also claimed that all knowledge was essentially acquired one way, through dialectics (through which he once deduced that there could be only seven planets in the solar system). Basically you’re trying to have a variant Millsian position totalize knowledge in a manner similar to Hegelian claims for the dialectic. I don’t think Mill would appreciate that (and I know Hegel wouldn’t).
    That leads us to your second proposed conclusion to your argument, that “our knowledge of reality is also unified,” a “seamless whole.” What’s odd about this is that your article does not argue this, simply asserting or implying it, in variant ways.
    Here’s one such moment I found particularly irritating:

    “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.”

    Because mathematicians “relax axioms” (derived or posited deductively), and physicists “relax” theoretical postulates (not axioms, BTW) derived from previous hypothesis and empirical research, does not at all mean they’re doing the same things, it means they’re not!
    BTW, non-Euclidean geometry does *not* “make sense,” it is counter-intuitive.
    Speaking about which – “intuition” – well, we’re not even going there, because your text makes it evident that you are uninformed as to the history of the problem of intuition.
    So basically we have an incomplete philosophical argument for a primitive empiricism offered as a totalization of knowledge based on shaky premises and lacking historical accountability.

    This is not convincing, I’m sorry.

    Beneath your somewhat bombastic prose, I recognize the passion; so I won’t say abandon your position. Rather I suggest further study, more precise language, rethinking your perspective (if you’re going to do epistemology, do epistemology), and more carefully structured arguments.

    (Oh, BTW, here’s another problem equating number with counting, and it is simpler, and more definitive, than the problem of higher mathematics: zero. Zero doesn’t count anything, it doesn’t even count ‘nothing.’ It can not be derived from sensory experience. [In fact it was derived from Hindu/Jainist metaphysics.] Yet it’s function in mathematics is undeniable. Problems such as this have to be accounted for, not “1+1=2” – in whatever universe we like to imagine.)

    (Oh, another BTW, concerning an argument that appeared on the thread, about “all bachelors are unmarried.” One commentator said that this was simply a matter of definition; that misses the point: It is true *because* is it is a definition, there’s no predication as “married” (or its negation), no object to designate as “bachelor” except as constructed by definition. Therefore definition is by nature the positing of premises for the sake of deduction. And while this may refer to “empirical reality,” the process is not itself empirical, and thus the issue falls into the domain of logic and epistemology, not the “seamless whole” of empirical knowledge.)

    (One last related BTW: one commentator raised the problem of induction, and this point was treated as trivial. The problem of “unmarried bachelors” shows us that the problematic relation between induction and deduction is far from trivial.)

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  28. Coel, you say that it is ” epistemologically identical”, except that one uses empiricism and plausibility arguments and the other uses axioms and logic. In other words, not similar at all.

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  29. Hi Stephen, Hi Coel,

    thanks for your responses. Just to be clear: I am not terribly bothered by the problem of induction and certainly do not that scientists should be bothered by it (we have philosophers of science for that). But I wanted to point out that Coel seems to have no other choice than agreeing that science (construed as empirical reasoning) cannot hope to address the problem. Now, there certainly are a lot of people who claim to have addressed it reasonably, so if that is the case then whatever these people did, Coel, by his own definitions, should not want to classify this as science.

    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.

    Don’t take that wrong, but it
    rather seems to me that you do not have much experience with what mathematicians actually do (outside of physics perhaps).

    From my experience, my colleagues from pure math are for the most part not bothered at all about the real world. It’s not so much that they actively go “against” the real world, rather than being convinced that they study a whole world of their own, wholly detached from the natural world.

    But even in applied math, while it is true that most of our models are inspired by “real-world applications”, the devil is still in the details. For when I study a system I am very often deliberately introducing assumptions that run contrary to the real-world. Heck, I want to prove stuff and reality is complex! What is more, in doing so I am led by structural insights (which assumption do I need to tweak to be able to prove a theorem) not by real-world facts.

    I find that the more important question in this context is: what makes it true, that a certain theorem holds. It can’t be a fact about the real-world (aka an empirical fact) given that I just introduced an assumption that does not hold in reality.

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  30. Selfawarepatterns,
    “quantitative logic” is not about ‘counting.’ (The distinction between measure and counting has not been addressed in either the article or the thread, but it is important. Hint: you can measure points on a graph, but that doesn’t necessarily count anything).
    I never cared for Frege, but his importance cannot be dismissed. And he is as far from ’empiricist’ as one could want. Logic as “The science of the most general laws of truth” is such exactly because it is not empirical (the hope of the proto-positivistsof the 1800s – including Russell, often championed as an empiricist, but actually an idealist – was precisely that they could develop a deductive logical argument for mathematics, irrespective of the empirical).

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  31. I define taking a shit as all the activities I do while excreta is leaving my body. Hence, Mathematics is a part of shitting. Of course I am not using a narrow definition of taking a shit. However, I require myself to be taking a shit for me to come up with conjectures.

    Of course, Mathematics involves the real world in coming up with conjectures and defining objects. If that makes it science, then astrology is science and so is literature. The requirements of justification and clarity have to be fore fronted in the definitions of science and mathematics. Mathematics requires a proof from axioms and conjectures, and hence there is no conclusion currently about the truth of the Riemann hypothesis. Science is dealing with the claims of truth about the real world and hence it cannot have a proof in the sense of mathematics.

    You can define everything which takes inspiration from the real world to be science. However, do you not see that there is a useful distinction to be made between these two activities on the basis of justification? So, let’s take your route and call everything which takes inspiration from the real world science. Then let me categorise science into various types:
    Subscience 1: requires a deductive proof from axioms and definitions
    Subscience 2: accepts propositions as true if they have not been rejected and have been established beyond reasonable doubt
    Subscience 3: rely on your past experiences to convince you of the truth of a claim (humanities)

    I am willing to work with these new names, but I think science, mathematics and humanities work better.

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  32. Hi Robin,

    You probably need to start with how you empirically verify the axioms of identity and non-contradiction.

    Build two models (or sets of models). Adopt the axioms of identity and non-contradiction in one model but not the other. Now test out how the models do for explanatory power, predictive power and parsimony. I would suggest that any model not adopting those axioms is going to be pretty useless in modelling the world.

    The above is a standard scientific method. Everyone accepts it for, say, choosing between two different variant models of gravity, but the point is that it works just as well for all ingredients of the model, including the maths and logic used.

    (For the axiom of non-contradiction see also note 6 of the OP.)

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  33. Hi Alexander,

    After posting this it immediately occurred to me what the reply will be: Math is also an approximation of reality, so the situation is the same.

    Yes, that is how I would reply. Concepts such as circles and straight lines are useful concepts that help us to model reality. In the same way, physics adopts concepts such as a point-like “particle” or a “field”, because they are useful models, and that sort of model used in physics is really the same thing as a mathematical concept.

    As far as I can tell, math starts out developing a system of relationships and then accidentally some of them are found to be useful in the real world …

    Historically, maths would surely have started with humans counting their sheep, counting days to the next full moon, estimating distances, weighing corn, bartering, and getting the walls of their house straight. Only later would it have been developed into the highly abstract system we have today.

    Theoretical physicists will often work with the abstract system where there is no immediate use to it. Most of cosmology, for example, is not much practical use.

    Science, on the other hand, starts out with the real world observations and only then has any incentive to propose a model.

    If I’m right about the historical development of maths (which I surely am!) then maths was the same. It is not the case that maths *started* with the logical axioms written down by Peano and Frege and others, and developed from there, rather Peano and Frege developed an abstraction of a real-world system that had been adopted because it was useful.

    There is no theory or model of non-existing but logically possible things in science waiting idly for that thing to pop up later.

    Yes there is! That is exactly what theoreticians do, they develop models based on axioms/laws. After that they (or, usually, other scientists) think about whether such things actually exist. A good example was Paul Dirac’s prediction of anti-matter, based on theoretical reasoning, which was then followed by finding anti-matter in the real world.

    But math was taught by sitting in a room deriving formulas from other formulas; never did the teacher demonstrate how to work with integrals and derivations (or complex numbers, for that matter) by showing us empirical evidence.

    When you’re teaching maths to really young children, basic counting and so on, you often do teach with real-world props. I can remember being taught with different coloured wooden “bricks” of different lengths, representing numbers, and if you placed the “5” brick alongside the “3” brick then it was the same length as the “8” brick.

    Similarly I can remember being taught geometry with drawing on papers with a rules and a compass, making triangles and circles, and measuring angles with a protractor. (And I’m also willing to bet that that’s how Euclid learned.)

    Later on, in school education, the empirical basics of maths are accepted, and subsumed into axioms, and from then on the maths is taught abstractly.

    But then much of science is often taught abstractly, in terms of “these are the laws, now let’s work with them”. For example if you’re taught about Newtonian gravity most school teachers will not start with a whole lot of empirical data, they just tell you the laws. This might not be the best way of teaching science, but it is how it is done much of the time.

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  34. Hi Mark,
    Thanks for your comments,

    One quibble – any fundamental axioms of mathematics that cannot be empirically justified should not be considered part of objective science. Right?

    It’s not quite that simple. In physics there are a lot of “axioms” and models that are either partially right or we don’t know whether they are right. Science is a process, with the task of considering competing models and picking the most useful ones. Some axioms of maths could have the same status: whether they are “useful” in the sense of reflecting real-world behaviour can be unclear. An example of that is the Axiom of Choice, which is debated by mathematicians.

    Consider the premise (or hypothesis) that these physics applicable fundamental mathematical axioms of our universe apply to all possible universes.

    Yes, to what extent axioms/laws are universal, or local and contingent, is a really interesting question. Some people consider that the divide would be between maths and logic (which are taken to be universal) and physics (which is taken to be contingent). However, I’m not aware of a decent argument for that claim. One could equally argue for some of maths/logic being contingent, and some of physics being universal.

    Of course from my point of view there is no real distinction between maths/logic and physics, and the issue of which axioms/laws are universal and which are contingent is something that we have to figure out empirically, as best we can.

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  35. Hi Antonio,

    The big problem I see is that if mathematics were empirically derived it would be also be empirically falsifiable. But it is not. In fact we have no idea of what a falsification of any mathematical truth would look like.

    Followed by:

    The key example here is the fifth Euclidean postulate […] When we found out that our world quite often does not respect the fifth postulate …

    Haven’t you answered your own question? You’ve just pointed to an example of falsifying a mathematical axiom, in the sense that “our world quite often does not respect the fifth postulate”.

    … we didn’t throw it out into the dustbin of failed theories, but we actually recognized its status as an axiom that has to be assumed as true in itself.

    What this means is that we recognised that the fifth postulate is not universally true, but we also find that often it is locally and contingently true in the sense of being a good-enough model of reality.

    The status of the fifth postulate is thus akin to that of Newtonian gravity, in that we know that it is not universally true, but often it is sufficiently locally and contingently true that it is a good model.

    Thus, the status of mathematical axioms and physical laws is, again, pretty much equivalent. (Of course people will recognise that Newtonian gravity is the weak-field limit of General Relativity, and amounts to assuming that the local curvature of space is small enough to be neglected — which is identical with adopting Euclid’s fifth postulate.)

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  36. I would like to wrap up by suggesting that the essay’s main positive argument in favour of its thesis fails due the the genetic fallacy.

    The fact that we abstract most the axioms of mathematics from empirical observations does not imply that mathematics is an empirical epistemology.

    Naturally humans tend to begin with concepts with which we are familiar and cannot easily go beyond the concepts that our minds are equipped to deal with.

    Furthermore much of mathematics is developed to contribute to science and so naturally much of it will be the mathematics that describes our world.

    But none of that implies that mathematics is itself is empirical.

    You have to look at the nature of mathematics itself. Mathematical truths are not verified empirically.

    The truths of mathematics do not depend on any physical fact – there could be no discovery about the universe which would falsify a mathematical truth.

    Would any proven theorem be any less true if there were no such thing as gravity? Of course not.

    These would be true if there were none of the other forces, no spacetime, or even dimensions, particles or quantum fields

    So mathematical truths cannot be falsified by empirical data, they are not verified by empirical data and they do not depend on empirical data or any physical fact of the universe in any way that anyone has ever determined.

    So in what way is mathematics empirical?

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  37. Robin, good try, but I also brought up modus ponens in a previous thread with Coel, and he insisted that yes, modus ponens is “true” only because it “corresponds” with reality. But of course it makes no sense to say that modus ponens is “true,” and I never heard from him how he would carry out the test…

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  38. Hi Robin,

    Consider modus ponens, is it validated empirically? No, in fact validating modus ponens empirically is a meaningless concept – it can’t be done. It is like empirically validating that there can be no married bachelors.

    Build two models (or two sets of models). Build modus ponens and the law of non-contradiction into one model. For the other model, don’t assume modus ponens or the law of non-contradiction. That means that, in that model, you may not use any reasoning that assumes, either explicitly or implicitly, modus ponens or the law of non-contradiction.

    Then take the two models, and try them out for explanatory and predictive power. See if you can predict the next solar eclipse using a model with no modus ponens and no law of non-contradiction. (Note, by the way, that you won’t even have counting numbers in this model, since the law of non-contradiction is one of Peano’s axioms, so I wish you luck in predicting the time and place of the next solar eclipse without any maths!)

    Isn’t it blatantly obvious that the models with modus ponens and the law of non-contradiction work much better in modelling the world?

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  39. Hi Robin,

    Because if string theorists genuinely do not care if they are doing physics or maths then they genuinely don’t care if any of their results are ever experimentally confirmed. I would contend that if they really are completely unconcerned about experimental confirmation then they are not doing science any more.

    But science is a communal process, not an individual one. Some people are theorists, some are experimenters, some build instruments, et cetera. These people can have different attitudes. Some string theorists may genuinely not care much about experimental confirmation, they can be thinking like mathematicians and exploring the maths of string theory for its own sake. If you didn’t like doing that sort of thing for its own sake you would not become a string theorist!

    Of course other scientists might then take those results and worry about whether they are experimentally confirmed — and yes, before anyone says, there will be people who care about both — and thus “science overall” cares about this.

    The knowledge of mathematics is not, and has never been that such-and-such axiom is true.

    Again, I disagree, if you think about the origins of maths it is all about real-world situations. Some people might not care whether the Axiom of Choice matches real-world behaviour (just as some string theorists may not care about the experimental confirmation of string theory) but the “collective” does care about those matters.

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  40. Hi schlafly,

    “1+1=2″ is not much of a theorem, and is more accurately the definition of “2”.

    Agreed. The question is then why has maths been defined in that way? My argument is that the whole definitional edifice of maths has been constructed because it is a good and useful model of real-world behaviour.

    Do you really want to argue that “1+1=2″ is a provisional result subject to empirical acceptance or rejection? Will you please tell us how this equation might be rejected?

    Build two models (or two sets of models). In one model adopt an edifice of maths with definitions such that 1+1=2. In the other model do not adopt that, adopt some other behaviour, or do without anything such (do without counting numbers if you like). (And mere re-labelling so that the same maths arises re-labelled is not sufficient here, you need an actually different logic.)

    Then test out your models for explanatory and predictive power in the real world. Try to predict the time and place of the next solar eclipse. See if you can do that without either explicitly or implicitly including anything that amounts to “1 + 1 = 2” or which is equivalent in some other notation, and without including some other axioms that entail 1 + 1 = 2. If you can do that I’ll be impressed!

    It seems obvious to me that the models including the 1+1=2 maths will work better. It also seems to me that this is exactly how, over history, humans have arrived at that mathematical edifice.

    Your closest example is string theory, but you must realize that most of that subject is viewed by outsiders as neither science nor mathematics.

    Really? Then how is it viewed?

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  41. Hi Coel,

    Many would say that tautologies are always true due simply to the way the meanings of their connectives add up. (And some might say that mathematical truths are similar sums of meanings.)

    Do you hold to the contrary that tautologies are empirical truths?

    If so, what do you see as the relation between the meanings of a tautology’s connectives and the tautology’s truth?

    Could a change in the empirical world turn a tautology false though the meanings of its connectives remain the same? If not why not?

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  42. Hi Madhav,

    You can define everything which takes inspiration from the real world to be science.

    I’d define “science” along the lines of seeking knowledge of the word, using the best tools we have to do that.

    Then let me categorise science into various types:
    Subscience 1: requires a deductive proof from axioms and definitions
    Subscience 2: accepts propositions as true if they have not been rejected and have been established beyond reasonable doubt
    Subscience 3: rely on your past experiences to convince you of the truth of a claim (humanities)

    The point is that these are not distinct. 2 and 3 are interdependent, and 1 is interdependent with 2 and 3. Science is also a process of scheming up models based on axioms/laws and seeing how well they fit data.

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  43. In my haste, I wrote my first paragraph to say exactly the opposite of my intention. First, the parenthesis, which, as directly following the first sentence, appears to imply that quantification has something to do with measure, was intended to be paragraphed separately, because it is a separate comment.
    Worse yet, I wrote ” you can measure points on a graph, but that doesn’t necessarily count anything” which is precisely opposite to what I wanted to say: you can count points on a graph, but that doesn’t necessarily measure anything.
    I wanted to raise, but not discuss, this issue: measure is an empirical practice involving counting, but clearly counting is not identical to measure, and can be used on purely mental constructions.
    The remark on Russell (which was unnecessary anyway) is controversial, since Russell stands opposed to the British Idealists of his own day; but what convinced me of Russell’s idealism is a (now little remembered) debate between Russell and Dewey in which Dewey tried to make a Peircean case that logic was developed out of the need to organize our thoughts in response to experience – an empiricist claim – which Russell rejected. In any event, Logical Atomism, Russell’s attempt to appropriate Wittgenstein’s Tractus into metaphysics, is clearly a Platonism.
    I mention that, because one problem with Coel’s article is that, if matters could be as clearly reducible as he claims, why the long history of debate? It’s not like these issues haven’t been raised before.

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  44. Hi ej,

    Just as I can grant that mathematics derived (originated) from counting, I can grant that all knowledge derives from (originates from, in the sense that the questions developing knowledge begin with questions about) sensory experience. But as I’ve just made clear, this is not at all the same as saying that “all knowledge *is* [sensory experience].”

    Yes, agreed. I’m arguing that knowledge derives from experience of the world, not that “all knowledge *is* [sensory experience]” (which I agree is not sensible).

    Now, physics seems to me to be a complex interweaving of mathematical deductions and empirical testing …

    Agreed.

    Oh, BTW, here’s another problem equating number with counting, and it is simpler, and more definitive, than the problem of higher mathematics: zero. Zero doesn’t count anything, it doesn’t even count ‘nothing.’ It can not be derived from sensory experience.

    You point out a lot of issues, but most of the time I don’t see why they are a problem for what I’m saying. I’m saying that both physics and maths are edifices adopted as models of the real world, because they work when applied to the real world. We include a zero in the models because it enables them to work better.

    Therefore definition is by nature the positing of premises for the sake of deduction.

    Agreed. Making definitions is part of constructing a model. What science does is then validate the models by testing them against reality. My whole point here is that mathematics also originated as a model of reality and is a model of reality.

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  45. Robin sums it up perfectly.

    All categories of human knowledge have their foundations somewhere in empirical reality. But we build on the knowledge, constructing grand edifices that tower far above their foundations. We use tools such as logic and semantic constructions to build the upper floors. Thus the upper floors can only be validated by the tools used for their construction. And in many cases the tools were not those of science.

    As Robin said:
    So mathematical truths cannot be falsified by empirical data, they are not verified by empirical data and they do not depend on empirical data

    Or, to make the case more obvious, take one of Siegfried Sassoon’s war poems(The Rank Stench of Those Bodies – http://bit.ly/1tnZQ4B). It is grounded in the reality of war but then look at the sentence construction, the rhyme and the metaphor. The power of the poem cannot be empirically derived from the facts of the war scene. Nor can the choice of words and metaphor. They are independent constructions of the human mind. The power of the poem is derived from the verbal constructions themselves. The poem can only be understood and felt on its own terms. Science plays no role here.

    And now Coel might reply – but we are talking about Maths, not poetry. Then I would remind him of his own words:
    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

    By the same line of reasoning(knowledge is a seamless whole, same basic ideas about evidence work in all subject areas…there are no other ways of knowing), his claim that Maths is part of science, must be extended to the claim that poetry is part of science. Good luck with that. It is called Reductio ad absurdum

    I think Robin has well and truly put the subject to bed.

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