Identity and difference in mathematics

[The Grey Man]

@Siouxsie has mentioned the Hegelian idea of 'negation of negation' in a thread about theodicy.

Is there a problem of evil?

https://www.themathesontrust.org/library/is-there-a-problem-of-evil Last year, I mentioned having asked some colleagues of mine a question similar to the Hardy Question: https://www.intpforum.com/threads/answer-the-hardy-question.28243/ What I did not mention at that time was that one or two...

intpforum.com

Hegel inherited from Spinoza and Platonic-Christian apophatic theologians the notion that, since every affirmation doubles as a negation of its own contradiction (p implies not not p), God must be beyond all affirmation, since he is one (or, to use perhaps a more precise Hindu term, non-dual) and therefore transcends all contradiction. I have long thought that this idea is applicable in the domain of mathematics by analogy between Creator and Creation.

Consider, for example, numbers. Modern philosophers from Nicholas of Cusa to Bertrand Russel have thought that numbers are fundamental to the structure of the cosmos, not to mention the Pythagorean cult of numbers and Jewish and Chinese numerology. Do the principles of negation by affirmation and unity in transcendence apply to the venerable idea of number? I think so.

It would be meaningless to speak of fractions without whole numbers and vice versa: without whole numbers, there would be nothing for fractions to be fractions of, and without fractions, whole numbers would simply be numbers unqualified. Similarly, irrational numbers are nothing other than fractions that cannot be expressed as a ratios between two whole numbers. Irrational numbers are therefore a negation of rational fractions, as fractions in general are a negation of whole numbers. In both cases, the identity of one species of number depends on its difference from another. On the other hand, notwithstanding its diversity of species, the genus of number is one.

Similar remarks could be made about geometrical entities: the sides and angles of a triangle are obviously different, but the angles describe the sides and vice versa, so that the triangle as a whole is one; a straight line and a circle are different, but they are also homogeneous insofar as a straight line can be defined as the circumference of a circle with an indefinitely great radius; and, to relate geometry to arithmetic, a number is different from a quantity of extension, but there is nevertheless a perfect analogy between the irrational square root of two and the length of the diagonal of a square. Such mathematical analogies are called isomorphisms.

Today, an entire mathematical discipline called Category Theory studies isomorphisms with the aim of furnishing a general account of relations or functions between different objects; and what are these functions if not 'negations of negation'? Could Category Theory, by its elucidation of unity in diversity by analogy, serve as a propaedeutic to the inescapably metaphorical dialectic of philosophy? Does Category Theory represent the vindication, in our time, of the inscription above the entrance to the Academy?

Let no one ignorant of geometry enter here.

 

[scorpiomover]

Inverses.

1a) We start with something. Call it "1".

Now invent +. Now we have 1, 2, 3, etc.

Now invent -, the inverse of +. It only applies to when (a - b) > 0.

Suppose that we expand the definition of numbers to include all numbers that are inversions of existing numbers. Now have negative numbers.

(1 - 1) = 0. Now we have zero.

1b) Make multiplication = add the same number n times.

Then make an inverse, divide. Whole number numerators can only be divided if they are multiples of the denominator.

Expand the definition to include almost all inverses, all except 0. Now we have rational numbers.

1c) Make power = multiply by the same number n times.

Try to invert polynomials of powers. Some solutions are not rationals. So: only some solutions are valid.

Expand the definition to include the solutions of polynomials that aren't rationals.

2) We expand the definition, to include the results that aren't in the original set. But to get those results, we have to use an inverse function. The function covers the entire set as its domain. But only part of the co-domain is covered. So to complete the system, you have to expand the inverse to cover the entire co-domain, which extends the function's range, which in turn means that the domain has to expand as well.

It's the inverse function that gives it away.

Say we have a set A. Say we have a function f:A->B. Say that g is the inverse of f, g:B->A. Then g: f(A) -> A. But g says nothing about B-f(A). So we can expand g to g' : B -> C, where A is a subset of C, and choose whatever values we want for g'(B-f(A)) that maintain the isomorphisms that we wish to preserve.

3) "Not" is the inverse function in logic.

 

[The Grey Man]

I think you've put your finger on it. Function, inverse function, domain, and codomain seem to be indispensable concepts in mathematics. Another is triadic synthesis: Category Theory is based on a composition rule that allows us to infer a function from A to C from functions of A to B and B to C, which is strikingly reminiscent of Aristotle's syllogism. How the the same patterns repeat themselves ever more elaborate ways, like musical motifs! History, too, seems to be governed by the law of analogy and isomorphism.

 

[sushi]

its all sets and numbers.