It's true, math is a social activity, but I bet a lot of it exists without a preponderance of mathematicians even being aware of it, much less agreeing it exists. — jgill
In your professional research, did you have the feeling that you were investigating aspects of truths outside yourself that you were trying to find out about? — fishfry
Have you ever seen the cauchy sequence of a non-computable real number? If I claim that that Cauchy sequence is for the number 42, how could you challenge that claim? — Ryan O'Connor
You may be right, but I'm of the view that we don't know exactly what we're talking about because there's more work to be done. — Ryan O'Connor
Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use him in support of your ideas, since that can only weaken your argument. — fishfry
Oh boy they're gonna gossip about the rest of us! — fishfry
I'm not the only one, Google around. And FWIW, I'm a crankologist. I enjoy reading math cranks and am familiar with the work of most of the prominent ones. — fishfry
That's his proper name, probably the one he was born with, and he's publicly called a crank. A little part of me cheers for the underdog — norm
You're right, he's a professor of math and he puts his ideas out there under his own name, and the likes of me throws rocks from behind my anonymous handle. Can't deny it. — fishfry
https://njwildberger.com/The pure mathematical community depends on these and other fancies to support a range of “theories” that appear pleasant but are not actually corresponding to reality, and “theorems” which are not logically correct. Measure theory is a good example –this is a subject in which the majority of “results” are without computational substantiation. And the Fundamental theorem of Algebra is a good example of a result which is in direct contradiction to direct experience: how do you factor x^7+x-2 into linear and quadratic factors? Answer: you can’t do this exactly — only approximately.
By removing ourselves from the seductive but false dreamings of modern pure mathematics, we open our eyes to a more computational, logical and attractive mathematics –where everything is above board, where computations actually finish in finite time, where examples can be laid out completely, and where we acknowledge the proper distinction between the exact and the only approximate. This is a pure mathematics which is closer to applied mathematics, and more likely to be able to support it. It also gives us many new insights, more precise definitions, and theorems which are actually …correct. — Wildberger
He must know how crankish he sounds — norm
So in the end you agree with the notion that existence is contingent on opinion, and you simply differ on which opinions count. You just lost the argument methinks.
And what if I find a metaphysician who, based on two years of dialog with me, clearly hasn't bothered to learn the most elementary facts of mathematics? Why should I trust that individual's judgment about anything? — fishfry
Where do they live? And what else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? Platonism is untenable. There is no magical nonphysical realm of stuff. And if there is, I'd like to see someone make a coherent case for such a thing. — fishfry
Well I don't understand. Contingent on what? If there is a Platonic realm after all, surely mathematical truths live there if nothing else. — fishfry
Wildberger is a nut, his math doctorate notwithstanding. He does have some very nice historical videos and some interesting ideas. But his views on the real numbers are pure crankery. You should not use him in support of your ideas, since that can only weaken your argument. — fishfry
The axiom of infinity lets us take all of the numbers given by the Peano axioms and put them in a set. That's the essential content of the axiom.
The Peano axioms gives us 0, 1, 2, 3, ...
The axiom of infinity gives us {0, 1, 2, 3, ...}
The former will not suffice as a substitute for the latter. For example we can form the powerset of {0, 1, 2, 3, ...} to get the theory of the real numbers off the ground. But we can't form the powerset of 0, 1, 2, 3, ... because there's no set. — fishfry
You're talking about how you'd like mathematics to be, but you do it entirely in a castles-in-the-air manner without regard to even a minimal understanding of the mathematical context. Philosophizing about mathematics is fine. But when the philosophizing concerns actual mathematical concepts, then, unless there is an understanding of the actual mathematics and the demands of deductive mathematics, that philosophizing is bound to end as heap of half-baked gibberish. — GrandMinnow
Ordinary calculus does use infinite sets. — GrandMinnow
The set theoretic axiomatization of mathematics is very straightforward, easy to understand, and eventually yields precise formulations for the notions of the mathematics of the sciences. — GrandMinnow
If you are sincerely interested in the subject, even from a philosophical point of view, you should learn the set theoretic foundations and then also you could learn about alternative foundations that bloom in the mathematical landscape. — GrandMinnow
And neither is there a Zeno's paradox with set theoretic infinity. — GrandMinnow
I wouldn't begrudge philosophical objections to the notion of infinity. My point though is that one does not have to be platonist to work with theorems that are "read off" in natural language as "there exists an infinite set". The axiom that is (nick)named 'the axiom of infinity' does not mention 'infinity' and, for formal purposes, use of the adjective 'is infinite' can just as well be dispensed in favor of a purely formally defined 1-place predicate symbol. — GrandMinnow
as you mention what you consider to be flaws in classical mathematics, as I said before, you have not offered a specific alternative that we could examine for its own flaws. — GrandMinnow
Then the real number 42 is the equivalence class that contains f and all g such that |f(n) - g(n)| --> 0 as n --> inf. — norm
we often think we are talking about numbers when we are really talking about talk about numbers. — norm
I understand your limit-based 'algorithm' but would there ever be an instant in time when you would be sure that it's not 42? — Ryan O'Connor
Yes, and I think we do the same about actual infinity. We don't conceive of actual infinity, we conceive of conceiving of actual infinity (using potentially infinite algorithms). — Ryan O'Connor
Hi ! I'll private-message you about that. — norm
Oh boy they're gonna gossip about the rest of us! — fishfry
I can't tell whether 0.999999...[?] is different than 1 until I finally find a non-9 in the expansion somewhere, so there's no bound on the check for equality. — norm
have you looked into Zeilberger? He's a maverick too, a bit of a finitist. — norm
If there is no way to reinterpret the Axiom of Infinity — Ryan O'Connor
When working with ZF, we are always dealing with finite statements. — Ryan O'Connor
isn't a forum like this a good place to discuss [half-baked ideas]? — Ryan O'Connor
Do limits require the existence of infinite sets? — Ryan O'Connor
Do you believe that there are any paradoxes related to the set theoretic axiomatization of mathematics — Ryan O'Connor
how much education should a person have before initiating a discussion on a philosophy forum like this? — Ryan O'Connor
What resolution of Zeno's paradox are you satisfied with? Limits can be used to describe a process of approaching a destination but they cannot describe arriving there. So how does one arrive at some new destination? — Ryan O'Connor
Are your and fishfry's posts in agreement? — Ryan O'Connor
videos — Ryan O'Connor
FWIW, I agree with Chaitin that noncomputable numbers are suspicious. I can't even show you one. I can only talk about them indirectly. But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis.You can't tell by inspecting the digits, but at least 0.999... is computable so you can make some assessments by comparing the algorithms used to generate 0.999... and 1. The same cannot be said about non-computable numbers, which is what I was getting at. — Ryan O'Connor
Again, set theory rises to the challenge of providing a formal system by which there is an algorithm for determining whether a sequence of formulas is indeed a proof in the system. So whatever you think its flaws are, that would have to be in context of comparison with the flaws of another system that itself rises to that challenge. — GrandMinnow
Not so, my friend. Norm is mathematically authentic, as are you and fdrake, and I will probably learn something from his posts, as I have from the two of you. — jgill
I'm of two minds about revealing anything about the expertise of math people on this forum. I realize the knowledge may intimidate some others and dissuade them from contributing their ideas. Or it might have the opposite effect of encouraging attacks on academia. Oh well, not a big deal. — jgill
The personal context you present is based in gross ignorance of the subject. I don't opine as to all discussions, but this discussion engenders an unfortunate trail of waste product - gross misinformation, misunderstanding, and confusion. Sometimes people who have an appreciation of the subject wish not to see such otherwise beautiful scenic trails left without being de-littered. — GrandMinnow
But if one does reject non-computable numbers, then R has measure 0, which completely breaks modern analysis. — norm
but no foundation has ever seemed 'just right ' to me. There's always some ugly weeds. In the end I'm a relatively carefree antifoundationalist who enjoys math as an excellent if imperfect language. — norm
IMO, there's a 'know how' at the bottom of things that perhaps can never be formalized or made explicit. — norm
Do you want to live in a country where the 'scenic trails' are exclusive to the 'privileged rich'? — Ryan O'Connor
You are trying to find a way to reject my ideas without understanding them — Ryan O'Connor
Can you give me an example of what would break down without non-computable numbers? — Ryan O'Connor
I like that you admit that there are ugly weeds. So you're just satisfied ignoring the weeds? But you must enjoy the philosophy to some extent, you're here after all? Actually, I'd love to hear what you think these weeds are... — Ryan O'Connor
Can you explain what lies at the bottom that you don't think can be explained? — Ryan O'Connor
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