# Are mathematical results influenced by the way we reason?

Intuitions of mathematicians, and the mathematics they develop, are ostensibly influenced by whether they primarily rely on visual_spatial and/or verbal_symbolic reasoning skills. Is it fair to say that mathematical activity of the majority of mathematicians for (at least) the past 300 years, was dominated by the use of their verbal/symbolic skills, and that this is reflected in the current body of mathematics?

I wish to provide some examples that need your participation, in order to (I hope) understand better my question.

Please observe the first diagram by using both your visual_spatial AND verbal_symbolic reasoning skills: Let X=1 (it actually can be any finite value > 0)

In that case:

The length of the black staircase = 2*(1/1) = 2

The length of the rad staircase = 4*(1/2) = 2

The length of the green staircase = 8*(1/4) = 2

The length of the purple staircase = 16*(1/8) = 2

The length of the blue staircase = 32*(1/16) = 2

The length of the cyan staircase = 64*(1/32) = 2

Etc. (there are infinitely many staircases with constant length 2).

Now, please observe the second diagram by using both your visual_spatial AND verbal_symbolic reasoning skills:

a=1/2 , b=1/4 , c=1/8 , d=1/16 , ... By doing so the following things are observed:

1) No infinitely many staircases with constant value 2 (for each staircase) are equal to √2 (the diagonal line) and this fact is written as 2>√2.

2) 2(a+b+c+d+...) is the result of the intersections of the diagonal black lines on the peaks of the infinitely many staircases, with the 2 sides of the diagram.

3) By (1) an (2) the convergent series 2(a+b+c+d+...)<2 exactly because (by using both your visual_spatial AND verbal_symbolic reasoning skills) it is inseparable of the fact that 2>√2.

4) By this inseparability 2(a+b+c+d+...) does not have an accurate sum (it has an accurate sum by persons that are using only their verbal_symbolic reasoning skills during their mathematical activity on this subject), but it has a non-accurate value < 2 (in case that it is observed by persons that are using both visual_spatial AND verbal_symbolic reasoning skills during their mathematical activity on this subject).

Here is the result of using only verbal_symbolic reasoning skills on the considered subject:

a=1/2 , b=1/4 , c=1/8 , d=1/16 , ...

S = 1/2 + 1/4 + 1/8 + 1/16 + ...

2S = 1 + 1/2 + 1/4 + 1/8 + 1/16 + ...

2S=(1 + 1/2 + 1/4 + 1/8 + 1/16 + ...) - S=(1/2 + 1/4 + 1/8 + 1/16 + ...) = S = 1

This result relies only on verbal_symbolic reasoning skills, and by doing so one is unaware that S<1 if observed by using visual_spatial AND verbal_symbolic reasoning skills (by this reasoning 2S<2 because S<1, so by subtracting S from 2S, one eliminates the fact (as observed by using visual_spatial AND verbal_symbolic reasoning skills) that S<1.

• By the way, the question as it posed is not really well suited for philosophy. Philosophy is not science and therefore such questions won't necessarily be answered from the position of scientificity. Philosophy works with examples, quotations and opinions. As opinion-based questions are not considered appropriate here, I don't see how an example can be given here. Therefore this question can only be appropriate here asking for a reference on the subject. Aug 22 '18 at 14:51
• What does it mean ? You have assumed a square with Side S=1. Obviously: 2S-S=S=1. So what ? Aug 22 '18 at 16:13
• You will not get much traction on this site with these kinds of posts. The problem is not with the questions so much as with the format. "Can mathematical results be influenced by the way we choose to reason?" would be a reasonable question, the influence of symbolic vs visual reasoning on mathematicians' work is also well documented. But the long-winded examples with invitations to observe and reflect on what is written, etc., are completely unsuitable for asking here. I edited your question to show what an acceptable question might look like, you can roll back the edit. Aug 22 '18 at 21:15
• You've proved that limits don't always respect arc length. Can you summarize your point? I couldn't actually understand what you're trying to say. Sep 16 '18 at 17:08

One way to put the dichotomy in a more philosophical or literary framework is to say that algebra is to the geometer what you might call the ‘Faustian offer’. As you know, Faust in Goethe’s story was offered whatever he wanted (in his case the love of a beautiful woman), by the devil, in return for selling his soul. Algebra is the offer made by the devil to the mathematician. The devil says: ‘I will give you this powerful machine, it will answer any question you like. All you need to do is give me your soul: give up geometry and you will have this marvellous machine.’ (Nowadays you can think of it as a computer!) Of course we like to have things both ways; we would probably cheat on the devil, pretend we are selling our soul, and not give it away. Nevertheless, the danger to our soul is there, because when you pass over into algebraic calculation, essentially you stop thinking; you stop thinking geometrically, you stop thinking about the meaning.

I am a bit hard on the algebraists here, but fundamentally the purpose of algebra always was to produce a formula which one could put into a machine, turn a handle and get the answer. You took something that had a meaning; you converted it into a formula, and you got out the answer. In that process you do not need to think any more about what the different stages in the algebra correspond to in the geometry. You lose the insights, and this can be important at different stages. You must not give up the insight altogether! You might want to come back to it later on. That is what I mean by the Faustian offer. I am sure it is provocative.

This choice between geometry and algebra has led to hybrids which confuse the two, and the division between algebra and geometry is not as straightforward and naïve as I just said. For example, algebraists frequently will use diagrams. What is a diagram except a concession to geometrical intuition?

Atiyah, M. (2001). Mathematics in the 20th Century. The American Mathematical Monthly, 108(7), 654-666.

Your title asks whether mathematical results are "influenced". That has connotations that if we reasoned a different way, the results would be different. That is not the case. It is more correct to say that our success at producing mathematical results is affected by the way we reason. Then, your main question asks about the "intuitions" of mathematicians, which are not at all the same thing as "results". I will assume that you are asking more generally about the way we "do" mathematics.

There are in fact many more modes of reasoning than you describe. Firstly, verbal and symbolic are quite different modes. This was why Boole, Russell and their kind developed symbolic systems or algebras for expressing arguments previously expressed only in words. Eric Clapton is famous for sometimes "talking" through his guitar music in private conversations - he had no other way to express his intuitions. And so on.

It turns out in practice that for anything beyond simple mathematics, symbolic manipulation is far the most powerful tool for solving problems and deriving new results.

Geometry saw a slow change from reasoning-by-drawing to reasoning-by-algebra, as the various algebraic methods emerged over centuries. Cartesian coordinates, trigonometry and the calculus slowly shifted its focus, though advances in descriptive and projective constructions continued. Meanwhile Euler's graph theory kick-started topology, which gained its algebraic form and completed the changeover about a hundred years ago.

Nevertheless, almost all mathematical expositions include diagrams and drawings of some kind or other as, although it may be easier to reason using algebra, it is far easier to understand what is going on through visual aids. Such expositions also include text introductions and conclusions, as the broader context is often easiest to express and understand verbally.

In summary, intuitions still arise and simple results can be obtained through all kinds of reasoning processes and simple results, and the mode of reasoning will be appropriate to the problem. However when it comes to following things up in any complexity, algebra reigns supreme.

• If you follow my question and my example, it is actually argues about reason with both our verbal_symbolic AND visual_spatial brain skills, in order to define the "bridge" among the simple and the complex, which, in my opinion, enables finer reasoning. Aug 11 '20 at 15:24
• I find your answer balanced and beautiful. Thank you Guy. Aug 11 '20 at 15:27
• Thank you. Would that all the Internet were such a rose garden. Aug 11 '20 at 18:49