What do you mean by existence? Have we proved the existence of the number 3? Of 1/3 = .333...? Of -6? Of $i$ such that $i^2 = -1$? Of the quaternion $1 + i + j + k$? Of $\aleph_3$? Of $\omega_1$, the first uncountable ordinal?

What does it mean (to you) to prove that a number exists?

What does it mean (to you) to prove that a number exists? — fishfry

Hmm, you've revealed that my original post was quite vague. I believe that an object exists if it is being computed. I exist as an artifact of the laws of nature being evaluated and the number 3 exists within my mind because I'm thinking about it right now. With this view, the number √2 (in its totality) cannot exist because it cannot exist in any finite computer. To be clear, I believe that a finite algorithm for computing √2 can exist, but the output of that algorithm (which I'm calling the number √2) cannot exist. 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?

But perhaps it is simplest if I take existence out of my original question: Have we really proved that √2 is an irrational number?

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. Is it proof by contradiction that you're concerned about?

It's interesting that you agree that sqrt(2) is computable, as are pi, e, and every other mathematical constant that anyone can name. Except for Chaitin's constant, which we can name but which isn't computable.

You seem to be applying a much stricter standard than even the mathematical constructivists. They would allow the existence of any computable number, since we can give an algorithm to approximate it to any desired degree.

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. That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you.

But why not? Why are you privileging decimal notation? The algorithm for pi is finite and expresses pi exactly. So what if the decimal representation's not finite? Why should that be the standard? If we have an algorithm, we have the number. That's the constructivist point of view.

For my own part, I take mathematical existence to mean anything that can be proven logically using the axioms of any given mathematical system. 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. And no mathematical objects exist at all in the physical world, since they're all abstractions.

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

Just checked. Here is a link to a discussion in Wikipedia about the proof that pi is irrational:

https://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational#:~:text=In%20the%201760s%2C%20Johann%20Heinrich,is%20a%20non%2Dzero%20integer.&text=In%201882%2C%20Ferdinand%20von%20Lindemann,irrational%2C%20but%20transcendental%20as%20well.

Not claiming I understand it.

Yes, Pythagoras gets credit for (but probably didn't personally have anything to do with) the proof that sqrt(2) is irrational. — fishfry

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. 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.

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

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. I think this sort of view is required if we are to avoid actual infinity. Otherwise, how would a constructivist answer the question: how many numbers are there? My response to such a question is 'how many numbers are where? In what computer?'

That's a terribly restrictive standard, for example 1/3 = .3333... doesn't exist according to you. — fishfry

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.

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

But won't there always be undecidable statements? It seems like your definition is too restrictive as it would be missing some truths.

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

Why is it necessary to have a number system which is complete?

Just checked. Here is a link to a discussion in Wikipedia about the proof that pi is irrational — T Clark

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.

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

Constructivists deny the law of the excluded middle. You might be interested in this. For my own part I don't have any affinity for constructivism although it's enjoying a resurgence lately due to the influence of computer science and computerized mathematical proof systems. Brouwer's revenge, I like to call it.

https://en.wikipedia.org/wiki/Constructivism_(philosophy_of_mathematics)

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

Different issue. Landing on side should be included in the outcome space. In high school statistics we were flipping coins once and a nickel started rolling on its side and slowing down and darn near landed on its edge, but it hit the wall and fell over. But I almost saw it happen.

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 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.

I think this sort of view is required if we are to avoid actual infinity. — Ryan O'Connor

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.

Otherwise, how would a constructivist answer the question: how many numbers are there? — Ryan O'Connor

Well then you're an ultrafinitist. You not only deny the existence of infinite sets; you deny the existence of sufficiently large finite sets.

I do actually have some sympathy for the ultrafinitist position, since it's the only mathematical ontology that is consistent with what's known about the physical world. 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.

My response to such a question is 'how many numbers are where? In what computer?' — Ryan O'Connor

Even the constructivists, with whom I've had many an interesting discussion in these very pages, believe in computable numbers. There is a countable infinity of them. Computable numbers, I mean, not constructivists.

The ultrafinitists don't put any particular upper limit on how large a number can be, only that there aren't infinitely many of them.

But you want to not only say that, but that whether a given number exists or not depends on whether someone's thinking of it. How can we ever determine that? It's an unverifiable standard. There is then no way to know whether any number exists and whether it still exists five minutes from now. It's impossible to hold such a view along with any kind of coherent ontology of numbers.

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

Why doesn't pi exist? It has a representation as a finite-length algorithm. By exist I mean mathematical existence of course, that's the only kind of existence I'm talking about.

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

Well in any sufficiently interesting mathematical system we are always missing some truths. That's just a fact. But at least it's not contingent. The Continuum hypothesis is always undecidable in ZFC. Now and five minutes from now and five million years from now. And in ZFC + CH, it's provable. Now and five minutes from now. We have logical certainty about what exists, unlike with your system in which we have to constantly poll 7 billion people and several hundred million computers.

Why is it necessary to have a number system which is complete? — Ryan O'Connor

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.

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

Of course LEM is always relative to a given universe of discourse. If pi is a real number that's not rational, then it's a real number that's irrational. Without the restriction to real numbers, you're right. It could be a sandwich. Time for dinner.

My argument is that just as a sandwich is not a number, perhaps neither is pi. Maybe it's an algorithm. — Ryan O'Connor

A = sandwich * r^2. Doesn't work for me. Arccos (-1) = sandwich. Nope.

A = sandwich * r^2. Doesn't work for me. Arccos (-1) = sandwich. Nope. — T Clark

The volume of a pizza of radius z and height a is pi z z a.

The volume of a pizza of radius z and height a is pi z z a. — fishfry

Are you proposing this as proof of the existence of God?

Are you proposing this as proof of the existence of God? — T Clark

Most definitely. With extra cheese.

First answer:
There's a way you can play this where it is a false dichotomy. But you have to go out into the desert and live on locusts and honey. If you are doing philosophy of math, you can basically say whatever you want. Personally I don't believe that odd numbers are real. (That's a joke, and it would be harder to make a case against the legitimacy of odd numbers IMO.) [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. ]

Second answer:
If you are asking a technical question, then it seems you have too options. Believe one or more anonymous posters who sound knowledgeable or find out which books are considered 'classics' in the field and see what you can make of them. I know something about the subject (but why should you believe me?) and I can say that lots of ink has been spilled on the issue by clever people. They exist, but not in such a intuitively pleasing way as natural numbers do.

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

Not that you should believe me as a single anonymous voice, but no one has to embrace the Platonic creed to get degrees in math. People who know how to shuffle symbols correctly can still argue about what it all really means. Wittgenstein was heavy on phil. of math, probably because it's a toy model of philosophy in general. What people even mean by Platonic Realm is not exactly specifiable, is itself a kind of 'irrational number' in the everyday shuffling of words. Approximation and vagueness is the rule, and integers are glowing exceptions to just about everything else that's human.

We separate two questions:

(1) Is there a real number x such that x^2 = 2?

(2) Supposing there is a real number x such that x^2 = 2, is that real number rational or irrational? (Note that 'irrational' simply means, by definition, 'not rational').

Proof supplied to answer (1) depends on certain axioms. Usually, these are the set theoretic axioms used to prove the existence of the real numbers as a complete ordered field. The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. Axiomatization is a desirable approach to mathematics, as it provides an explicit objective algorithmic standard by which anyone may judge of a purported mathematical proof whether it is indeed a correct proof, as opposed to subjective standards such as what happens to be or not be in the mind at any given time of some human being or another. So, if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms?

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. However, as to proof by contradiction, the proof that sqrt(2) is irrational is of this form [let P be any proposition]: We wish to prove it is not the case that P (in particular, we wish to prove that it is not the case that the sqrt(2) is rational). We suppose it is the case that P. We then derive a contradiction. We conclude that it is not the case that P. That is a constructive proof form.

That is not to be confused with the non-constructive proof form: We wish to prove it is the case that P. We suppose it is not the case that P. We then derive a contradiction. We conclude that it is the case that P.

As to the overall contradiction form, the irrationality of sqrt(2) is of the former, constructive, form.

He did not prove that √2 is an irrational number. — Ryan O'Connor

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.

won't there always be undecidable statements? — Ryan O'Connor

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? By polling among the billions of known existing people (and examining just completed computer computations) whether they have such and such mathematical answers or computations in their minds or in their computer output at some given time?

Why is it necessary to have a number system which is complete? — Ryan O'Connor

This was answered by another poster. 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.

And we can do exact arithmetic using any rational number — Ryan O'Connor

Arithmetic, sure. But you haven't shown how to do exact calculus without irrational numbers.

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

Mathematical platonism (roughly put) is the view that mathematical objects exist independent of consciousness of them and that mathematical propositions are true or false independent of conscious determination. Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does not require subscribing to mathematical platonism.

Understanding the sense of mathematical existence statements - such as the existence of irrational numbers - does require subscribing to mathematical platonism. — GrandMinnow

This 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.

I can show sqrt(2) exists as others mentioned, as real number such that $x^2 = 2$, once I've formalized the construction of the real numbers and shown their completeness.

We can simply make up an arbitrary symbol $\sqrt 2$ with the property that $(\sqrt 2)^2 = 2$, and then consider the collection of all rational numbers $a + b \sqrt 2$ where $a$ and $b$ are rational. We will find that we've invented a set of numbers that obey all the rules of a field (I can add, subtract, multiply, and divide with all the usual properties) that contains a square root of 2. This works very nicely.

In fact this is exactly how we introduce the complex numbers to students, as the set of all $a + bi$ where $a$ and $b$ are real and $i^2 = 1$.

We can formalize the above idea by starting from the rational numbers, forming the ring of polynomials with rational coefficients, and mod out the ideal generated by the polynomial $x^2 - 2$. The resulting object is a field in which 2 has a square root.

In other words the two previous paragraphs show that if you believe in the rational numbers, you can easily adjoin to them a square root of 2. That is: if the rationals exist then so does the square root of 2. One does not need any Platonic realm to perform these symbolic constructions.

If someone asks if sqrt(2) as I've defined it has mathematical existence, I just point to any textbook on real analysis or abstract algebra. I do not need a book on metaphysics! All the mathematicians in the world agree that these constructions are valid and that's what gives sqrt(2) mathematical existence.

Now let me give an objection to what I said, one raised by @Metaphysician Undercover when we had this convo a while back. And that is, that mathematical existence is now contingent on what people say. My idea isn't any better than @Ryan's idea of polling all the humans to see if they're thinking of sqrt(2) at this very instant. My standard is to poll all the mathematicians in the world to see if most of them are willing to agree that sqrt(2) exists. I'd be the first to agree that my criterion for mathematical existence has some problems.

But a Platonic world where non-physical things exist? That seems untenable. So we're left with the opinions of mathematicians.

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'.

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

Doh! That would have saved me a lot of typing! LOL.

Numbers do not exist! Humans have invented numbers as a way of expressing certain concepts. The concept of one apple plus one apple equals two apples is fundamental, but is only a concept which humans may imagine. To any other animal on the planet earth, there is no such thing as one apple or two apples, there is only apples.

Constructivists deny the law of the excluded middle. — fishfry

I believe that the only measurable states of a proposition are true or false so in one sense I accept the Law of the Excluded Middle. Where my view deviates from the norm is that I believe there can also be an unmeasured 'potential' state where a proposition is neither true nor false (akin to Schrödinger's Cat). And returning back to numbers, numbers which are not a part of any computation are in this unmeasured potential state.

Different issue. Landing on side should be included in the outcome space. — fishfry

Is it a different issue though? Isn't 'not a number' a reasonable option to be included in the outcome space of the proof √2.

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

Assume for the moment that our universe is like a computer simulation. Wouldn't your existence be contingent on the simulator 'thinking' about you? To me, it seems reasonable to think that, like us, numbers are contingent. Why must everything eternally and actually exist?

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

Why do we need all mathematical objects to actually exist? And moreover, why do we need to know that all numbers actually exist. To me it seems sufficient to know that you have the potential to keep counting the natural numbers, we don't need to actually count to 'the end'...in fact we can't. Perhaps if we fully embrace potential infinity and potential existence, we will find that we don't need actual infinity or the platonic realm.

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

Consider Euclid's Theorem. What of the following did Euclid actually prove?

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. Nevertheless, the math is the same, it's the philosophy (the interpretation) that is different. I believe ZF and Peano arithmetic just need to be reinterpreted.

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

While I don't believe in the existence of infinite sets, I wouldn't attach myself to the finitist label. Finitists usually have an uphill battle trying to establish mathematics as rich is infinite mathematics. I simply think we shouldn't interpret ZF in certain platonic ways.

Why doesn't pi exist? It has a representation as a finite-length algorithm. — fishfry

Pi the infinite digit number cannot exist. Pi the finite-length algorithm can certainly exist.

Well in any sufficiently interesting mathematical system we are always missing some truths. — fishfry

With your view these truths are missing. With my view the 'missing' statements are not missing at all, they're fully accounted for in the unmeasured 'potential' state. It's just that some statements, like 'this statement is false' must permanently remain in the unmeasured 'potential' state.

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

If we pluck the irrational numbers off the number line, we are not left with holes in between the rational numbers...we are left with lines in between the rational numbers. With this view, we don't have to believe that infinite 0-D points can somehow be assembled to form a 1-D object. Instead, lines (or more generally, continua) are fundamental, not points. Think of how you draw a graph: you start with a piece of paper (a continua) and you draw a grid on it. At each intersection (point) you label it with coordinates (numbers). Everything about this is finite. In between the points/numbers lies a continua. But somehow we think about it all backwards. We think that we start with infinite points and numbers and then they someone assemble to form a continua. It's because of this thinking why calculus seems so paradoxical. I don't believe that the intermediate value theorem is entirely false, it just needs to be reinterpreted to apply to continua instead of points.

The volume of a pizza of radius z and height a is pi z z a. — fishfry

Ha! Love it!!

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

I'm not okay with that, which is why I truly appreciate this discussion. I also don't consider myself an infinity-denier, after all I'm a huge proponent of potential infinity (I only reject actual infinity). And I also believe Cantor's work was extremely important, I just think it needs to be reinterpreted.

Approximation and vagueness is the rule, and integers are glowing exceptions to just about everything else that's human. — norm

Or could 'Approximation and vagueness' just be the norm because we haven't fully figured it out? Math has changed so much in the past ~100 years since Cantor, what reason do we have to think that all of the foundations have been set? There are still way too many paradoxes to think that.

The real numbers as a complete ordered field provide a foundation for the mathematics of the physical sciences. — GrandMinnow

Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. Real 'algorithms' on the other hand are very useful. When I use pi in an equation I use it to refer to a potentially infinite series which I dare not try to calculate...so I keep it in algorithmic form.

if you reject certain set theoretic axioms, then we may ask: What are your alternative axioms? — GrandMinnow

I hope that my views are largely in agreement with ZF and that ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics.

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

Aren't you beginning your proof with an assumption, that irrationals are numbers?

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

I understand why we want it to be a number, but that doesn't mean it is. After all, when we write it out explicitly we always write it as some algorithm. Why can't it simply be an algorithm? Our classical intuitions have us wanting to be able to precisely measure any coordinate, for example, the coordinates of the points where y=0 and y=x^2-2 intersect. But with quantum mechanics, our intuitions have changed. With the uncertainty principle we know that there is a fundamental limit to the accuracy with which the values for certain pairs of quantities can be predicted. Is it possible that there is a similar limit to which certain pairs of coordinates can be measured in mathematics?

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

My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. The undecided state is not a defect of my view, it is a feature. Returning to quantum analogies, it is like how the unmeasured superposition state of a particle is a valid state for the particle.

. 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

My view is in total agreement with the foundations of calculus. In fact, I believe that when calculus was reformulated based on limits (and potential infinity) to banish infinitesimals that we didn't go far enough because we failed to banish real numbers (which are inseparably tied to actual infinity). We should have replaced real numbers with real algorithms and interpreted calculus to be not a mathematics that outputs numbers, but a mathematics that outputs processes.

fishfry, norm and GrandMinnows, thanks for your detailed and educated feedback!

there can also be an unmeasured 'potential' state where a proposition is neither true nor false — Ryan O'Connor

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.

There is actual work in formal constructivist, finitist, and computationalist, or multi-valued mathematics, but you don't reference or commit to any specific such formulation. So we don't have a basis for evaluating the merits of your notions compared to those of classical mathematics on the level playing field of "I'll show you my system, explicitly, without vernacular vagueness and you show me yours."

Isn't 'not a number' a reasonable option to be included in the outcome space of the proof √2. — Ryan O'Connor

Again, please show your system of axioms and rules by which one may make an evaluation of such circumstances.

Why do we need all mathematical objects to actually exist? — Ryan O'Connor

If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem.

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

The justification is from axioms from which we prove that there exists an infinite set. You are free to reject, or be uninterested in, those axioms, but in the meanwhile you haven't said what your axioms are. And you are even free to reject the axiomatic method itself, or be uninterested in it. But then this conversation would be at a hard impasse between, on the one hand, mathematicians who understand the benefit of formal axiomatics and require that mathematical proposals be backed up not just by homespun philosophizing, and, on the other hand, you.

I believe ZF and Peano arithmetic just need to be reinterpreted. — Ryan O'Connor

ZF and PA are formal systems. They are interpreted by the method of models. Whatever you have in mind by a reinterpretation of a formal system is not stated by you. However for you to state such a thing would require that you do know the basics of mathematical logic that is the context of such systems.

I simply think we shouldn't interpret ZF in certain platonic ways. — Ryan O'Connor

Fine. We don't need to. However, the mathematics itself stands whether the mathematician regards it platonistically or not.

Real numbers are never used in applied mathematics. Every number that we have ever used in a calculation is rational. — Ryan O'Connor

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.

ZF just needs to be reinterpreted. This is largely an issue of philosophy, not mathematics. — Ryan O'Connor

ZF is a formal system. In ZF we prove that there exists a system, which we denote as 'the real number system' and we prove that it is a complete ordered field.

Aren't you beginning your proof with an assumption, that irrationals are numbers? — Ryan O'Connor

Yes, as I stated explicitly, "Supposing there is a real number x such that x^2 = 2."

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.

Why can't it simply be an algorithm? — Ryan O'Connor

If you show us your actual system for mathematics, then we could evaluate its heuristic advantages or disadvantages compared with classical mathematics. But just saying "it's an algorithm not a number" is a an informal thesis, not an argument.

My view is that we don't need to 'decide' every mathematical statement because 'undecided' is a valid state. — Ryan O'Connor

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.

My view is in total agreement with the foundations of calculus. — Ryan O'Connor

No it's not. Clearly.

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

This reminds me of the construction of the real numbers from Cauchy sequence of rational numbers. Equivalence classes are required though, because there are infinitely many approximations of the 'same' real number. For instance, consider f(n) = 1/n and g(n) = -1/n. Both converge to the rational number 0 and are representatives of the same real number 0. Errett Bishop went around the use of equivalence classes somehow. I can't remember how, and I'm not a specialist, but you'd probably really like the spirit of his constructive mathematics. 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. A mathematician can always retreat to formalism, etc.. I distinctly recall conversations with one mathematician who disliked philosophy altogether. In math (IMO), you really can know that you are correct if you don't mind not knowing what it is you are correct about. [In general, we don't know exactly what we are talking about, but math tempts us to forget that.]

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?

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.

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

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.

My standard response is that math is a formal game, like chess. A position is legal if and only if it follows from the rules, there's no right or wrong to it, nor any deeper reason.

But I must admit that math isn't really that simple. 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.

Bowling balls fall down, and that's forced on us too, but bowling balls are physical. Mathematical objects are purely abstract entities, yet the facts about them seem absolutely true independent of their discovery.

I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you?

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 hinted at this in my first reply to the OP. I'm amplify that answer here: We are social beings, profoundly interested in one another. We want to be respected, and for intellectual types that involves our words being respected.

Anyone can make up whatever philosophy or mathematics they like, but they are highly unlikely to be taken seriously, largely because they are highly unlikely to create anything impressive by starting from nothing (or rather by starting from inherited commonsense, a cage that the less uneducated won't even see as one.)

The broader question is: why is peer-review valued and important? Even a brilliant individual is just one little short-lived human being, likely too in love with themselves to be sufficiently self-critical. Envy and competition keeps people grudgingly honest, and thousands of minds together can cover far more intellectual terrain and see into one another's blindspots.

I haven't got a good answer. A lot of smarter people than I don't have a good answer either. Have you? — fishfry

There is a special field of study which delves into the nature of being, existence itself, and this is called metaphysics. Metaphysicians, being trained in this field, are best able to say whether something exists or not.

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

My ideas are so many steps away from a formal mathematical theory and in any one of those steps my ideas can be revealed to be inconsistent or useless. With that said, it's unreasonable to expect a formal theory to be perfected in isolation. And while my ideas are far from publishable standards, I do think they're at a level where they can be discussed on a forum. So please allow me to pick your brain, and please demolish my ideas. All I can guarantee to you is that you are dealing with someone incredibly receptive to criticism so the probability of going in circles is likely far less than what you expect.

please show your system of axioms and rules by which one may make an evaluation of such circumstances. — GrandMinnow

I don't have axioms for you and to be honest, I don't have the technical skills to do much of the mathematical heavy lifting. I'm an engineer not a mathematician nor a philosopher. But I still think I can contribute. When Descartes developed analytic geometry, I suspect that he didn't present axioms but instead presented an intuitive way of thinking that proved incredibly useful (I'm particularly referring to Cartesian coordinates graphs). I think I could do something similar.

I believe it was his work which catapulted us to our actual infinity point-based world-view. When we draw a graph, we think that it is completely filled with points. It is a 'whole-from-parts' view where a continuum is constructed from infinite points. But consider this alternate 'parts-from-whole' view. We start with a continuum, perhaps a 2D square whose dashed edges correspond to x=∞, x=-∞, y=∞, and y=-∞. This continuum has no points, only the 'pseudo-points' (∞,∞), (∞,-∞), (-∞,∞), and (-∞,-∞). Draw curve from (-∞,∞), passing through the interior, and ending at (∞,∞). Label this curve y=x^2-2. Next, draw a somewhat horizontal curve through the interior and crossing y=x^2-2 at 2 points. Label these points (?,0) and (?,0). Finally label the starting pseudo-point (-∞,0) and the ending pseudo-point (∞,0). This graph does not have the infinite points to give it a fixed geometry. Instead, it can be thought of a topological system, whereby the relationship between what was actually drawn is maintained through continuous deformations. This covers just an overview of where I'm coming from, but let me just say that so many paradoxes (especially Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity. Instead, this graph has limitless potential to be further refined by adding additional lines.

If it's an object, then it exists. At the very least, in formal terms, there is an existence theorem. — GrandMinnow

I argue that we cannot talk about existence without including a qualifier: actual or potential. An object can potentially exist. In my graph example, I can think of potentially infinite points that could be introduced should I decide to add more lines, but until I do so, those points only potentially exist. And so in the context of that graph the only number that actually exists is 0 (and perhaps 2 if you're including the function's definition).

The justification is from axioms from which we prove that there exists an infinite set. — GrandMinnow

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? For example, 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. I hope but cannot demonstrate that we can replace the actual infinities in ZF with potential infinities. Cantor was a 'Whole-from-Parts' guy so he built up set theory from points. What if we instead took a 'Parts-from-Whole' view?

the mathematics itself stands whether the mathematician regards it platonistically or not. — GrandMinnow

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.

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 think your comment is a result of me not communicating myself properly. All that I'm saying is that when I use pi precisely I do not evaluate it, I keep it as...pi. I consider this to be a real algorithm. But when I actually evaluate it to produce an actual number, the number that I produce is always a rational number.

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

Let's set this aside for now since I'm not in a good position to debate with you about what exactly is proven by the axioms.

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

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. With a 'Parts-from-Whole' view it's easy to talk about this stuff...they're just unmeasured potential objects. In a way, by embracing incompleteness nothing gets left out.

No it's not [in total agreement with the foundations of calculus]. Clearly. — GrandMinnow

Limits have a precise meaning in a 'Parts-from-Whole' view of calculus - they describe potentially infinite processes. If you want the area under a curve, approximate it with a set of smaller and smaller rectangles, to no end. There's no need to rationalize how lines can be assembled to create an object of area. In this example, if we work with potential infinity, the objects of study already have area, because we are not studying points...we are studying continua.

This reminds me of the construction of the real numbers from Cauchy sequence of rational numbers. — norm

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?

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

Thanks for the recommendations. Yes, there are a few experts (e.g. Norman Wildberger) who have problems with real numbers, but they're few and far between. His criticism is quite technical but I agree with you that it's a philosophical issue.

In general, we don't know exactly what we are talking about, but math tempts us to forget that. — norm

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.

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

My view is that we can only think objects into existence internally. For example, a computer program can simulate a reality where internal to that reality the flying spaghetti monster is real. But a computer cannot do any amount of computing to make the flying spaghetti monster real external to the simulation.

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

At what point did we prove that it was a number?

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

I like to think that there exist truths independent of consciousness, whether it's certain axioms of mathematics or the laws of nature. But I think '5 is prime' is a contingent truth...

thousands of minds together can cover far more intellectual terrain and see into one another's blindspots. — norm

I like that idea that together we have no blindspots.

Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover

Perhaps metaphysicians have an important voice, but I'm more inclined to say that philosophers of mathematics and philosophers of physics are best equipped on this matter.

Metaphysicians, being trained in this field, are best able to say whether something exists or not. — Metaphysician Undercover

Ahhhhh, so we shouldn't poll the general public as @Ryan suggests; nor the mathematicians, which I suggest; but rather the metaphysicians! Well that certainly makes a big difference. /s

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?

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

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.

But I think '5 is prime' is a contingent truth... — Ryan O'Connor

Well this I don't understand. Contingent on what? If there is a Platonic realm after all, surely mathematical truths live there if nothing else.

e.g. Norman Wildberger — 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 himin support of your ideas, since that can only weaken your argument.

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

No. The existence of an inductive set is specifically the content of the axiom of infinity. If all we wanted was a procedure for cranking out infinitely many numbers, Peano would suffice. 0 is a number and if n is a number then so is Sn, the successor of n. That gives us each of 0, 1, 2, 3, 4, ... without end.

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.

Also I'm not sure what you intend by writing, "actually exists." We only mean that an infinite set has existence within our theory. There's nothing "actual" about it, of course. Personally I doubt that any sets at all have actual existence. I can see the apple on my desk but I confess I don't see the set containing the apple. Sets are strictly an abstract formal system. Existence is relative to whatever the axioms say. Perhaps you didn't intend for "actually exists" to be different than, "mathematically exists," in which case never mind this paragraph.

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.

Ordinary calculus does use infinite sets. As I mentioned, one can provide finitistic or other alternative axiomatizations, but, as I said, to fairly evaluate the advantages and disadvantages of such alternatives, we would have to know really what those axiomatizations are. You should understand this point well: 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. Meanwhile, as best I can find, many finitistic alternatives are either much more complicated, harder to grasp, and possibly fail rigor.

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.

I'm not up to untangling all of your comments, but here are some points:

it's unreasonable to expect a formal theory to be perfected in isolation. — Ryan O'Connor

I didn't say that initially in a discussion the formal aspect has to be perfect. It just has to be reasonably coherent and credible.

When Descartes developed analytic geometry, I suspect that he didn't present axioms — Ryan O'Connor

I haven't read the mathematical papers of Descartes, but I suspect that he presented some basic principles and reasoned deductively from them. Then, over centuries, the deductive principles and methods of mathematics became more and more sharpened, as we eventually articulated the notion of formality as recusiveness and algorithmic effectiveness.

Zeno's paradox, derivative paradox, dartboard paradox) are not paradoxical with this view because this view is void of actual infinity — Ryan O'Connor

And neither is there a Zeno's paradox with set theoretic infinity.

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

We already have that concept. It's called 'recursive enumerability'.

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

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.

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

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.

Mr. Public: One of the drawbacks of the current vaccines is that you have to stick a needle in the arms of people and many people don't like that.

Scientist: True. So what's your alternative?:

Mr. Public: I think we can do it with pills instead.

Scientist: What's your formulation? Where are your trials?

Mr. Public: I don't have those. I'm just approaching it from philosophy.

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

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. A single practitioner can have an original idea, one that assumes mathematical existence at that moment.

Wiki: "Existence is the ability of an entity to interact with physical or mental reality."

On the other hand, it does take that kind of recognition to establish the importance of a mathematical idea in the mathematical community.

Once again, a topic of limited interest to practitioners. :smile: