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.
T: Why?
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.
SM: Why?
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.
***
Patrice Ayme
Tags: Finite Logic, Metamathematics, Non Aristotelian. Finite math, Peano Arithmetic, ZFC
Patrice Ayme's Thoughts 
November 5, 2013 at 1:08 am |
Great job writing this up!!
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November 5, 2013 at 1:15 am |
Thanks Thad! And welcome to the comments! The NON A world is opening up…
PA
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November 5, 2013 at 5:29 am |
I clicked the ‘Like’ button not because I understood what I read, far from it, but because you have given me an incentive, once November is behind me, to crave your patience in explaining this. It seems like a profound thing to understand before my very finite life comes to an end!
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November 5, 2013 at 2:54 pm |
Thanks Paul. Several people told me directly it’s hard to use the “like” button, or to comment (wordpress registration can be tricky). Although I like basically all your posts, none of the “likes” has gone through, for weeks… (It takes long to “load”, weirdly; a bit like the New York Times I have subscribed to for decades, and which tells the public I am NOT “verified”)
You made me think about how one could put Non Aritotelian Logic, let’s say, poetically. In a way, it’s the logic of limits. It’s much more realistic than what’s reigning now. Ecologically, it’s what we need.
Good luck with the novel.
PA
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November 6, 2013 at 4:21 am |
My patience will be at your disposal. I do things that going to NON A is a major mental change. In a way, it introduces the Quantum where it belongs, at the core of the mind.PA
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November 6, 2013 at 2:24 am |
Yikes.
I hope you won’t mind very much if I keep my pet infinite’s pelt to hang over my fireplace. I rather liked the creature. It never harmed anyone and was happy to provide an airy perspective to my clogged mind.
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November 6, 2013 at 4:09 am |
Dear Dominique! Glad to see you manifest again. I am delighted to see you disagree. Because you make an interesting point: you regret the infinite beast. I should rather say, the infinity beasts. Because there is a whole hierarchy of them.
However, there is nothing to regret. You see, with this, well, master stroke, matters are getting enriched, not impoverished.
On the surface, it looks like much of mathematics will implode. And, indeed, it will. However, on second thought, all i did was to make the logic of mathematics MORE COMPLEX. So getting rid of the infinities jungles, is actually opening new logical dimensions.
It unclogs minds by gaining dimensions.
“Never harmed anyone”? I am sure of the opposite. Many generations of mathematicians have struggled with problems related to ever more byzantine versions of infinities that have proven more or less intractable.
Many traditional problems may fall easily from the new approach. Others will look insignificant. After all, insignificance is what happened to most of Euclidean geometry.
The Onion apprach, what I call META (which is used for the construction of my version of the naturals and rationals), also solves all the logical paradoxes (as it blocks, per force, all self referentials).
This has an effect even on some people’s theory of consciousness, as they use, rather naively, self reference as the description/root of consciousness.
PA
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November 6, 2013 at 5:09 am |
Must stop reading this post and the comments last thing before turning the bedside light out. Find myself thinking about what’s written for an infinitely long time before going to sleep. 😉
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November 6, 2013 at 6:16 am |
That’s the whole problem with infinity: never ends, sucks down everything into the hole… the primordial cave? I will be coming on something that will touch on Sagittarius A* tomorrow, BTW, speaking of black holes…;-)!
PA
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November 6, 2013 at 3:18 pm |
Dear Patrice, I did not say I disagreed. I am in no position to agree or disagree. My math level never went beyond that of a good French engineering school – that is, early nineteenth century. I enjoyed it a lot but was well aware that much wider venues were being explored.
I would never presume to cast an opinion on such issues – “sutor, nec ultra crepidam”. Knowing what you don’t know is better that thinking you know all.
My point, as you well perceived, was about the discomfort that may come from having the carpet of common assumptions whipped out from under mankind’s feet, as already happened several times. When it has to be done, so be it, but it carries a price. And on that price, I can comment, even if I have no way of knowing if it buys me a new Grail or a vial of snake oil.
Man’s mind is unique among animals in its abilities, but also in its needs.
As an example: man’s abitity to store and process knowledge is several orders of magnitude ahead of other primates, however evolved – and with it goes man’s addiction to cramming himself full of knowledge. Be it Homer’s works (compulsory knowledge for educated Ancient Greeks, from start to end) or the Bible or baseball scores, or the fifty nuances of green some Amazonian tribes can name, or all visible stars. An African traditional healer has to spend seven hard years memorizing before graduating and setting up shop – like a surgeon.
Another need, I think, is that of an immediate, personal link to the perception of transcendence. Religion once fulfilled this need, and it lost much of its appeal when the infinite provided a cleaner, less gory conduit. The urge is strong; when physicians state the Universe is finite, they hasten to add that there may be an infinity of universes along – to the great relief of most, despite the lack of any tangible consequences. By and large, the human mind is claustrophobic and dislikes boundaries, even though some are agoraphobic and will crowd into sects to fulfill their own craving for narrow mental venues. Once again, I judge neither.
My own contribution, therefore, limits itself to an intuition that the exploration of an infinite-less world (as different from a finite one) need not rule out the continued use of the infinite as a notion. You say that Euclidian geometry faded into irrelevance. There I heartily disagree. What it lost is its claim to exclusivity – which it never seriously held, since nobody ever claimed Euclid’s postulate to be an axiom, and it was only a matter of time before other venues would be explored by Riemann et al. Within its better-defined field of validity, it remains valid and relevant.
Wouldn’t it be a pity if Pascal’s definition of the infinite (observe of “de-finition of the in-finite” is in itself a fascinating expression) was to fade into nothingness? “A sphere, the center of which is everywhere, and the circumference, nowhere”. Elegance is never irrelevant.
PS Blame WordPress for my absence. It now seems to work OK. Crossing fingers!)
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November 6, 2013 at 6:17 pm |
Dear Dominique: A very well thought out and interesting comment. I will start at this point with technical answers, the more advanced stuff will come later, with more time attributed.
The French “physicien” translate as “physicist”, not “physician”, the later meaning MD, Medical Doctor.
The universe thing in present day mainstream physics is OBVIOUSLY COMPLETELY FALSE. It’s just a collective hallucination, a herd phenomenon. Caused by LACK of imagination considering the Quantum, the Universe, and their relationship.
Among other things, the so called “multiverse” or “cosmological inflation” don’t even respect energy conservation. Actually they don’t respect UNIVERSE CONSERVATION. It’s a mistake worthy of toddlers.
Yes, I have had big problems with WordPress too. I actually lost my finished version of Obamascare, thanks to WP (the one I published was just from a few scarps put together). People have complained to me they could not comment, and I myself have been unable to “like” any post from anybody.
OK, more later on more substantial stuff.
PA
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March 31, 2014 at 12:07 am |
[…] is part of my general, Non-Aristotelian campaign against infinity in mathematics and beyond. The nature of mathematics, long pondered, is […]
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