Arising_uk wrote:Sorry, I'm loving this thread as whilst I can understand very little of it I'm enjoying the little philosophy of mathematics that pops up that I can understand and I'm really scared to be breaking in here
Oh no please by all means jump in. We need fresh meat. I'd be happy to explain any points of curiosity or confusion. Nobody on math-oriented websites discusses whether .999... = 1. It's not a serious dispute. If I have any purpose in being here at all it's to try to explain the viewpoint of modern math to anyone who's interested. Modern in this case meaning after 1850 or so when Augustin-Louis Cauchy invented the epsilon business that's bedeviled students ever since.
but I'm at a bit of a loss as to what you mean here about convincing someone as my remedial mathematics has the symbol "+" as something like "and" and "2" as shorthand for "1+1" and "4" as shorthand "1+1+1+1" so I'm at a loss as to how "2+2=4" can ever be 'wrong' or needs to be explained as it says "1+1+1+1=1+1+1+1"? Are you trying to ground numbers in a logical base? If so I thought this had already been done by Russell and Whitehead?
R&W are no longer relevant. Their entire thesis that math could be reduced to logic (this is called logicism) was totally destroyed by Gödel.
Today we model the natural numbers (and everything else that depends on them, including the real numbers) on set theory. In parts of contemporary math, set theory has been replaced by category theory, which can be called a structuralist approach. (I know more about category theory than I do about structuralism). There are also new approaches to foundations like type theory, which actually goes back to Russell. So perhaps he'll have the last laugh. Type theory II, the Revenge of Russell. It's a thing in modern math but I don't know much about it. It's big in computer science.
And before that someone called Peano had sorted the numbers?
A very interesting point, the difference between Peano and set theory. Peano set down some axioms for the natural numbers. He based everything on the idea of a successor function
that inputs 0 and outputs 1, inputs 1 and outputs 2, etc.
So with Peano we have all the natural numbers 0, 1, 2, 3, 4, ... However we do NOT have a completed set of them. We have no container that encompasses them all. Peano's axioms describe Aristotle's potential infinity.
In set theory, one of the most important axioms is the axiom of infinity
. The axiom of infinity says that there is a set
that models the natural numbers. That's the von Neumann definition. Zero is the empty set, 1 is the set containing 0, and so forth.
A set of natural numbers is Aristotle's actual
infinity. It was Cantor's bold and brilliant breakthrough to assume he had a completed set of the natural numbers, and then derive further logical consequences from that. It is precisely the axiom of infinity that distinguishes math from physics. We have no evidence in physics that there are any infinite collections. In math, infinite sets are taken as given.
Essentially all .999... = 1 opposition eventually comes down to objecting to the axiom of infinity. The Extremefinitism site (which I label a crank site, but I love cranks so I'm perfectly willing to read them for entertainment value) explicitly objects to the mathematical assumption of infinity.
It's very relevant that the Extremefinitism site rejects the axiom of infinity and .999... = 1. Those two positions go hand in hand.
I note for the record that finitism itself is a perfectly valid philosophical stance. The Extremefinitism guy is a crank because he thinks he's a finitist but doesn't actually know what he's talking about. If he actually studied the subject he claims to be blogging about he'd have a much better site.
There is an intermediate viewpoint called intuitionism or constructivism. As I understand it, intuitionism is a little mystical (points come into existence through the active selection of an observer); and constructivism is computational: a point exists if and only if it can be generated by an algorithm. So (as I understand it, which isn't much) an intuitionist accepts pi = 3.14159... because we can write a computer program to crank out the digits. But they do not accept the so-called noncomputable numbers
, which are strings of digits that can not possibly be generated by any algorithm because they are essentially random.
So the intuitionist/constructivist real line is shot full of holes, but it's still infinite.
I should mention that earlier I noted that the hyperreal line is shot full of holes too. But the hyperreal line is much richer than the standard line. The hyperreal line has all the reals PLUS the infinitesimals. The constructivist real line is restrictive. It has the computable real numbers but not the noncomputable ones.
It's perhaps philosophically interesting that the standard real line is complete (every nonempty set bounded above has a least upper bound); and both the restrictive (constructivist/intuitionist) and the expansive (hyperreal) variants are not complete. Like Goldilock's porridge, the standard real line is "just right."
As I mentioned earlier there are in fact philosophies of math called finitism and ultrafinitism that deny the axiom of infinity. It's a perfectly legitimate position. In fact it is logically consistent to deny the axiom of infinity. When you do, you get the exact same system as described by Peano. You get most of number theory, including mathematical induction. You just don't get any infinite sets, no Cantorian set theory, no transfinite numbers.
The biggest problem with finitism is that it's very difficult to get the theory of the real numbers off the ground. But a hundred years from now who's to say what intellectual ideas will be in vogue. Anything's possible.
Ooo! Just wikkied and seen that C.S.Peirce had axiomatised them as well! Wow!! As I love that guy even tho' he was also way above my weight of understanding.
Ah C.S. Peirce. I've heard a little about him. He apparently anticipated the structuralist approach to mathematics. His point is that the continuum, if it means anything at all, can not possibly be made of discrete points as it is in set theory. And as it turns out, the category theory revolution in contemporary math is evidently a rediscovery of the ideas of Peirce.
In fact in the categorical account of set theory, sets don't have any internal structure. We don't ask what is "inside" of sets. Rather we only care what relations a given set has with other sets. There are no more points.
Evidently Peirce was a century ahead of his time and modern math is just starting to catch up. I wish I knew more about him too.
You raise great points. Everyone is welcome here as far as I'm concerned. Especially devotees of the Flying Spaghetti Monster.