# Math understanding: intuitions and proofs

I think that cogsci can help me to study a problem in philosophy of math. Consider this case (adapted from M. Detlefsen, Brouwerian intuitionism):

Lucy has the kind of understanding of a given math subject S that we typically associate with the master mathematician. John has a superb ability to manipulate the axioms for S according to logical means.

Problem: is there a significant difference between Lucy's knowledge of S and John's knowledge of S? Is it true that the second kind of knowledge does not imply the first? Is it true that Lucy understands S to a certain significant degree), John does not understand S to a certain significant degree.

Are there works in cogsci that can help me to study this problem?

I have found a relatively old article (Arbib, A piagetian perspective on mathematical constructions), but nothing else.

Thanks very much.

• If this is taken as a logical argument then it is completely broken. It is clearly missing some premise along the lines of "Master mathematicians have an ability to manipulate axioms". In its current state you neither imply anything on Lucy and John, nor you can infer anything given the details you have provided. – Izhaki Dec 7 '15 at 0:42
• @Izhaki A master mathematician does not manipulate axioms like you do in a formal system (e.g. Hilbert-style calculus) to deduce theorems. Even when they communicate their result to others they do not provide a deduction in a formal system. Usually master mathematicians do not even know how to use a formal system. Most of them do no even know what an Hilbert-style calculus is. However, many works on the foundation of math rely on formal systems. So one can imagine the two cognitive agents above and wonder whether and how their epistemic command of S is different. – MatteoBianchetti Dec 7 '15 at 1:57
• @Izhaki One more note: I am not presenting any argument. I am just giving a scenario and I ask whether there is work in cogsci that deals with the problem of characterizing the different understanding of S by Lucy and by John. Thanks. – MatteoBianchetti Dec 7 '15 at 2:08
• "axiom manipulation" ability has its bearing in cognition. From your comments I also understand that such tool isn't used by master mathematicians. Forgive my ignorance, but in any case I think it'll help others if you elaborate on the way master mathematicians work or think. – Izhaki Dec 7 '15 at 2:31
• @Izhaki. I have been intentionally vague about the way master mathematicians work, because it is where I need help from cogsci. They often say that rely on visualizations, intuition, conjectures, constructions, more or less standard proof-techniques, etc. And they often say that simple symbol manipulation (not even merely proving theorems) does not provide understanding. What is that they exactly do and what is that thing that they call understanding? Here is where I am asking cogsci for help. – MatteoBianchetti Dec 8 '15 at 9:07

## 2 Answers

I think it is important here to be clear about what John is really capable of doing. If all he is able to do is manipulate the axioms of S correctly, this actually does not get him very far, for at least two reasons. (This is also why theorem proving computer programs have significant limitations.)

1) Suppose we give John a specific theorem P of S and ask him to prove P. Can he do it? Not necessarily. There is a set of logical steps based on the axioms that proves P, so it seems within John's reach. But one problem here is that of computational complexity. If John proceeds by just, in an undirected fashion, trying to derive new facts of S from the axioms in the hope of eventually arriving at P, the number of paths that he can go down will increase exponentially and it may take longer than the lifetime of the universe for him to arrive at the result. Master mathematicians in the field of S, on the other hand, have sharp intuitions about what is likely to be an effective path to a proof of P.

2) Mathematicians do more than resolving previously specified open problems. They also try to generate important new problems. A mathematician who is not very good at formal proofs may nevertheless make a significant contribution by correctly conjecturing a very interesting result (even if someone else needs to prove it).

1) and 2) overlap somewhat; for example, a master mathematician may conjecture a lemma L in S that is extremely useful for proving P.

If John is not able to do 1) and 2), then it's not clear he is really very skilled at S. If he is, then it seems that maybe he isn't so different from Lucy after all; somewhere he has the right intuitions. Maybe Lucy is more consciously aware of her intuitions, perhaps leading into a discussion about what it is like to experience those intuitions. But many of the intuitions of master mathematicians don't come from conscious processing; certain ways of proceeding just feel right to them but they can't necessarily explain why.

Some master mathematicians have written on the process of mathematical invention and on some methods of plausible reasoning in mathematics. Here are some references:

Jacques Hadamard. An Essay on the Psychology of Invention in the Mathematical Field.

George Polya. Mathematics and Plausible Reasoning Volume I: Induction and Analogy in Mathematics. Princeton University Press, 1954.

George Polya. Mathematics and Plausible Reasoning Volume II: Patterns of Plausible Inference. Princeton University Press, 1954.

Henri Poincaré. “Intuition and Logic in Mathematics”. In: La valeur de la science. Available here: http://www-history.mcs.st-andrews.ac.uk/Extras/Poincare_Intuition.html.

You might also find some interesting papers by David Tall, whose field of research is mathematical thinking: http://homepages.warwick.ac.uk/staff/David.Tall/

As suggested by present, in these paper you can find little on symbolic manipulation and a lot on how to approach new and complex mathematical problems with analogy, intuition, and other similar cognitive tools.