What does it mean (to you) to prove that a number exists? — fishfry
But perhaps it is simplest if I take existence out of my original question: Have we really proved that √2 is an irrational number? — Ryan O'Connor
But perhaps it is simplest if I take existence out of my original question: Have we really proved that √2 is an irrational number? — Ryan O'Connor
Yes, Pythagoras gets credit for (but probably didn't personally have anything to do with) the proof that sqrt(2) is irrational. — fishfry
But you want to strengthen that standard to say that a number only exists when we can not only give an algorithm for it, but that we can execute the algorithm to completion given the physical constraints of the universe. — fishfry
That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you. — fishfry
For my own part, I take mathematical existence to mean anything that can be proven logically using the axioms of any given mathematical system. — fishfry
That includes all the computable numbers and all the noncomputable ones as well, which after all are necessary to ensure the completeness of the real numbers. — fishfry
Just checked. Here is a link to a discussion in Wikipedia about the proof that pi is irrational — T Clark
He only proved that √2 is not a rational number. He did not prove that √2 is an irrational number. Yes, I'm concerned with proof by contradiction. — Ryan O'Connor
If I flip a coin and tell you that it is not heads that is not proof that it is tails because it may have landed on its side. — Ryan O'Connor
Actually, I want to strengthen it even more so to say that a number only exists when it is being computed. If there is no computer currently thinking about 3 right now then the number 3 does not exist as an actualized number. It only has the potential to exist. — Ryan O'Connor
I think this sort of view is required if we are to avoid actual infinity. — Ryan O'Connor
Otherwise, how would a constructivist answer the question: how many numbers are there? — Ryan O'Connor
My response to such a question is 'how many numbers are where? In what computer?' — Ryan O'Connor
I believe that the decimal representation for 1/3 cannot exist but nevertheless the number certainly can. For example, it is LL on the Stern-Brocot tree. And we can do exact arithmetic using any rational number using the Stern-Brocot tree. — Ryan O'Connor
But won't there always be undecidable statements? It seems like your definition is too restrictive as it would be missing some truths. — Ryan O'Connor
Why is it necessary to have a number system which is complete? — Ryan O'Connor
I have no doubt that pi is irrational (i.e. not a rational number). But a sandwich is also not a rational number. My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm. — Ryan O'Connor
My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm. — Ryan O'Connor
The mainstream approach to giving the number √2 existence requires us to assert the existence the Platonic Realm (which I equate with an infinite computer), but extraordinary claims should be backed by strong evidence, for which I have come across none. Thoughts? — Ryan O'Connor
He did not prove that √2 is an irrational number. — Ryan O'Connor
won't there always be undecidable statements? — Ryan O'Connor
Why is it necessary to have a number system which is complete? — Ryan O'Connor
And we can do exact arithmetic using any rational number — Ryan O'Connor
The mainstream approach to giving the number √2 existence requires us to assert the existence the Platonic Realm (which I equate with an infinite computer), — Ryan O'Connor
Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does require subscribing to mathematical platonism. — GrandMinnow
Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does require subscribing to mathematical platonism. — GrandMinnow
I meant to write "[...] does not require [...]." I edited my post just now upon reading your post and realizing that I mistakenly left out the word 'not'. — GrandMinnow
Constructivists deny the law of the excluded middle. — fishfry
Different issue. Landing on side should be included in the outcome space. — fishfry
I see your point, but then existence becomes contingent on what everyone's thinking about and/or computing. 3 might or might not exist depending on what 7 billion people and a few hundred million computers are thinking or computing at this exact instant. What kind of standard for existence is that? Not one that would have much support from ontologist, I'd wager. — fishfry
But your particular flavor of ultrafinitism is untenable, granting existence to only those things that someone is thinking about or computing at any given instant. Under such a philosophy we can never say whether a given number or mathematical object exists. — fishfry
If you deny the mathematical existence of the natural numbers you not only deny ZF, but also Peano arithmetic. That doesn't let you get any nontrivial math off the ground. Not only no theory of the real numbers, but not even elementary number theory. You may be making a philosophical point but not one with much merit, since it doesn't account for how mathematics is used or for actual mathematical practice. — fishfry
Well then you're an ultrafinitist. You not only deny the existence of infinite sets; you deny the existence of sufficiently large finite sets. — fishfry
Why doesn't pi exist? It has a representation as a finite-length algorithm. — fishfry
Well in any sufficiently interesting mathematical system we are always missing some truths. — fishfry
Because otherwise the real number line has holes in it. The intermediate value theorem is false. There's a continuous function that's less than zero at one point and greater than zero at another, but that is never zero. Of course the constructivists patch this up by limiting their attention to only computable functions. The constructivists have answers for everything, which I never find satisfactory. — fishfry
The volume of a pizza of radius z and height a is pi z z a. — fishfry
You can say whatever you want and call it philosophy. And if no one agrees and you are truly OK with that, then you win. I mention this because of experience with infinity-deniers, anti-Cantorians, etc. — norm
Approximation and vagueness is the rule, and integers are glowing exceptions to just about everything else that's human. — norm
The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. — GrandMinnow
if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms? — GrandMinnow
Perhaps (I don't opine for the purposes of this particular post) proof supplied to answer (2) may be called into question constructively by rejecting the dichotomy that a real number is either rational or it is not rational. — GrandMinnow
The proof that there is a real number x such that x^2 = 2 comes later in the history of mathematics. It is found in many a textbook in introductory real analysis. This is desirable, for example, so that we can easily refer to a particular real number that is the length of the diagonal of the unit square. — GrandMinnow
Do you think your version of non-axiomatic mathematics "decides" every mathematical statement? There are undecidable statements in the mathematics of the plain counting numbers themselves. Undecidability is endemic to the most basic mathematics even aside from the real numbers as a complete ordered field. So how does your version of non-axiomatic mathematics "decide" all mathematical questions? — GrandMinnow
. I would add that a complete ordered field is the ordinary foundation for calculus, which is used for the ordinary mathematics for the physical sciences. — GrandMinnow
there can also be an unmeasured 'potential' state where a proposition is neither true nor false — Ryan O'Connor
Isn't 'not a number' a reasonable option to be included in the outcome space of the proof √2. — Ryan O'Connor
Why do we need all mathematical objects to actually exist? — Ryan O'Connor
Interpretation 1: Any finite list of primes is incomplete.
Interpretation 2: There exist infinitely many primes.
These two interpretations may seem equivalent but they're not because the second makes an unjustified leap to assert the existence of an infinite set. — Ryan O'Connor
I believe ZF and Peano arithmetic just need to be reinterpreted. — Ryan O'Connor
I simply think we shouldn't interpret ZF in certain platonic ways. — Ryan O'Connor
Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. — Ryan O'Connor
ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics. — Ryan O'Connor
Aren't you beginning your proof with an assumption, that irrationals are numbers? — Ryan O'Connor
Why can't it simply be an algorithm? — Ryan O'Connor
My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. — Ryan O'Connor
My view is in total agreement with the foundations of calculus. — Ryan O'Connor
After all, when we write it out explicitly we always write it as some algorithm. Why can't it simply be an algorithm? — Ryan O'Connor
his I would disagree with. One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. No Platonism needed. We've seen this standard applied to irrational numbers, negative numbers, transcendental numbers, complex numbers, quaternions, transfinite numbers, and many other now-familiar mathematical objects.
In order to demonstrate that sqrt(2) has mathematical existence, I do not need to posit a mystical Platonic realm in which sqrt(2) lives. If I did, I might be challenged: What else lives there? The baby Jesus? The Flying Spaghetti Monster? Pegasus the flying horse? No, I don't need to sort all this out just to know that sqrt(2) exists. — fishfry
Why give special status to " a preponderance of mathematicians", granting them the capacity to determine the existence of things? Isn't there a preponderance of Catholic theologians who believe in Jesus, and a preponderance of Pastafarians who believe in the existence of the spaghetti monster? Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in? — Metaphysician Undercover
Why give special status to " a preponderance of mathematicians", granting them the capacity to determine the existence of things? — Metaphysician Undercover
You ask a good question, and one I can't answer. This is the only thing you've ever said to me that has made me stop and think, and for which I have no good answer. — fishfry
I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you? — fishfry
The context of modern logic and mathematics involves formal axiomatization. Anyone can come up with all kinds of philosophical perspectives on mathematics and even come up with all kinds of ersatz informal mathematics itself. But that does not in itself satisfy the challenge of providing a formal system in which there is objective algorithmic verification of the correctness of proof (given whatever formal definition of 'proof' is in play). I mentioned this before, but you ignored my point. This conversation is doomed to go nowhere in circles if you don't recognize that what you're talking about is purely philosophical and does not (at least yet) provide a formalization that would earn respect of actual mathematicians in the field of foundations. — GrandMinnow
please show your system of axioms and rules by which one may make an evaluation of such circumstances. — GrandMinnow
If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem. — GrandMinnow
The justification is from axioms from which we prove that there exists an infinite set. — GrandMinnow
the mathematics itself stands whether the mathematician regards it platonistically or not. — GrandMinnow
That is patently false. Of course, usually physical measurements, as with a ruler, involve manipulation of physical objects themselves and as humans we can't witness infinite accuracy in such circumstances. But that doesn't meant that mathematical calculations are limited in that way. — GrandMinnow
I gave that as a supposition in the context of separating the two questions I mentioned, But, meanwhile "there is a real number x such that x^2 =2" is proven from the ordinary axioms. — GrandMinnow
Fine. Then your remark to the other poster about undecidability is not pertinent. By the way, in classical mathematics a statement that is undecided is not undecided simpliciter but rather it is undecided relative to a given system. — GrandMinnow
No it's not [in total agreement with the foundations of calculus]. Clearly. — GrandMinnow
This reminds me of the construction of the real numbers from Cauchy sequence of rational numbers. — norm
Have you looked into Brouwer? Or Chaitin's Metamath: The Quest For Omega? In short, I think some experts have problems with the real numbers, but these problems are philosophical/intuitive rather than technical. — norm
In general, we don't know exactly what we are talking about, but math tempts us to forget that. — norm
Why do you think that mathematicians, simply by believing in something have the capacity to grant that something "existence", while for other groups of people, simply believing in something is insufficient for the existence of what is believed in? — Metaphysician Undercover
Well, either a number can be expressed as a ratio of two integers or not. If it can be the number is rational and if can't be it's irrational. When there are only two choices, the fallacy of the false dichotomy can't be committed. — TheMadFool
5 is prime, and that seems to be true independently of the opinions of people. Mathematical truth has a necessity that's forced on us in some way nobody can understand. — fishfry
thousands of minds together can cover far more intellectual terrain and see into one another's blindspots. — norm
Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover
Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover
I like to think that there exist truths independent of consciousness, whether it's certain axioms of mathematics or the laws of nature. — Ryan O'Connor
But I think '5 is prime' is a contingent truth... — Ryan O'Connor
e.g. Norman Wildberger — Ryan O'Connor
Axioms are not my strength, but could we perhaps reinterpret the Axiom of Infinity to assert the existence of an algorithm for generating an infinite set, without requiring that the infinite set actually exists? — Ryan O'Connor
it's unreasonable to expect a formal theory to be perfected in isolation. — Ryan O'Connor
When Descartes developed analytic geometry, I suspect that he didn't present axioms — Ryan O'Connor
Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity — Ryan O'Connor
with a few lines of code I can create a program to list all natural numbers even though it is impossible for the program to be executed to completion — Ryan O'Connor
I believe that if we abandon Platonism by replacing the actual infinities with potential infinities that the mathematics will stand. This is what I mean when I say it's a philosophy issue, not a mathematical issue. — Ryan O'Connor
Division by zero can be handled by the Fregean method of definition. And I addressed Godel previously; you don't know what you're talking about with regard to Godel. Morevover, 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. 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.With the 'Whole-from-Parts' view we can't put brackets around everything and call it math. There is no set of all sets. We cannot talk about division by zero. We can't avoid Gödel statements. — Ryan O'Connor
One can take the viewpoint that symbolic math is a human endeavor; and that a thing has mathematical existence whenever a preponderance of mathematicians agree that it does. — fishfry
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