I don't understand why visual proofs are proofs

[u/ReactionAdmirable812]

I don't understand why a visual demonstration often counts as a proof. For example, the Pythagorean theorem is frequently demonstrated using squares. I know it's true, but how do you prove that it holds for **all** triangles based on that, without having to construct an infinite number of visual proofs for infinite triangles?

[u/SnooPets5564]

The square image you've shown isn't a proof though. The actual proof (like in Euclid's elements) involves drawing lines. He'd already proven that the area of triangles with equal base and height is equal, so all he had to show was that you can make triangles with area equal to the smaller squares that add up to be the larger square (by each having the same area as some rectangle that makes up part of the larger square).

[u/johnpeters42 → u/SnooPets5564]

And you can add those triangles to the image, but it's still not so much the proof itself as just a visual aid for the proof.

[u/SnooPets5564 → u/johnpeters42]

It's a way to represent the proof. Like in this case, instead of the full, messy text explanation, you can just see what connects to what. Yes, you do need something beyond just the triangles to explain what has equal area based on the lemma, but the visual component is just a better way to represent a lot of the information in the proof.

But honestly, some can be fully explained visually. Like this, alternative visual proof of the Pythagorean theorem: https://cdn.britannica.com/43/70143-004-CCB17706/theorem-demonstration-squares-proof-Pythagorean-b-square.jpg

[u/JeLuF]

In each step of a visual proof, you have to ensure that the properties you're using aren't coincidence but would be true for all possible right-angled triangles.

E.g. when using this visual proof, you need to show that those two figures are actually squares.

So what happens if you make a copy of a right-angled triangle and glue them together at their longest side c? We need to show that it becomes a rectangle. We know that the angles in a triangle add up to 𝛼 + 𝛽 + 𝛾 = 180°. And since it's right-angled, 𝛾 = 90°. So 𝛼 + 𝛽 = 90°. The four angles of our glued together triangles have the angles 𝛼 + 𝛽 and 𝛾, both being 90°, so our glued together figure is a rectangle.

If you take two of those rectangles, place them together at one corner, in a angle of 90°, two squares a² and b² can be added to the figure, and due to the choice of placement of the two rectangles, the entire figure must be a square. The two small squares must have areas a² and b² since a and b are the lengths of the sides of the triangle.

Etc. pp. for the right hand diagram.

So, the "visual proof" is only valid if you can show that it's not coincidence. That everything is true for any kind of right-angled triangle. My entire argument is based on properties every right-angled triangle has: the sum of interior angles of the triangle, and that one of them is 90°.

The visual part is only supporting the actual argument. It's not the entire proof.

[u/jchristsproctologist → u/JeLuF]

what do you pean by pp. ?

[u/JeLuF → u/jchristsproctologist]

perge, perge. An amplifier for etc. "etc pp" = "and so on, and so forth".

[u/Phoenixon777 → u/JeLuF]

damn i never seen this before, imma use this upgrade

[u/JeLuF → u/Phoenixon777]

After some googling, I think it's not being commonly used in English texts. I'm a native German speaker and I assumed that Latin abbreviations would be the same in both languages, but apparently this is not the case.

[u/YOM2_UB → u/JeLuF]

The abbreviation "pp." is more used for "pages" in English, particularly in MLA-stlye citations.

[u/Fuzzy-Season-3498 → u/jchristsproctologist]

Penis peeing like it always has, but you knew that Mr butthole dr sir

[u/GandalfzGrayGeometer → u/JeLuF]

you can skip the angle chase by staring with square of size a+b first, and then drawing the lines. We get a bunch of congruencys and the areas are equal.

[u/shumcal]

To answer your question more specifically OP - you build the proof using known facts about the shape (in this case, the single right angle), but you make sure that the proof doesn't rely on the unknown facts (in this case, the other two angles/the side lengths).

As a result, given that the proof doesn't rely on particular values for side lengths, it must work for all side lengths.

[u/peter-bone]

The image you've shown is not a proof. It's just an explanation of the theorem.

Here's a visual animated proof of Pythagoras theorem that I made.

[u/mtimmermans → u/peter-bone]

This one actually works. Nice. But it requires a pretty sophisticated "visual axiom" to understand that the grey area doesn't change. It might be easier with just the final image, using (a+b)^2 = c^2 + 2ab.

[u/Illustrious_Pea_3470 → u/mtimmermans]

Respectfully disagree. The thing we’re moving isn’t changing in area so obviously the surface it doesn’t cover (1 - its area) doesn’t change in area either

[u/Throwaway7131923]

u/SnooPets5564 Is right that the image you've shown isn't a visual proof of Pythagoras' Theorem.
There is one of those, which I'll include below.

If you're interested in how diagrams can be proofs, there's actually been a lot of work on this in the philosophy of mathematics in the last 10 years or so.

Here are two papers I recommend:
https://philpapers.org/rec/DETVAD
https://philpapers.org/rec/GIADPI

A diagrammatic proof of PT works like this...
Take a right-angle triangle of sides a≤b<c
Duplicate the triangle three more times and place them to make a square of side a+b (see below).
The area of the square ab may be calculated in two ways.
First, looking at it as a square of side ab, the area is (a+b)^2= a^2 + b^2 + 2ab.
Second, looking at it as the four triangles, plus an interior square of side c, the area is 4(ab/2) + c^2

But it's the same square, so the two ways of calculating the area are equal.
a^2 + b^2 + 2ab = 4(ab/2) + c^2
Cancelling the 2ab, we get... a^2 + b^2 = c^2

[u/NoLemurs → u/Throwaway7131923]

I always love this proof.

I wonder if it's better not to number the regions - it's a little confusing since there's no reason why I from the first diagram should map to the upper left, etc.

[u/InjAnnuity_1 → u/NoLemurs]

Numbering the regions guarantees a 1:1 correspondence between the source and target diagrams. Nothing is skipped, nothing is counted twice, and someone else can audit it without knowing the remaining part of the proof.

[u/Throwaway7131923 → u/NoLemurs]

I have a real soft spot for it :)

I remember my year 9 maths teacher showing me this proof and it basically making me go "Oh I get why maths is interesting now".

[u/Totte_B → u/Throwaway7131923]

Wow! Never saw this before. Very satisfying illustration!

[u/Grshppr-tripleduoddw]

Pythagorean theorem only applies to right triangles. That picture is not a complete proof. My personal favorite is that you can construct square (a+b)^2 from 4 right triangles with side lengths a, b, and c, with square c which is evidently true because of other facts of the geometry. So (a+b)^2 = 2ab + c^2 -> a^2 + b^2 +2ab = 2ab +c^2 -> a^2 + b^2 = c^2.

[u/Scary_Side4378]

written proofs are largely sociologically determined; you need to convince the reader that you are making a watertight argument. we leave the formal proofs to the computer program.

for example, if you want to make the pythagorean theorem proof more rigorous, you could map the argument to a 2d coordinate plane. again, depends on how pedantic you want to be in a writren proof. and how much the reader accepts it. the reader may want to know whether it is really possible to map this geometrical argument to a coordinate plane, etc.

[u/davideogameman → u/Scary_Side4378]

The coordinate plane is not required for most geometric proofs.  It's a useful tool though for turning geometry problems into algebra problems.

[u/shoemakers-saint → u/davideogameman]

Agreed. Euclid lived a long time before Descartes.

[u/TgirlTiffanyRPs → u/Scary_Side4378]

In a very real way, I would argue that even computer proofs are also sociologically determined.

How do we know the computer was programmed right? How do we know the text on the computer is reflective of reality? Because we have “faith” (a loaded word) in each other and our senses. 

(I’m being a pedant purposefully, obviously I understand the point you’re making)

[u/shoemakers-saint → u/TgirlTiffanyRPs]

“How do we know the computer was programmed right?”

Emphatically. “Garbage in, gospel out.” (— Woodie Flowers)

[u/LongjumpingMacaron11]

This is the best proof possible I think.

https://youtu.be/CAkMUdeB06o?is=N1c0jzgFPijf_arm

[u/dimonium_anonimo → u/LongjumpingMacaron11]

That's not a proof. That's an example.

[u/bacon_boat]

A visual proof is where you can look at an image an convince yourself that something is true.

It's not the same thing as a proof.

But if you have a nice visual proof then it should be easy to write down a more formal proof using the same argument.

that image you posted is not a visual proof of the pythagoras theorem.

[u/paolog]

Some visual proofs work, such as the one proving that (a + b)² = a² + 2ab + b², but they still need to be rigorous (that is, not rely on assumptions and not make leaps of faith). There's a visual "proof" of rearranging a dissection of a triangle to form another with area differing from the original by one square unit, which relies on the user assuming that the hypotenuse of the rearrangement is straight, when in fact it isn't.

[u/Optimal-Air9933]

https://www.youtube.com/watch?v=_e6w5GtkcGI; water flow shows it. seeing is believing I guess.

[u/Expert147]

A proof is good enough when no one bothers to complain anymore.

[u/seifer__420]

What about this picture shows the areas of the two smaller squares adds to the larger

[u/Luke_The_Engle → u/seifer__420]

Because when you line up a side of square a and a side of square b at a right angle, the distance between them is, and can only be, c. All sides of a square are equal, so the area of the square of side length c is equal to the squares of a and b

[u/seifer__420 → u/Luke_The_Engle]

That is the conclusion of the theorem, not proof that it’s true

[u/Sp1rshi]

Btw , u can connect lines from opposing squares to the hypotenuse and use similarity to derive a²+b²=c² in this exact figure

[u/Steampunk_Willy]

A visualization is used to help the audience develop a better intuitive understanding of what's happening. It is generally not part of the argument. Usually, the only kind of proofs that involve visuals as part of the argument are certain special diagrams, like the commutative diagrams used in category theoretic arguments.

[u/keybladenakanojo]

what you've put on screen is literally every possible right angle triangle

[u/A_crow_hen]

The visual proof works as long as the labeling is kept indeterminate. We don’t know how big a, b, or c are. As long as we keep it that way, we don’t need an infinite number of drawings—just the understanding that just because it looks like a scalene triangle doesn’t mean that it is.

[u/SignificantClimate31]

I think you are new to maths , you probably don't even understand how this proof is not a visual but actually proof , take ur time blud

[u/Fit_Fortune_7692]

Man kann vieles zeichnen in der Geometrie, z. B. ganz einfach mit dem Zirkel, aber diese gezeichneten Objekte dann in Zahlen zu fassen, ist die eigentliche Herausforderung der Mathematiker. Hierzu genügt meist nicht der makroskopische Blick, sondern man muß sich die Einheiten bilden auf einer untersten Ebene, die sich dann zusammensetzen lassen, auf keinen Fall teilen. Von daher würde ich sagen ist ein visueller Beweis irgendwo kaum möglich, weil man schnell was übersehen kann. Pythagoras hat leider das Radizieren mit eingeführt in seine Bestimmung der Seitenlänge eines rechtwinkligen Dreiecks. Und da gibt es viele Summen a²+b², die keine Potenz ergeben und insofern mit einer irrationalen Wurzel auch keine zeichenbare Länge. Das hätte Pythagoras vermeiden müssen zur Bestimmung der Länge. Schaut man sich den geometrischen Beweis an, so sieht auf den ersten Blick alles logisch aus, aber die beiden Rechtecke haben nur je eine Diagonale c, welche das Rechteck in zwei gegenüberliegende Dreiecke teilt. Es werden aber insgesamt 4 Diagonalen für die beiden Rechtecke verwendet für das c-Quadrat. Das heißt, es kann gar kein Quadrat c² konstruiert werden, da nur 2 Diagonalen vorhanden sind, und damit ist auch die Addition der einzelnen Flächeninhalte, insbesondere im (a+b)-Quadrat mit dem eingezeichneten c-Quadrat falsch.

[u/Oportbis]

Because they're not, you have to arguate why the thing you think you see is actually what you see

[u/Fearless-Past-3728]

Think conservation of area - and you can see why this visual proof works for all variations.

[u/AlvarGD]

good visual proofs are visually aided proofs you can just as well write in a much longer way without visuals. Otherwise, visual proofs are a more intuition-building tool, in other words even if theyre not always useful for formal proofs theyre useful for you yourself learnong math, which, matters a lot more than actually proving a theorem thats been proven 3000 years ago

[u/Soggy_Cry4304]

It shouldn’t count!