
The Horgan Surface and the Death of Proof - headalgorithm
https://blogs.scientificamerican.com/cross-check/the-horgan-surface-and-the-death-of-proof/
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usgroup
This is also well studied in Philosophy.

Even not that hard math proofs call on intuition pretty quickly, and I think
it’s a legit question to ask when does it basically become empirical proof.

To my understanding maths has a common basis in set theory but set theory
itself is just a common basis for maths (ie as opposed to “the truth”), but
even so proofs are typically not given via set theory especially if the
theorem is complicated.

The idea is that mechanising proof and making it more algorithmic will make it
more like what the layman thought proof is , but it at the same time raises
the question, what is this all for? Ie maths is what mathematicians do. I
think it admits to no clearer a definition. So it’d probably make sense for
mathematicians to decide, what precisely is the role of proof?

The author talks about this in addressing computer aided maths and the general
recruitment of applied mathematicians to solve problems whilst caring less to
advance “the science” whilst doing it. Chomsky spoke a lot about this in
criticising the use of probabilistic models in science as ultimately scarcely
contributing to causal understanding, which in itself is separate from being
able to predict phenomena well. Ie you can do practical things without ever
knowing what’s going on under the hood, but science wants to know what’s going
on under the hood and so prefers different approaches to its advancement.

~~~
tome
> Even not that hard math proofs call on intuition pretty quickly

Can you give an example? Nothing jumps to mind from my years of studying
mathematics.

~~~
mgsouth
Q: If I have a rectangle "a" wide and "b" high, how do I find the width "c" of
another rectangle which has the same area but is "d" high?

A: I'll be pedantic and take it in baby steps:

    
    
      Given: area = height * width
      Given: area(c, d) = area(a, b)
    
      c * d = a * b               Because equivalencies of
                                  equivalents are equivalent
    
      (c * d) * x = (a * b) * x   Equivalencies multiplied
                                  by the same thing are
                                  equivalent
    

Q: Wait, where did "x" come from?

A: Just bear with me...

    
    
      x = 1 / d                   Because I say so
      (c * d) * (1 / d) = (a * b) * (1 / d)   Substitution again
      ...                         Blah blah blah
      c = (a * b) / d             Tadah!
    

Q: What if "d" is zero?

A: (Chuckle) Division by zero isn't defined.

Q: Why not?

A: Well, we want division to be defined as the inverse of multiplication.

Q: Inverse?

A: If x * y = z, and we know x and z, what is y? x * y = z => z / x = y.

Q: So why can't we divide by zero?

A: Because if x is zero then y could be _anything_. It's not so much that
division by zero is evil as it is that multiplying by zero does not have a
unique inverse, and disallowing multiplication by zero would be just too
weird.

Q: Why don't we just say that y has a whole slew of possible values?

A: Because we don't. Look, this is simple arithmetic, not some weird group
thing. You wanted _one_ equivalent rectangle and I gave it to you. If d is
zero there's no c that could be big enough to equal a * b. (Before you ask,
"infinity" is not a number.) Well, I suppose you _could_ pick a number if...

Q: ... a or b is zero. I have no idea if a or b or d will be zero. Can I still
use "c = a * b / d?"

A: Don't you have to wash the car or something?

[edit: consistent use of "c" and "d"]

~~~
pavel_lishin
> _Q: What if "d" is zero?_

You can avoid chuckling by pointing out that this is a nonsense question. A
rectangle which has a height of zero isn't a rectangle at all - it's a line.
And lines don't have areas.

(You can continue chuckling after someone asks why lines have no area, I
guess.)

~~~
mgsouth
I'm not aware of a formal definition of rectangle which excludes zero height.
Sounds like an intuitive leap :)

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jobigoud
So this is like a Parker Square for surfaces.

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resters
Does anyone know of any mathematical objects that have been created
_intentionally_ to defy intuition?

~~~
Twisol
Oh sure. Topology has a wonderful cavalcade of such objects. Wikipedia
mentions a book that collects many
([https://en.wikipedia.org/wiki/Counterexamples_in_Topology](https://en.wikipedia.org/wiki/Counterexamples_in_Topology)).

A particularly nice example is the Alexander horned sphere.
[https://en.wikipedia.org/wiki/Alexander_horned_sphere](https://en.wikipedia.org/wiki/Alexander_horned_sphere)

~~~
JJMcJ
There is also a book "Counterexamples in Analysis", and other counterexamples
books.

