What accounts for humans’ math ability?

KF: Well said! gpuccio

BO'H (attn GP): I am speaking to the substantial core of the disciplines, what in the main builds the knowledge base. Mathematics and the natural sciences simply do not work in the same way, even through the former will be applied in the latter, often as model building. mathematics rests on axiomatic systems and bodies of proofs deriving from such, that is its strength. The sciences rest on observation and inference, framing theories as models that seek to be accurate to nature and at least empirically reliable across the span of observations. Very differing kinds of warrant apply. GP seems to be speaking in more details than I can at the moment given what I am facing. KF kairosfocus

Bob O'H: Just a note about this at #29: "Maths is much more than proving theorems, it’s also about building structures (whether they be models, algebras, logical systems or something else). Theorems show that the structures are correct." Math is about building structures, but there is no limitation to what you can build, because mathematical structures are completely independent from the outer world. IOWs, you can build anything you like. There are not "correct" or "incorrect" structures. There are only "consistent" or "non consistent" structures. Non consistent structures are usually refused in mathematics. But any consistent structure can be accepted. IOWs, the only requirement for a mathematical (or logical) structure is that it must not allow to prove at the same time one proposition and its opposite: A and not A cannot be proven both true in a consistent system. Theorems do not show that the structure is correct: they are tools to derive by deduction all the possible logical implications of the structure. Only if a series of correct theorems can prove that A and not A are true at the same time, we can say that the whole structure is non consistent. That's how Godel proved that, for any mathematical system, we have to choose between consistency and completeness. Guess what is the usual choice? :) gpuccio

Bob O'H: Just a final note, to summarize what I think. Let's say that we accept that mathematical knowledge is vastly innate, IOWs a product of the human mind. IOWs, we accept that it is not empirically derived, and that its objects of knowledge are not observable facts (the platonic concept of mathematics, accepted by many). Now, we have two systems: a) The world out there b) The mathematical world independently generated by the human mind. The problem here, what I call "the big question", is the following: Is there any need a priori that the laws that shape the world out there have a very good correspondence with mathematical tools generated by our human mind? And the simple answer is: no. No, unless we accept that both things were generated by a similar process or agent using similar principles. IOWs, we could as well have: a) An outer universe A which works according to its principles, or no principle at all, completely different from our poor mathematical tools. and: b) An inner mathematical world, completely satisfying to the human mind (or at least to the minor subset of the mathematicians'minds), but completely useless in describing what happens out there. This is an absolutely possible and even likely scenario, if the two worlds have no real connection in their origin. After all, many times whole systems generated by the human mind have demonstrated to be completely ineffectual in describing or predicting what happens in the outer world (including many superstitious convictions about how to win a lottery! :) ). So, although I appreciate your thoughts, I believe that the big question remains. Of course, you are absolutely entitled to believe differently :) . gpuccio

Bob O'H at #26:

Described is important here. The description of a phenomenon is not the same as the phenomenon itself.

Of course. But if you can effectively describe a phenomenon in some way, there must be a good correspondence between formal properties of the phenomenon and formal properties of your description. The map in not the territory, but a good map needs to use a language that is appropriate for the territory.

So maths describes the universe, but that does not mean that the universe follows the mathematics: it isn’t solving differential equations and the like.

It means that the universe has a mathematically describable structure. Of course the universe is not "solving differential equations": its designer solved them, and implemented the solution in the laws that shape the universe. You are confounding the results of the work of the mathematician with the mathematician itself.

I use Markov chains and Gaussian distributions when modelling ecological communities, but that doesn’t mean the universe is Markovian or Gaussian. But those mathematical constructs work well as models of the real world.

This would require a long discussion, but I will try to give at least a brief answer. Probabilistic models (excluding the quantum context) are a mathematical way of describing with some efficiency deterministic systems that we cannot realistically describe in a deterministic way. It's always "hidden variables stuff" (excluding the quantum context). But the true laws that govern the system are the deterministic laws. Now, a deterministic laws is not the same as a probabilistic model. Newton mechanics or the gravitation theory, in any of its form, are not probabilistic models: they try to explain observed facts with a strict mathematical description. They are models for observed facts, because as we know they cannot be proved, but only accepted as best inferences, but the model itself, the tool, is deterministic, not probabilistic. So, if we accept that the model itself is the best inference, at least for the moment, we are also accepting that some absolute laws which has a form comparable with the mathematical form of the model is at work out there. And the big question comes again: why are there laws at work out there that have some form comparable to our mathematical tools?

I don’t see that QM is conceptually different, it’s still a model of the universe.

QM is conceptually different, unless you are part of the very restricted group of post-einsteinians who still believe in a "hidden variables" interpretation of QM. QM is a strictly deterministic set of laws, that describe the evolution of the wave function. It is exactly like newtonian mechanics, or gravitation law. A strict mathematical construct that can be applied to the outer world. Then, there is the probabilistic aspect as soon as the wave function collapses. OK, but again, unless we are the last einstein fans in QM thought (and I am certainly not one of them), probability has here a very different meaning: it is not a way of describing at best a system with hidden variables that we can some day find out; it's an intrinsic part of the deterministic theory. Such intrinsic probability is a law itself, not only a way we describe systems. It is a law at work out there, only its mathematical form is different from a strictly deterministic law: it is a strictly probabilistic law, which determines what happens. Again, it is not necessarily the law as we express it (the map is not the territory), but something that must have a strong formal correspondence with our law. And let's remind here that QM laws (including the probabilistic component) are the most efficient mathematical tools that we have ever developed to understand and predict observable facts that happen out there. gpuccio

Bob O'H at #27: "Mathematicians don’t only use induction in inductive proofs. They also use it in the way scientists do, to make conjectures." Not the same thing. Mathematics is made of proofs and theorems. Conjectures are only a minor part, of relatively scarce importance unless they are proven (or, like in the case you mention, disproved). I can agree that making conjectures and disproving them by counterexamples is in a sense an application to mathematics of the inference procedure used in empirical sciences, but you seem to miss a couple of important points that make the difference: a) As I said, making conjectures and disproving them is only a minor component of mathematical thought. The bulk of mathematics is proof, either by deduction (in most cases) or by mathematical induction (which is not the same thing as empirical inference). Conjectures can be made, but mathematicians are however trying to prove them. If they are falsified by some empirical computation, like in the case of Lender and Parkin, that just means that mathematicians will stop trying to prove them. But, unless and until they are proved, conjectures are not really a part of mathematical knowledge. b) The situation is opposite for empirical sciences. Empirical sciences cannot prove anything. There are no empirical theorems which can be proved by deduction or mathematical induction. The only knowledge that is left in empirical sciences is inference, in particular the choice of the best available inference, or empirical falsification (if and when the Academy decides it can accept it). IOWs, theories are the only form of empirical cognition, while conjectures are only an accessory and not really important part of mathematical knowledge. That's a big difference, and you seem to miss it, or to greatly underemphasize it. gpuccio

Also, model-making is not at the core of mathematics, it is an application, which is often indeed inductive.

Thank you, so we actually agree. Or will do once you realise that applied mathematics is actually mathematics too. Maths is much more than proving theorems, it's also about building structures (whether they be models, algebras, logical systems or something else). Theorems show that the structures are correct. Bob O'H

BO'H: Mathematical induction is not inductive in the sense commonly used in science. One proves that case n implies case n+1 then case 1 or 0 etc. This then entails a chain. Now, that mathematicians may use induction in forming conjectures is not at all the same as that such are the backbone of mathematical, axiomatic systems. In sciences, inductive frames are the backbone of theories etc. I am astonished that some would seem to see this as a controversial statement. Also, model-making is not at the core of mathematics, it is an application, which is often indeed inductive. Arguably a theory is often a model that we hope may be close enough to true. And of course we look to empirical observational and experimental data for confirmation of evident empirical reliability. Model making in short is typically application of Math to science, engineering, management or economics etc. We do not look to model making to claim demonstration of theorems etc. KF kairosfocus

gpuccio @ 25 (& kf @ 20): Mathematicians don't only use induction in inductive proofs. They also use it in the way scientists do, to make conjectures. They can then test them, in the same way that scientists do (e.g. the work behind this paper). As I pointed out @ 16, there is even a branch of mathematics called experimental mathematics. (there's also a branch called Lie Theory, but I'm not sure I trust any of those results...) Bob O'H

gpuccio @ 24 -

Laws shape the outer world, and those laws can be described only by a very sophisticated use of mathematics, which seems to be just a product of the mind.

Described is important here. The description of a phenomenon is not the same as the phenomenon itself. So maths describes the universe, but that does not mean that the universe follows the mathematics: it isn't solving differential equations and the like. I use Markov chains and Gaussian distributions when modelling ecological communities, but that doesn't mean the universe is Markovian or Gaussian. But those mathematical constructs work well as models of the real world. I don't see that QM is conceptually different, it's still a model of the universe. Bob O'H

Bob O'H: I agree with KF (post #20) that the induction used in some forms of mathematical proof is completely different from the inference in empiric sciences. They are two very different processes. That said, I think that inductive mathematical proof still makes mathematicians a little uncomfortable, even if they have to use it here and there. So much are they rooted in deduction! :) gpuccio

Bob O'H: I appreciate your point of view, but I have to disagree. It's not only a question of "regularities" and "formal ways" (whatever those concepts may mean out of a mental perspectives. The example of QM, that I gave, is specially intriguing, because quantum concepts are really distant from all our intuitive perceptions of the outer world: they cannot really be grasped outside of a very sophisticated use of complex mathematics (literally complex: see the necessity of complex numbers, which are, by definition "not real"), and the mathematics that is implied is definitely beyond the grasp of most people. And yet, QM is the most successful theory of the outer world that we have been able to develop. I don't know how the concept of wave function, the controversial concept of its collapse, and many other aspects of basic QM theory, can be described only as a "formal way to describe regularities". And what is a "formal way", after all? It's only a mental conception. A formal way is much more than a regularity, but even a regularity has no reason to exist if there is no mind to define it. Laws shape the outer world, and those laws can be described only by a very sophisticated use of mathematics, which seems to be just a product of the mind. So, the big question remains, IMO. If it's not "thinking God's thoughts after Him", then how do you explain it? gpuccio

GPuccio @19 - the universe has regularities, and mathematics is a formal way of describing regularities. So e can come along and use the bits of mathematics that we need to describe the world we inhabit. There are other mathematical constructs we could use, but they don't explain the world so well, so we discard those. Bob O'H

KF: Indeed, the old "thinking God's thoughts after Him" remains, I believe, the only reasonable explanation :) gpuccio

GP, because it explores in a systematic way the logic of structure and quantity, which affects physical objects through the logic of being. Which is pointing to the central significance of mind in the world, or indeed any possible world. KF kairosfocus

BO'H: mathematical induction is in fact deductive in character. So is mathematical disproof of a suggested claim by reductio ad absurdum. The knowledge base and key claims of science address the world of experience and observations, and the pattern of warrant that dominates is of inductive character. In mathemacics we are characteristically starting with axiom systems and deriving results from these through the deductive logic of entailment or implication. . KF kairosfocus

Bob O'H: OK. But the big question remains: how is it that a discipline generated by the human mind to investigate conscious insights of the human mind (mathematics), is so efficient and precious in explaining observed facts (which are certainly not generated by the human mind)? See quantum mechanics, for example... gpuccio

gpuccio -

As I see it, the difference is not in the methods, but in the stuff that is investigated.

Yes, I agree. It's also why science and organised crime are different, to use Feyerabend's example. Science does use deductive reasoning as well, but the step it has to take that maths does not is the link to observed data, and to unobserved data (e.g. through generalisation or prediction), and it's here where different sorts of reasoning are needed. Bob O'H

Bob O'H: The things you say may be true, more or less, but I don't think that they are really relevant for the basic difference between mathematics and empirical sciences. As I see it, the difference is not in the methods, but in the stuff that is investigated. Empirical sciences deal with observed facts, and try to explain them. To do that, they use mathematical tools. However, the explanation itself is never deductive, and is always an inference. Of course, the mathematical structure that is used in the explanation can use deductive methods. On the contrary, mathematics does not explain observed facts. In itself, it is about specific cognitive insights of the human mind. And it is essentially deductive. OK, we have the use of algorithms and computers in modern mathematics that helps investigate some problems, but the essential nature of the problems remains mental and the essential method used remains deductive. So, the point IMO is that mathematics originates from the human mind, while empirical sciences originate from the attempt to explain observed facts by tools that originated in the human mind (like mathematics). Of course, I am aware that not all mathematicians or philosophers of mathematics agree with the "neo-platonic" view of mathematics, but I think that many of them do agree with it. There are, of course, those who believe that mathematics, like empirical sciences, originates from observed facts. I, like Penrose and many others, believe differently. gpuccio

kf - you are aware that (a) not all of science is inductive, and (b) mathematicians also use induction to falsify hypotheses? There are times when, in science, we can look at all possibilities and see if they conform to our theories. If mathematics isn't science then it's because mathematics and science investigate different things, not because their methods are fundamentally different. BTW, there's even a branch of mathematics called experimental mathematics. Bob O'H

Headlined: https://uncommondesc.wpengine.com/animal-minds/is-mathematics-a-natural-science-is-that-important/ kairosfocus

BO'H: While I now have far weightier matters on my mind, I pause. Inferences of inductive character as described are the heart of scientific methods and reasoning. Mathematics, since the days of the ancient Geometers, has been deductive. The number and colour of swans is an indefinite value stretching into the future, the unobserved past and involving unobserved cases in different places --- you CANNOT inspect all swans, so inferring whiteness on the pattern was inductive and failed in the 1700's. A conclusion based on exhaustive inspection of a finite, definite set of cases, is simply not the same as such induction. And note, the issue pivots on reasoning pattern. Mathematics simply does not work in the way Natural Sciences (much less social and psychological/ behavioural ones) do. Indeed, it is the great gap in the naturalistic account, dealing with abstract entities and logical relationships that then by force of how logic affects possibilities and necessities of being, constrain what may be in an instantiated world. The logic of structure and quantity stands athwart the rush of evolutionary materialism and challenges it with cases that cannot be avoided, but whose full significance can be suppressed. For instance, absent a responsible, rational, free, morally governed mind, the integrity of thought and value on truth and doing it right required for Mathematics is fatally undermined. And that brings with it all of that stuff about how can minds be free, and how can minds be morally governed. Where, mere programmed mechanical necessity and/or chance variability do not account for rational, responsible freedom. Cannot account for it. That is where men like Euler speak still, and not just in 0 = 1 + e^i*pi or the like, an expression that shows the deep coherence across huge swathes of Mathematics that in the course of their development did not at all need to come together like that, as far as we know. Which is extremely suggestive on the core nature of the roots of the world. KF kairosfocus

kf - empirical does not mean "inference to best explanation or generalisation from a consistent pattern, inductive argument", so you're shifting the goalposts, and it is nonsense to suggest that complete enumeration means that something is not empirical - "all swans are white" is a statement that can be tested empirically by looking at all swans (if all swans were white!). Bob O'H

Are human mental abilities, including our innate mathematical abilities, evidence of some kind of preadaptation? It appears that Alfred Russel Wallace (who Darwin credited with co-discovering natural selection) thought maybe it was, which is probably why his contributions to theory of evolution has been relegated to historical obscurity. In his book, The Devil’s Delusion, David Berlinski summarizes Wallace’s view.

[T]here is no evident distinction, Wallace observed, between the mental powers of the most primitive human being and the most advanced. Raised in England instead of the Ecuadorian Amazon, a native child of the head-hunting Jivaro, destined… for a life [in] the jungle, would learn to speak perfect English, and would upon graduation from Oxford or Cambridge have the double advantage of a modern intellectual worldview and commercially valuable ethnic heritage. He might become a mathematician, he would understand the prevailing moral and social codes, and… [even] might find himself a BBC commentator, explaining lucidly the cultural significance of head hunting…

Berlinski then points out that…

“ the idea that a biological species might possess latent powers make no sense in Darwinian terms. It suggests the forbidden doctrine that evolutionary advantages were front loaded far away and long ago… [p 158 & 159]

john_a_designer

The material universe is finite and discrete. Mathematics is infinite. Therefore, the material universe is not all that exists. Furthermore, anything that can do math cannot have a material origin. Thus, it is impossible to explain our ability to do math by evolution. This is one of those things that is so obvious materialists just ignore it. EricMH

BO'H: First, sorry, I misread -- missed the not for some reason. Second, proof of a finite result by complete enumeration is not the same as an empirical, inference to best explanation or generalisation from a consistent pattern, inductive argument. It is like proving a logical conclusion by truth table based examination of cases instead of doing the algebra of logic. KF kairosfocus

kf - Eh? How can you say to me "[b]y your admission, you are a mathematician", when I've explicitly said @3 that I'm not. I gave an example @5 where mathematics was done empirically. You just ignored that comment. So you're not doing so well on reading what I wrote. Bob O'H

JAD, yup, that's why I noted as at 4 above. KF kairosfocus

The question here is very simple: How could a mindless process, like Darwinian evolution, produce a brain and mind capable of abstract mathematical thought so profound, as Galileo, Kepler and Newton discovered, it can be used to unlock the mysteries of the universe? Materialists are unable to answer that question. But furthermore, they aren’t even honest enough to admit they can’t. john_a_designer

BO'H: By your admission, you are a mathematician. You therefore full well know the difference between Mathematical and Scientific methods. You know that mathematics has an emphasis on axiomatic systems and on results derived therefrom by logical, step by step proofs, accumulating into what is now a huge body of knowledge. You know full well that proofs are generally not done by empirical examples and making an inference to generality or to the best current explanation or the like. KF kairosfocus

Mathematics is precisely not a discipline in which theoretical constructs are empirically tested and are taken as a sort of weak form, provisional knowledge due to empirical reliability.

Eh? The four colour map theorem was a theoretical construct that was empirically tested: they narrow down the possible maps, and then used a computer to literally try every combination. Other postulates are certainly "a sort of weak form, provisional knowledge due to empirical reliability", because not every combination can be tested, and no other proofs are available (e.g. Golbach's conjecture). Bob O'H

Folks, the fundamental problem with an evolutionary materialist account of Mathematical ability is that Mathematics requires responsible, rational freedom to identify and follow chains of highly abstract reasoning. All that stuff about jumped up apes with excess neurons in a world driven and controlled by blind chance and/or mechanical necessity simply cannot account for such freedom. Indeed, notoriously, it leads to self-referential incoherence, undermining the very theory itself, as say Haldane pointed out so long ago now:

"It seems to me immensely unlikely that mind is a mere by-product of matter. For if my mental processes are determined wholly by the motions of atoms in my brain I have no reason to suppose that my beliefs are true. They may be sound chemically, but that does not make them sound logically. And hence I have no reason for supposing my brain to be composed of atoms. In order to escape from this necessity of sawing away the branch on which I am sitting, so to speak, I am compelled to believe that mind is not wholly conditioned by matter.” ["When I am dead," in Possible Worlds: And Other Essays [1927], Chatto and Windus: London, 1932, reprint, p.209. (NB: DI Fellow, Nancy Pearcey brings this right up to date (HT: ENV) in a current book, Finding Truth.)]

Of course, Pearcey and many others have kept this issue in circulation down to today. We need to recognise that a self-undermining, self-falsifying frame of thought destabilises reasoning beyond that point. Through, the challenge that implications following on falsity at best may be empirically testable and shown reliable in a domain of thought, but are inherently prone to error, due to ex falso quodlibet. Indeed, this is a root of the Mathematical proof technique of reducing what one wishes to eliminate, to absurdity. KF PS: I have also recently seen someone describing Mathematics as a "Science" -- a sign of Scientism's attempt to monopolise all serious knowledge. Of course Scientism is itself self-referentially incoherent. However, more importantly, Mathematics is precisely not a discipline in which theoretical constructs are empirically tested and are taken as a sort of weak form, provisional knowledge due to empirical reliability. We need sterner stuff, rooted in logic and coherence, driven in the end by self-evident first principles of right reason. For example, number itself pivots on distinct identity, e.g. A vs ~A leads to 1 and 2 etc. Indeed, this pattern of being rooted in logic is part of why Mathematics plays the role of a plumbline in considerations on scientific endeavours. We need the logic of structure and quantity (including space etc) to be a standard of reference. That we can do Mathematics is a sign. kairosfocus

BTW, Denyse, how do insulting stereotypes of mathematicians help ID? Do you think Dr. Dembski & Prof Marks will appreciate the your insinuations about them ("some brilliant young nerd is fooling around with transcendental numbers, unable to make a living or find a girl")? Actually, will their wives appreciate it? FWIW I'm not a mathematician. But don't tell my department head. Bob O'H

Dembski & Wells write:

The main problem, however, is that none of [the Evolutionary Hypotheses] provides a detailed, testable model for assessing its validity.

Which, by and large, is a fair criticism. (I think work could be done on this, e.g. looking at the genetic basis of mathematical ability, and seeing what other aspects of intelligence it correlates with. Or looking at whether people find mathematicians of the opposite gender attractive). But then the authors write:

On the other hand, from an intelligent design perspective, mathematics is readily viewed as an inherent feature of intelligence and rationality. Moreover, the fact that the mathematical theorems we prove mirror the deep structure of physical reality suggests that intelligence is fundamental to nature and not merely an accidental or historical byproduct of blind material forces.

Which raises the obvious question - what is the detailed, testable model for assessing its validity? My prediction (based on a sufficiently detailed testable model of ID): rather than provide one, Dembski & Wells will take a leisurely boat ride down the Isis. Bob O'H

Well, the math thing is an interesting specific developed talent, but what playing the violin? It ain't got no frets. You either know EXACTLY where your fingers go, or you're wrong. One or the other of the primitive tribes had developed, prior to flood of Western culture, a system of counts by body parts (including elbows and knees) whose last "number" was "many". And for a REALLY big herd of antelope, you could double down with "MANY many". So the Sun is many many yards from Earth. A mile is also many many yards. But then who counts distance in miles? I was told once that in modern California the distance "just down the road" means "you don't have to stop to EAT during the drive". I imagine "just down the savanna" is still a popular yardstick for explaining distances. I have to feel sorry for the math whizz sitting around a campfire and trying to invent names for, oh, "many antelopes distributed among 'elbow' hunters". And there wasn't much chance that he would EVER meet another human who also had an interest in the IDEA of numbers. But I also believe that the numeric coprocessor was always there. It just took 900,000 years to come up with methods for coding the query so the coprocessor could kick in. vmahuna