When should I say "nothing is as it seems"?

Question

"Intuition" is the best friend and worse enemy of mathematicians! Sometimes using intuitive arguments could be very helpful to understand the nature of a phenomenon. Many of the deepest true conjectures were just an intuitive argument at the beginning. (e.g. Fermat's Conjecture)

On the other hand in some realms of mathematics it seems nothing is as it seems! The theorems are against our natural intuition and a naive approach could lead one to bad mistakes. (e.g. Naive set theory and some parts of geometry)

Also in some relams of mathematics there is an uncertain situation. For example some consequences of the Axiom of Choice like existence of a basis for each vector space, are completely intuitive and some other consequences of this axiom like existence of a well-ordering on every set, are completely strange.

I am hesitated what I should say to my students about the general role of their naive intuition in their mathematics.

Question 1. Which one of the below options are better to say to a student? What are possible advantages/disadvantages of each option?

• Your naive mathematical intuition is your best leader in the mathematical research. Follow it often but be careful for some rare anti-intuition exceptions.
• In mathematics nothing is as it seems! Don't trust on your naive mathematical intuition at all. Just follow the logical reasoning not intuitive guessing. Almost all important mathematical theorems are against natural intuition. This is the point which makes them special and important.

Question 2. If the real situation is a combination of both of the above options, is there any criterion to determine what situation is true in a particular field of mathematics? In other words, can we say that a particular mathematical field (e.g. Linear Algebra) is intuitive and another one (e.g. Set Theory) is anti-intuition? If yes, how can we determine it?

Answer 1

TL;DR

Intuition is one of the things that distinguishes humans from machines. We should use it, especially to guess what we would like to prove. However, then, we ought to prove our claims.

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Introduction:

I would like to start with dividing mathematical problems into two groups:

• straightforward problems, where mechanical work is enough,
• complex problems, where there has to be something more in the proof.

Intuition comes with a cost of introducing errors, and while dealing with problems from the first category we should avoid it unless it outperforms the cost of manipulating overcomplicated expressions. However, while solving a problem from the second group we have to use intuition, there is no other way (well, we could ask someone or do a brute-force search, but in most cases this is infeasible).

Quite long digression about automated theorem proving:

There is a whole research area called automated theorem proving which tries to make computers prove theorems. It is worth noting that there are results, like Robbins conjecture which eluded humans, but were proved by machines. However, there is a vast number of results that are inaccessible to machines (existence of such problems is implied, for example, by undecidability of the halting problem).

I would like to emphasize one aspect which makes theorem proving hard. It is not exactly right, because right now the use of many various heuristics allows computers to solve a multitude of practical problems. Nevertheless, the core issue did not disappear and shows itself in more complicated cases.

The aforementioned issue is that, if the proof behaves in some non-monotone way, the computers are unable to guess the appropriate middle formula that would connect the premises and the thesis. For example, the proofs are usually of the form

$$\textit{premises} \to \textit{something} \to \textit{thesis}.$$

The proof would be straightforward if one could deduce something going forward from premises or going backward from thesis (perhaps doing both). However, it might happen that the inference looks like

$$\textit{premises} \to \ldots \to p \land (p \to q) \to q \to \ldots \to \textit{thesis}.$$

Observe that there is a some form of non-monotonicity between $p$ and $(p \to q)$, namely, the weaker $p$ is, the easier it is to prove it from the premises, but the harder it is to prove $(p \to q)$; the stronger $p$ is, the easier is for $(p \to q)$ but the more difficult for $p$. Independent of whether we go forwards from premises or backwards from thesis, we might have to check an impossible number of paths before we decide if the theorem is true (produce a proof) or not (produce a counterexample).

On the other hand, if we had to prove two theorems, first $\textit{premises} \to p \land (p \to q)$ and the second $p \land (p \to q) \to \textit{thesis}$, then it would be much easier (provided there's no similar situation inside the subparts).

Why it matters:

There is a professor who said, that it is the modus ponens / rule of detachment where the genius in the proof hides.

The hardness is to know the middle lemma that makes the proof work. In other words, the difficulty lies in what we try to prove, all the rest is straightforward (of course, that does not mean easy). A great example of this phenomenon are theorems which we have to strengthen to make them easier prove (see here for some concrete examples).

However, it is the intuition that tells us what to prove. It is a necessary component of any research that deals with more that just a straightforward tasks. It is the best leader we have – we don't have anything else.

Conclusion:

I think it is great to use intuition and to encourage the students to use it. Every time I can, I encourage them to train their intuition. Nevertheless, there is an important part: we have to actually prove our results. Intuition might be great, but it will not substitute for a proof. There is nothing wrong with having bad intuition if we learn from our mistakes. Once we will start forming a proof, we gain more and more understanding that will correct our intuition.

Some suggested that they alternate between trying to prove the hypothesis and finding a counterexample. From my perspective both actions deepen our understanding and indirectly correct our intuitions.

Intuition should be trained and that can't be done without using it. Perhaps our solution will be inferior or suboptimal, but thanks to that, next time our approach we will be better. On the other hand, if we were to keep using the same techniques again and again, we won't advance as much.

To answer your second question, I don't believe there is any general way to determine how much intuition do you need in any particular domain. I would say experience is the best you can do, but even then the new generations might have a whole different view, i.e. new tools and techniques might allow for new points of view that might turn everything upside-down. In fact, if you understand some result deep enough, it is intuitive (despite how non-intuitive it might seem to others).

I hope this helps $\ddot\smile$

Answer 2

I think it is important to develop mathematical intuition in one's work, but not necessarily of the naive sort.

Of course, one's intuition may turn out, ultimately, to be incorrect. An important outcome in this case is not to assume that you have bad intuition, but instead to think about how you can make the end result sync up better with what could be reasonably guessed.

Borrowing terms from (Piagetian) developmental psychology, the (unpleasant) experience of acquiring new information that does not accord with one's existing cognitive schemas is known as disequilibrium; the process of updating one's schematic understanding so as to fit in this new information more naturally is known as equilibration, and this is when learning occurs.

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Allow me to give a concrete example: Ask secondary school students to use their intuition to guess the percentage of entries in a $10 \times 10$ multiplication table that are odd. In all likelihood, they will guess $50\%$; this is probably based on some intuition around numbers alternating between odd and even.

However, the actual answer is $25\%$. Why? A proof is that a product of two numbers is odd iff both numbers are odd; in this case, we have five choices for odd numbers ($1, 3, 5, 7, 9$) and so a total of $25$ out of the $100$ entries will be odd. With regard to intuition, one might note that taking products can introduce new factors, and a number is only odd if you never get a factor of $2$; this is tough. Alternatively, one might note that among the four cases eveneven, evenodd, oddeven, and oddodd, only $1$ of the $4$ gives an odd number; so seeing $1/4 = 25\%$ feels right intuitively.

At a higher level, suppose you are a teacher thinking about other nonstandard problems to pose with the multiplication table. How many distinct entries are there in an $n \times n$ table? This question looks pretty reasonable, and one can quickly guess how to start it: check small values of $n$. But now step back and use your (Number Theoretical) intuition: the multiplication table is a way of encoding composite numbers and prime numbers (e.g., a number $p$ is prime iff it appears exactly twice in an "extended" multiplication table). By updating one's intuition to conceive of the table as being related to primality, it comes as less of a surprise that this question is far too difficult for a secondary school student. (In fact, what we have here is the Erdos multiplication table problem; it is quite hard: MathOverflow link.)

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As for distinguishing between different areas of mathematics by using intuition as a lens, I do not think this is a fruitful approach. It may be that some people are primed to have certain ways of thinking, or modalities of thought, cf. Facet 4 in the following reference:

Gruber, H. E., & Wallace, D. B. (1999). The case study method and evolving systems approach for understanding unique creative people at work. Handbook of creativity, 93, 115.

But sometimes one's way of thinking differing from the general trend in an area of mathematics can lead to a breakthrough. For a concrete example, consider that Paul Cohen managed to prove the independence of the Continuum Hypothesis (from ZFC) with a line of thought (related to "decision procedures") that directly contradicted Godel's work. (MathOverflow link.) In the end, it was not Cohen's set theoretical intuition that allowed him to tackle Hilbert's first problem, but rather intuition developed elsewhere. That said, other Mathematical Logicians and Set Theorists, once they equilibrated in response to Cohen's work, were quickly able to push forward the notions introduce by Cohen. For more on this history, see:

Kanamori, A. (2008). Cohen and set theory. The Bulletin of Symbolic Logic, 351-378.

In this case, we have advances in set theory due both to those with mathematical intuition from elsewhere, as well as due to those with mathematical intuition specifically from set theory and mathematical logic. Though I only offer this one example here, I am quite sure that others abound throughout the many areas of mathematics.

Answer 3

I would be inclined to choose my words more delicately. Namely, it's not that intuition-per-say is misleading, but that untrained or inexperienced intuition is... untrained and inexperienced. Of _course_ many reasonable, normal things (in this complicated world) will be surprises to a naif. That doesn't mean they're truly counter-intuitive.

I would advocate always taking as preliminary position that one must trust one's intuition, all the while recognizing the possibility that one is mistaken/naive. To tell people to distrust their in tuition I think is a very bad thing, especially insofar as it tells them that they should, in effect, accept anything from a sufficient authority, without much questioning. When this becomes suspension of critical faculties, it is obviously methodologically disastrous.

I would also disagree with the examples of naive set theory and so on. Mostly naive set theory is perfectly fine in practice, in applications to mathematics. The fact that it can be over-revved, pushed beyond safe limits, should not be construed as counter-intuitive. Yes, it is interesting to see how to axiomatize (=set rules) to avoid seeming paradox... but I note that it is very difficult to arrange rules sufficiently encompassing so as to prevent all stupid choices.

So, the real point is that intuition is not static. It is trained, not innate.

And I think that it's not exactly "definitions" (or axioms, nearly the same thing) that train intuition at a most fundamental level, but examples, especially the examples that show both the good side, and the bad side, of issues eventually iconified into definitions or axioms.

Given the historical fact that almost universally examples have preceded "definitions", and that definitions tend to have more explanatory power than do definitions (which always leave me immediately asking myself what situation the dang definition was trying to avoid or get into... why not just tell me directly, instead of being coy?), I strongly feel that definitions are in effect a summary of important examples, and make scarce sense and should not be given prior to examples... and, by that point (as I am fond of saying) the formal definitions, the main theorems (which were what we wanted all along, if only we admitted it up front) and their proofs (illustrated more simply in examples) should be almost entirely clear.

Answer 4

For the first question, without a doubt neither. You should explain to them that mathematical intuition is developed through working with definitions. When they first encounter something they shouldn't guess, and they should always work with the definitions in front of them.

When they sit to revise the material, or solve their homework assignments, open the notes from class and look at the definitions. When sitting in the exam, write the relevant definitions (for what you are given, and what you need to show) in the draft pages.

Of course, my experience is that the more abstract the field is, the more it is important to work with the definitions. What is true for a first course in logic and set theory, might not be true for the first course in linear algebra. But in both cases it is important to work with the definitions, and by doing so develop intuition.