Are visual proofs mathematical proofs?

>>8043036
Yep

>>8043036
it depends

>>8043036
no. never.
visual drawings give a good idea of things, but unless you actually write it down, it's really easy to trick yourself

>>8043037
>>8043481
>>8043485
I feel there is a conflict in opinions.

>>8043036
They're an acceptable proof in synthetic geometry, because that's about proving inherently visual things and there's no way to get around that, other than maybe using the rigorous autism developed for geometry in the late 19th century. But that's not accessible to most people doing geometry.

For any other field, no, a diagram is not a proof. At best it's a heuristic to help make you believe something.

>>8043508
>mathematics of the 19th century
>not accessible for most people doing geometry

????????????????/
write down your geometry or it's nothing but hot air

That particular visual "proof" you posted seems fishy. How does ""rearranging""" translates into mathematical language?

>>8043710
lenght are conserved in case of translation, so i guess it's fine

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>>8043036
Absolutely.

Symbolic manipulation of formulas are visual manipulations of variables under certain invariance laws, such as commutativity.

Visual proofs of the pythagorean theorem are visual manipulations of triangles and squares under invariance laws, such as invariance of area under translation.

HOWEVER, OP's image is not a proof, as it is not clear that the left square must be equal to the right square.

The attached gif, however, would be a legitimate visual proof of the pythagorean theorem even without the accompanying words.

>>8043739
Nevermind, I'm a moron.

OP's image is identical to the gif.

OP's image is a valid visual proof.

>>8043036
Of course. That is why it is obvious why things like the integers exist. We can write them as strokes on a board. It is also why the real numbers do NOT exist!

File: HCfGOYp.gif (178KB, 624x519px) Image search: [Google]

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[math]1+2+\ldots+(n-1) = \binom{n}{2} [/math]

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[math]F_1^2 + F_2^2 + \ldots + F_n^2 = F_n F_{n+1} [/math]

>>8043739
The real question is are proofs that can't be made visual made up bullshit?

>>8043036
Not equivalently so not at all.

File: sf56R.png (119KB, 3390x1563px) Image search: [Google]

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[math] \arctan{\frac{1}{3}} + \arctan{\frac{1}{2}} = \arctan{1} [/math].

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>>8043037
>>8043481
>>8043485

>>8043508
>For any other field, no, a diagram is not a proof. At best it's a heuristic to help make you believe something.
nice personal opinion

>>8043789
you can break down many higher dimensional things and abstract situations into graphical representations, it would just take you a really really really long time and wouldn't help you wrap your mind around it at all.

people need to stop cringing at abstraction in modern math. its fucking useful and allows us to solve bigger more complex problems than before.

>>8043036
Not usually, no.

>>8043036
>symbollic proofs aren't visual proofs

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>>8043739

>>8043036
All mathematical proofs are social constructs, so if you can visually convince other mathematicians that its correct, its a proof

>>8045421
I am familiar with what happens the missing area in the triangle illusion, but cannot figure this one out.

Can someone explain what is going on?

>>8043036
Yes but you can't just draw you need to use postulates and theorems to proof every thing you draw

>>8045726
If the details of the proof can be sufficiently inferred, given mathematical maturity, from the visual proof, it is a proof.

Proofs don't list every single detail, and this is especially true of proofs in advanced math. There are many, many details that the reader is expected to fill in, due to the reader's expected mathematical maturity.

Visual proofs are no different. E.g. in >>8043821 the reader is expected to fill in the detail, prove to himself, that the inscribed figure really is a square. If the proof were written, it would state this without proof, and the reader would be identically obliged to fill in the proof of this himself.

>>8045732
Yes. what I was saying is that drawings do count as proof as long as they're backed by axioms, to answer OP's question

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As >>8045721, nevermind. I'm sure the same sleight of hand as the missing-square-in-the-triangle puzzle is being employed. The original cut is not a perfect straight line, and when the pieces are re-arranged, there is very slight underlap and overlap that is filled in imperceptibly.

>>8044503
kek

>>8043508
>For any other field, no, a diagram is not a proof.
You clearly haven't read Hatcher's "Algebraic Topology". Many of the proofs are literally just pictures, beginning with chapter 0.

The whole book is online. Look at page 2. The proof is literally just a picture.

https://www.math.cornell.edu/~hatcher/AT/AT.pdf

Many of his other proofs are literally just pictures as well.

>>8045758
Interesting, thanks for the read

A proof is only ever a sequence of statements. Pictures can be used to aid understanding of the proof but should not be regarded as part of the proof. Proofs are formulated in logic. The statements about the picture should make logical sense without the picture.

Definitely, there is even a book called "Proofs without words". It's a nice little book. I think I have it here somewhere.

>>8045758
I wouldn't consider the thing on page 2 a proof so much as giving the idea of a proof. Those kind of pictographic arguments can cause problems in more delicate situations.

Semantics.

A computer scientist in proof theory, say one coding up programs for automated theorem proving, would not consider any proof in a math journal on number theory or differential geometry to be a formal proof. It's skipping easy calculations and assuming lots of theorems implicitly.
Mathematical proof is a fluid concept, dependent on who's doing the math.

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I have literally never seen a formal proof of the countability of the rationals.

The only proof ever presented has been the attached picture.

This is because giving an explicit formula for this bijection and proving that it is a bijection would hugely obfuscate its visually presented obviousness.

But my question is why is it not a proof?

I could, for example, make you a great proof of the "Squeeze Theorem" using drawings of circles. If you accept the objects I've drawn to really be circles, then it's true. It's also easy enough to translate it to statements on paper, but that's not the point. Why is my proof not a proof if it's just the drawings with explanations that clearly show something?

I will admit though that it seems very hard to prove something that uses the word "all" with pictures and without using contradiction.

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The area under a cycloid is three times the area of the generating circle.

Aren't drawings necessarily just special cases?

>>8046269
Obviousness is for intuition. You can right what is on that picture in no time.

>>8043036
Starting from the drawing.

Area of Square = (a+b)2 = a2 + 2ab + b2
Area of Square = c2 + 4 * ((a * b)/2)

Substitute Second Equation into first one

a2 + 2ab + b2 = c2 + 4 * ((a * b)/2)
a2 + 2ab + b2 = c2 + 2ab
a2 + b2 = c2

idk 2bh but i'm sure that they're friggin gay as frig

>>8045758
I didn't find any proofs done with pictures. The fuck are you talking about.

>>8045266
>I choose to devote my life to practicality

okay pleb

>>8046269
your post is absolute idiotic shit
if you've never written it down, you've never done the fucking proof
the bijection is easy as fuck to write. if you can't you don't understand the proof and you're just fooling yourself

>>8043508
>synthetic geometry
>visual
Are you retarded? Do you not know what synthetic geometry entails?

>>8046289
Mamikon's visual proof is a thousand times better than this, anon.

When we fucking prove that through any two points of space only passes one single line we're actually fucking visualizing the symbols we use when writing the proof (even though we're not using a drawing of two dots and a line, but logically proceeding through some old boring conveniently shaped [s]lamps[/s] axioms).

So what's the damn counterargument?

(Yes, I'm mad.)

In my graph theory course I've had it written on my submissions explicitly that 'pictures aren't proofs' (though they would later retract that statement seeing that I did accompany a written proof to the diagram).
So, if nothing else, it's up to whoever is marking your papers. :^)

>>8043036
Any visual proof can be written out, but not all written proofs can be visualized. I think mostly the visual is just a good way to make it intuitive, but it lacks rigor without the accompanying text.

>>8045721
watch the upper right piece that breaks off and goes to the left

>>8046304
>>8046698
No time? Easy as fuck?

Go ahead. Define that particular depicted bijection explicitly. You have a generous 30 seconds.

>>8046269
>What is Cantor pairing function
Jesus Christ, anon.

>>8050488
I know of the Cantor pairing function.

Its inverse can be used to inductively give a bijection by passing over pairs that are not coprime.

But this does not give the depicted bijection. It gives a different bijection.

Regardless, a person must respond to >>8050480 by constructing something like the Cantor pairing function from scratch in under 30 seconds (no looking it up), otherwise it's not "no time" or "easy as fuck".

You must also prove that your pairing function is the claimed bijection.

>>8046269
>I have literally never seen a formal proof of the countability of the rationals.
(x,y) -> 2^x * 3^y
x -> (x,1)
By Schröder–Bernstein theorem, there exists a bijection.
QED

>>8050503
Correct, that is a formal proof.

I guess I had in mind, "using the visually depicted bijection". But whatever.

>>8043036
I took geometry one in college so allow me to show you my superior intellect.

>right

Let square QWER have sides of length a + b

Assume QW is a side of square QWER, if not pick two vertices so that it is a side and call those points Q W

Define a coordinate system over line QW and from that define the point T so that distance QT = a and TW = b

Now find the other vertex that froms a side with W and let this one be E. Define a coordinate system over line WE and a point Y so that distance WY = a and distance YE = b

Find the other vertex that froms a side with E and call this one R, repeat the process and call this new point U.

Find the other vertex that makes a side with R and from our constructed square this will always be Q and find the new point I by the process already described.

Now construct square TYUI and label the length of its sides as c.

From this construction it is implicit that Q, T and I are not colinear and therefore triangle QTI is defined with sides QT which has already been said to have length a, side IQ which is constructed to have length b and side IT which is also a side of square TYUI which means it must have length c.

3 other different triangles are implicitly defined in this construction just aswell.

So it is not a visual proof, just a proof with a kinda long setup.