It’s a NON ARISTOTELIAN WORLD:
Tyranosopher: Finite Logic should be called Non Aristotelian Logic. As I will show.
Simplicius Maximus, a contradictor: I have two objections to your finite math madness. First it makes no sense, and, secondly, even if it did, it would be pointless.
Tyranosopher: I love contradictions. I squash them, then drink their juicy parts. OK, bring it on. Let’s start with the contradiction you found. A French contributor, Paul de Foucault, already made the objection that m/0 = infinity.
Sounds good. However, it violates Peano Arithmetic (PA). PA is the arithmetic common to all metamathematics. But for mine, of course. (I violate much, with glee, including the pairing axiom!)
In PA, a.0 = 0 is one of the two axioms defining multiplication. So we see that if x = m/0, we would have x.0 = m. In other words, m = 0.
That’s not surprising: a number called “infinity” is not defined in PA.
Simplicius Maximus: OK, fine. Here is my objection. It’s well known that the square root of two is irrational. Even Aristotle knew this, but you apparently don’t. And then you give the world lessons about everything. You are a charlatan.
T: What do you mean by irrational?
SM: Ah, you see? It means square root of two cannot be equal to m/n, where m and n are integers. Let’s abbreviate square root two by sqrt(2). Irrational means the expansion of sqrt(2) never ends.
SM: Here is the proof. Suppose sqrt (2) were rational. That means: m/n = sqrt (2). Let’s suppose the terms m and n are as small as possible. That’s crucial to get the contradiction.
T: Fair enough.
SM: Now, square both sides.
T: That means, more exactly, that you contrive to multiply the left hand side of the equation by m/n and the right hand side by sqrt(2).
SM: Happy that you can follow that trivial trick. That gives us the equation: mm/nn = 2.
T: As sqrt (2) sqrt (2) = 2. Indeed. By the way, you made an unwarranted assumption, so I view your reasoning as already faulty, at this point.
SM: Faulty? Are you going mad?
T: I will dissect your naïve error later. But please finish, Mr. Aristotle.
SM: Call me Aristotelian if you wish. Multiplying both sides of the equation by nn, we get: mm = 2 nn. That implies that m is even. Because if m were odd, m = 2u + 1, then mm = 4uu + 4u + 1 , the sum of an even number (4uu + 4u) plus 1… And that, the sum of an even number with one, is odd. Hence m = 2a.
But then 2a2a = 2 nn, or: 2 aa = nn. Thus n is even (same reasoning as before: the square of an odd number cannot be even). So we see that both m and n are even, a contradiction, as we assumed m and n were the smallest integers with a ratio equal to sqrt (2).
T: This proof is indeed alluded to in Aristotle, and was interpolated much later into Euclid’s elements. The official Greek mathematicians did not like algebra.
SM: I see that, although you don’t know math, you know historiography.
Tyranosopher: I do know math, I’m just more rigorous than you, august parrot.
Simplicius Maximus: Me, a parrot? Me, and 25 centuries of elite mathematicians who are household names, dozens of Field Medalists are also of the avian persuasion? How can you be so vain and smug?
Tyranosopher: Because I’m smarter.
SM: Really? Smarter than Aristotle?
T: That’s an easy one. People like Aristotle spent a lot of time, all too much time, with politics, not enough with thinking. OK, let’s go back to your very first naive mathematical manipulation. You took the square of both sides.
SM: Of course I did.
Tyranosopher: You can’t do that.
SM: Of course I can.
Tyranosopher: No. In FINITE math, a = b does not imply that aa = bb.
T: Because aa could be meaningless. It could be too big to have meaning. It’s a added to itself a times. If, as we compute aa, we hit the greatest number, #, we must stay silent, as Wittgenstein would have said.
In FINITE math, the infinite set of integers N does not exist. Only what can be finitely constructed exist. Because there is no way to construct the set N, as it would be infinite (if it existed; that’s a huge difference between what I propose, and what David Hilbert proposed). In my system, integers and rational numbers are constructed, according to the principles I exposed in META, layer by layer, like an onion.
SM: Wait. There are other proofs of the irrationality of square root of two.
T: Yes, but it’s always the same story: at some point, multiplication is involved, so my objection resurfaces.
SM: OK, all right. Let me go philosophical. What’s the point of all this madness? Trying to look smarter because you are so vain, at the cost of looking mad? Do you realize that you are throwing out of the window much of modern mathematics?
T: Calm down. Entire parts of math are left untouched, such as topology, category theory, etc. My goal is to refocus all of math according to physics, and deny any worth to the areas that rest on nothing.
All too many mathematicians have engaged in a science as alluring as the counting of angels on a pinhead in the Middle-Ages.
SM: Dedekind said: “God created the integers, and the rest was man’s creation.”
T: Precisely, God does not exist, so nor does the infinite set of the integers, N. This will allow mathematicians to refocus on what they can do, and remember that there is a smallest scale, and it would, assuredly change the methods of proof, in many parts.
SM: Such as?
T: Take the Navier Stokes fluid equation: one has to realize that, ultimately, the math have got to get grainy. This would help physics too, including all computations having to do with infinities.
SM: You are asking for a mad jump into lala land.
T: We are already in lala land. Finding the correct definitions is even more important than finding the correct theorems (as the latter can’t exist without the former). The reigning axiomatic theory, ZFC (Zermelo Fraenkel Choice) requires an infinite number of axioms. What’s more reasonable? An infinite number of axioms, or my finite onion?
The answer is obvious. It’s a NON ARISTOTELIAN WORLD.
In my not so humble opinion, the consequences are far reaching.