# 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. – rus9384 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 ? – Mauro ALLEGRANZA 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. – Conifold Aug 22 '18 at 21:15