The Chomsky paradigm states that all languages obey certain laws or conditions which ultimately are a function of the physical properties of the brain. Is there a similar constraint on mathematics? Is it thought that there is universal (the universe being humans) structure of mathematics that is the result of the structure of the human brain? If so, is this the reason that math works so well at predicting observations? In other words, since our rules of mathematics arise from the structure of our brains, and it is the same brains making observations based on mathematics, it makes sense that they'd match? Might an alien mind not observe such a good correlation between human mathematics and the results of human experiments?

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    This doesn't sound like a linguistics question so much as a philosophy-of-mathematics questions . . . – ruakh Dec 22 '13 at 21:13
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    Hmmm...yes, this is a tricky one. Personally i find the question interesting, and i think this is the sort of question generally that (formal) linguists tend to be interested in, despite it being outside of linguistics proper. I'm happy to leave this open - i think the link to linguistics is there. – P Elliott Dec 22 '13 at 22:03
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    This is more or less Kant's theory about mathematics. – fdb Dec 22 '13 at 22:09
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    Chomsky's ideas of universal grammar were about natural languages. The language of mathematics, chemistry, programming, etc. don't come under this. Human brain or not, water will remain H20 or H-O-H no matter what kind of brain describes it at the atomic level. Similarly, as Turing and Church proved, all algorithmic machines can be, at best, Universal Turing Machines, i.e. there will always be problems that are undecidable by such automatons. – prash Dec 23 '13 at 5:44
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    And here's where the distinction between formal and natural language becomes relevant. – P Elliott Dec 24 '13 at 0:42

Is it thought that there is universal (the universe being humans) structure of mathematics that is the result of the structure of the human brain?

Yes, if we are speaking about basic mathematical concepts, like integers and basic spatial relations. It is now well established, for instance, that infants already possess a concept of number. A good place to start reading about this is the work of Stanislas Dehaene.

If so, is this the reason that math works so well at predicting observations?

Why yes, for basic reasons of natural selections, we would expect our basic sense of number to provide a description of the external world which is congruent with our other senses.

If so, is this the reason that math works so well at predicting observations?

It seems to me that with this question, you envision a much stronger thesis, one which is usually attributed to Kant, as user fdb correctly pointed out. The reason is that the current consensus view that basic mathematical concepts are hard-wired in our brains applies only to very basic mathematical concepts. In fact, I seem to remember reading studies suggesting that not only subtraction with large numbers was not hard-wired, the truth was that our brains are hard-wired to make mistakes while subtracting large numbers (at least when numbers are represented in the usual way). So beyond elementary concepts, the mathematical language developed through the patient working of mathematicians, sometimes (or even often) struggling against their intuitions. Such a struggle will inevitably be extremely sensitive to cultural norms and preconceptions.

A famous example is the case of negative number, which were truly accepted in the West only 1500 years after the beginning of mathematics as a science, a clear indication (to me, at least) that there are not (in any self-evident way) a reflexive property of the way our brains are organized.

Might an alien mind not observe such a good correlation between human mathematics and the results of human experiments?

It is of course hard to imagine a truly alien mind, but if you have some background in mathematics, you can envision a historical and conceptual development of mathematics truly different from what happened in any human civilization that we know of while remaining meaningful and intelligible to us. Consider for instance how an alien brain perceiving the world mostly though echolocation might envision geometry (my intuition, very feeble of course, would be that such a mind could consider blurring of periodic signals, and hence convolution and Fourier's theory, as much more intuitive than euclidean geometry). For a simpler example, consider an alien mind whose hardwired concept of number would lend itself more naturally to a modular interpretation than to a natural number interpretation (they would be counting in Z/nZ for some n, in effect, rather than in N), and if this seems too strange a thing to you, consider that this is essentially how computers count.

On the other hand, the idea that even a strikingly different conceptualization of mathematics may lead to a re-evaluation of the correlation between mathematical formalization and experiment is too kantian for my taste: after all, our predictions tend to be quite accurate, and I don't see how perceiving the world in an alternate way could change that. In my opinion, the reason mathematics model the physical world so efficiently is that the physical world is by and large simple: a large number of phenomena can be correctly described using a handful of principles (minimize something) and (in fact equivalently, by Noether's theorem) a couple of symmetries. So descriptions of the physical world will involve simple rules, and the abstract study of simple rules is (almost by definition) what we call mathematics. A good way to discriminate the (mostly pragmatic) position I here defended and the kantian position is to notice that the record of mathematics is actually quite mixed: it is great for celestial mechanics (the historical impulse for the kantian position) and physics more generally, but mostly abysmal for biology. This could be quite mysterious from a kantian position, but is crystal clear from a pragmatic one: celestial mechanics is the by-product of essentially a single law, biology is the by-product of billions of years natural selection sieving trough random changes and so is very unlikely to admit a formalization through simple rules.

Finally, because I would feel it inappropriate to clog linguistics.stackexchange with speculation on the cognitive and philosophical content of mathematics, let me mention that one of the core principle of current minimalist linguistic theory is that syntax is the by-product of the necessity of the structures constructed by a single operation (Merge) operating recursively to be legible by the so-called interfaces (for instance, the phonological interface). N.Chomsky, to which this belief about syntax is due, has contemplated in several publications (e.g here) the possibility that some higher arithmetical capabilities of the human brain (those going further that the elementary ones discussed above) might stem from a recycling of this merge operation in other cognitive context: informally, he suggests that perhaps human beings are so talented at manipulating recursive abstract structures (natural numbers, algorithms...) because our linguistic ability is based on a recursive operation.

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  • "... the current consensus view that basic mathematical concepts are hard-wired in our brains applies only to very basic mathematical concepts" That's misleading. If those basic structures are enough to build all of mathematics, as axiomatic set theories do, with a basic machinery, e.g. SAT solver with a brood force hypothesis generator for starters, and if proofs are becoming part of the machinery via Curry-Howard Correspondence, I'd say that higher level structures crystallize. Also, every idiot can count to one. NNs are DEQ solvers. Was that known three years ago? – vectory Jan 13 '19 at 7:20
  • @vectory The truth value of the sentence "higher level structures crystallize" is irrelevant to the passage you quote: the point is that there is no consensus that these higher level structures are hard-wired in the human brain, whereas there is by now a consensus that basic concepts are so hard-wired. Perhaps the consensus is wrong - who knows? - but it is a fact that there is a at present a consensus that addition of small numbers is hard-wired, and no consensus that representations of the absolute Galois group are (in fact, that applies already for subtraction of largish numbers). – Olivier Jan 14 '19 at 10:58
  • @Oliver I'm calling into question the meaning of hard wired, not the merit of the answer. I have the fantasy that consciousness bootstrapping, the "first thought", happens in the womb, forming a sense of one. It's thinkable that DNA hard-codes the blueprint for an Arithmetic Logic Unit or other Finite State Machine, but I do think that prenatal emergent properties still qualify as hard coded. If I'd be critical I'd say that you confuse arithmetic for algebra, and an important question is, do very basic mathematical concepts include a concept of irrationality, pun very much intended. – vectory Jan 14 '19 at 11:42

I think that ultimately, this is a question without a definitive answer. But it's not a bad one to ask. My feeling is that it contains the germs of a downfall of the 'Universal Grammar' thesis, since unlike language, mathematics can fairly clearly distinguish between the axiomatic and lawful. So while there are many ways in which some mathematical relativity can be conceived, it has no impact on the universal foundations of mathematics. You don't need to look to aliens. The ancient Greeks saw the world through geometry and saw their arithmetic as being reducible to it (I'm grossly simplifying, of course), and we see the world in exactly the opposite way, reducing geometry to arithmetic. However, these two universes are completely equivalent, something that cannot be said for two languages.

However, this first part of the answer takes at face value the very broad reading of the 'Universal Grammar' assumptions. However, I'd say that when read more narrowly (viz when we look closely at the actual principles posited), I'd say that there's no reason to assume that something like this would hold for mathematics. When you look at the grammar of any language, the universal principles posited by the UG theorists only govern a tiny fraction of all the rule-ful patterns present in a language. But with mathematics, you can derive the vast majority of its rules from the underlying axioms. Therefore, I suspect that the Lakoff/Nunez thesis has much more to say about this: http://www.amazon.co.uk/Where-Mathematics-Comes-Embodied-Brings/dp/0465037712.

Note: I should declare an interest here. I'm an ardent anti-Chomskean (http://metaphorhacker.net/2010/08/why-chomsky-doesnt-count-as-a-gifted-linguist/) and I translated one of Lakoff's books, so I certainly come with some theoretical baggage.

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  • Universal Grammar is only unfallen for those who have drunk the complete Kool-Aid. Most linguists think it's a crock. And I agree with Lakoff and Núñez that mathematics is a metaphorical construction, so that it's as much a part of the brain and body as any other metaphors. Nothing special about mathematics here, and no general mathematical rules derived from that, either. Hersch and Davis's The Mathematical Experience is probly a good idea to read, along with Lakoff and Núñez. – jlawler Dec 23 '13 at 21:41
  • What are we taking universal grammar to mean here? I take it to mean the set of properties that all languages have in common, and i take the universal grammar thesis to mean that this set is non-empty - Maybe these properties aren't linguistic-specific. I get the impression that most linguists believe that something like this is almost trivially true, whether they call it UG or not. – P Elliott Dec 24 '13 at 0:46
  • @DominikLukes I don't really understand your answer, or how it relates to the UG hypothesis wrt NLs. All i can really get from it is that mathematics and NLs are different things. Big surprise there. You say: " these two universes are completely equivalent". It seems to me that this precisely isn't true. We have a completely different (and arguably better) understanding of the natural world than the Greeks did. We also have, e.g., non-Euclidean geometry, which the Greek never even conceived of. – P Elliott Dec 24 '13 at 0:52
  • @PElliott Sorry, I was ranging a bit farther afield than prudent. 1. What I meant but 'completely equivalent' is to say that anything that can be expressed in geometry can be expressed in arithmetic and vice versa (and I'm aware that this is not always practical outside a quite limited universe of expressions). 2. By 'universal grammar', I mean the set of particular principles formal linguists refer to under that label. Most definitely not 'common properties' of all languages. Everyone admits to some here but they are completely incommensurable across theories. – Dominik Lukes Dec 24 '13 at 12:09
  • @DominikLukes Right, thanks for the clarifications. The principles that formal linguists refer to under the label of 'UG' are just (falsifiable) hypotheses about what those common properties might be. – P Elliott Dec 24 '13 at 15:57

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