What would you say the meaning is? Just curious. — jgill

A quarter counterclockwise turn in the plane. That's the simple meaning. I was probably too harsh with my criticism of her video though, it's an excellent summary of the use of complex numbers in physical science. Just missed the mathematical essence IMO.

A quarter counterclockwise turn in the plane. That's the simple meaning — fishfry

That's what happens when multiplying a+bi by i.

I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. :nerd:

That's what happens when multiplying a+bi by i. — jgill

Right, that's the answer to "does the square root of -1 exist?" Just as the number 5 can be interpreted as stretching a line segment by five units; and multiplying by -1 preserves the length and reverse the direction of a line segment; multiplying by i rotates the segment a quarter turn counterclockwise. And if you do it twice in a row, you get the same effect as multiplying by -1. This in my opinion is what they should be explaining to every high school student. But they don't. And apparently they don't explain it to physicists either!

I play in the complex plane all the time, and I have always visualized figures and imagery and motion. Even created what might be considered art in the process. — jgill

So what does the L mean in your equation earlier? Not familiar to me.

So what does the L mean in your equation earlier? Not familiar to me. — fishfry

Give me a few moments. See the Gabriel's thread.

"There exists a unique x such that x^2 = 2." — GrandMinnow

The problem of course, being that it is debatable whether there is such an x.

E!x x^2 = 2

is a theorem of ordinary mathematics.

Anyway, I made my point that existence is not a predicate..

Actually, I don't think you have.. You simply used "exists" as a verb, and verbs refer to actions which must be predicated of a subject to say anything truthful. So "there exists..." really doesn't say anything meaningful because you haven't properly identified the subject referred to when you point with "there".

"There exists an object that has the property that its square is equal to 2" is perfectly fine English.

E!x x^ — GrandMinnow

Hello,

here the sight with layer logic instead of classical logic:

In layer logic you have to give a layer to the statement that should be possibly true:
In layer k the statement E! x^2 = 2 is true,
in another layer m the statement E!x x^2 = 2 may be also true, but with another x (say y).
So we can have x^2 = 2 is true in layer k and x^2 = 2 is false in layer m and y^2 = 2 in layer m.

Therefore the square root of 2 in layer logic is most probably not one irrational number
but many rational fractions in different layers (times).

Yours
Trestone

Isn't denying the existence of sqrt of 2 on the grounds that it isn't a computable number a bit like denying the existence of a "heap" of sand on the grounds that a "heap" isn't derivable from a granular definition of sand?

The "Sqrt 2" is at the very least, pragmatically useful as a moniker for the hypotenuse of a certain class of visually recognisable triangle, and it should be remembered that we have as much empirical justification for labelling the hypotenuse with sqrt 2 as we have for labelling its other sides with "1".

Zeno's reaction to his paradox is also similar to yours, in his conclusion that the existence of motion is impossible on the grounds that motion cannot be constructed from positional information. But the converse is also true: a position, in the logical sense, cannot be constructed by slowing down motion. Motion and position are irreconcilable concepts pertaining to mutually exclusive starting conditions of a system and mutually exclusive choices of disturbance of the system by an observer thereon after. Each concept can only informally represent the other as a "limit" that can only be approached but never arrived at.

The constructivist isn't forced into believing in the literal existence of hypertasks as the platonist might insists, rather the constructivist only needs to deny the existence of a universal constructive epistemological foundation.

I didn't write "E!x^". It doesn't make sense. I wrote "E!x x^2 = 2".

Hello GrandMinnow,

sorry, the citation at the beginning of my writing was an error and can be ignored.

Yours
Trestone

Isn't denying the existence of sqrt of 2 on the grounds that it isn't a computable number — sime

Didn't read back to find the source of the quote, but sqrt(2) is certainly computable. For example you can use a standard iterative procedure.

Why would you think that someone who has not studied existence would know as much about existence as someone who has studied existence? — Metaphysician Undercover

Because there are SOME subjects in which more study implies more knowledge; and others where it doesn't.

If you study more math you know more about math than someone who doesn't study math. Medicine, physics, and history are in this category.

But take, say, baseball. A non-athlete spends his life studying the game. Reads books on strategy and tactics, knows all the statistics, can name all the players on the 1928 Philadelphia Athletics and their batting averages. Another person knows nothing of the history of the game but has been playing all their life and has been toiling in the minor leagues for years (think Bull Durham). Who knows more about the game? I would say the practitioner and not the student.

Existence is the same. If someone's been existing for a few decades they know as much about existence than a philosopher. The philosopher knows the history of what great thinkers have written about the subject. But philosophy does not confer actual knowledge of its subject; only knowledge of what others have said.

Philosophers don't necessarily lead better lives than others, nor are they more moral, and they most definitely don't know any more about existence than the general public.

In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence.

yes, sorry, i should have clarified that i was referring to number in the extensional sense, i.e. as an obtainable state of a calculation. This is because i understood the OP as questioning the existence of sqrt 2 in this extensional sense.

Naturally, Sqrt 2 exists intensionally in the constructive sense of an algorithm that generates any finite cauchy sequence of computable reals, whose limit x fulfils the axiomatic definition

sqrt 2:= x : x^2 = 1.

Furthermore, it is constructively provable that the limit is irrational in the sense of being separated from any computable rational number. And the sqrt of 2 intensionally exists in this sense, irrespective of whether the limit of this process of calculation is axiomatically accepted as being a number in the extensional sense.

Of course, normally when we assign numbers to physical lengths we aren't resorting to logical construction, nor even necessarily to calculation, which means that our practical use of numbers is logically under-determined. We could for instance, declare the hypotenuse of unit length triangles to be "real" lengths that correspond to extensional numbers relative to some novel logical construction of numbers, in which the unit lengths of the other sides of the triangle only exist intensionally as limits in this number system.

In my opinion, constructive mathematics should permit a plurality of axiomatic systems to represent our use of numbers. That way we don't have to make arbitrary a priori decisions as to which numbers only exist as limits.

Philosophers don't necessarily lead better lives than others, nor are they more moral, and they most definitely don't know any more about existence than the general public.

In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

:up:

"There exists an object that has the property that its square is equal to 2" is perfectly fine English. — GrandMinnow

I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated.

Existence is the same. If someone's been existing for a few decades they know as much about existence than a philosopher. The philosopher knows the history of what great thinkers have written about the subject. But philosophy does not confer actual knowledge of its subject; only knowledge of what others have said. — fishfry

All I can say is, wow! This is an unbelievable opinion coming from an educated person like yourself. Would you also say that a person who has been breathing for a few decades knows as much about breathing as a biologist?

So going to university and studying a subject of study only provides one with what other great thinkers have said about that subject, but it doesn't provide one with any knowledge of the subject? It only provides one with what those who've studied that subject, say about the subject? What about studying mathematics, wouldn't this be the same thing? Studying mathematics doesn't provide any real knowledge of mathematics, only what others who have studied the subject say about the subject. What do you think knowledge of a subject consists of, if not what those who study the subject say about the subject?

In particular, a philosopher who knows hardly anything about mathematics is in no position whatsoever to comment on mathematical existence. Many philosophers of mathematics are in this position. They simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

The problem with this perspective is that "mathematical existence" means something completely different than "existence" in the philosophical sense. The op does not ask about "mathematical existence", it asks about "existence". If it asked about the mathematical existence of irrational numbers there would be nothing to discuss. Clearly irrational numbers are used by mathematicians therefore they have mathematical existence.

The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. That would be self-evident. So mathematicians who hardly know anything about existence, yet think they do because they know something about mathematical existence really seem to have very little to say about the philosophical question of whether certain mathematical objects which obviously have mathematical existence, have existence.

I didn't say it isn't perfectly fine English. I said you haven't properly identified the subject signified with "there", to which "exists an object" is predicated. — Metaphysician Undercover

Whatever you have in mind linguistically is irrelevant since the sentence is linguistically perfectly correct.

Even more simply:

There is a unique object whose square is 2.

or

A unique object exists such that its square is 2.

or

An object exists such that its square is 2 and no other object is such that its square is 2.

or

A unique object exists such that it has the property that its square is 2.

Etc.

All perfectly grammatical and sensible English.

[Many philosophers of mathematics] simply don't know enough math to comment intelligently on the subject of mathematical existence. — fishfry

My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind?

I don’t regard professional mathematicians as being trustworthy or helpful with regards to the literal truth of their subject, for several reasons.

1. They can be expected to exhibit a political bias towards inflating ontological claims in mathematics , in the same way that professional chess players can be expected to inflate the importance of chess and exhibit bias against other games. Traditionally this is bias is evident in the continued prevalence of classical logic in the justification of mathematical claims and the high percentage of mathematicians who are platonists, a position that is troubling to any engineer or computer scientist who unlike pure mathematicians have actual responsibilities regarding the physical actualization and interpretation of mathematical results.

2. The subject of ontological claims and commitments is the domain of logic rather than of mathematics, and the classical mathematician isn’t logic savvy, unlike today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset.

The problem with this perspective is that "mathematical existence" means something completely different than "existence" in the philosophical sense. — Metaphysician Undercover

I agree.

The op does not ask about "mathematical existence", it asks about "existence". — Metaphysician Undercover

I believe I asked the OP what they meant by existence and don't believe I've gotten an answer.

If it asked about the mathematical existence of irrational numbers there would be nothing to discuss. Clearly irrational numbers are used by mathematicians therefore they have mathematical existence. — Metaphysician Undercover

Ok then you are in agreement with my point. I only speak of mathematical existence. But why the square root of 2? How about the number 3? That has no more claim on existence than sqrt(2). That's why the OP is confused. They think 3 exists but not sqrt(2). I think they both have mathematical existence, and as far as "existence existence," I'd like to see a coherent argument one way or the other. I take no position at all. Clearly numbers don't have the same claim to existence as rocks or fish.

The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. — Metaphysician Undercover

This thread hasn't even begun to touch on the subtleties of that subject. I've seen no decent arguments one way or the other. And if that's what the OP really cared about, they'd have asked if 3 exists. Once you bring in sqrt(2) you are talking about mathematical existence.

That would be self-evident. So mathematicians who hardly know anything about existence, yet think they do because they know something about mathematical existence really seem to have very little to say about the philosophical question of whether certain mathematical objects which obviously have mathematical existence, have existence. — Metaphysician Undercover

It's "above their pay grade" as Obama would have said. So make an argument. Do you think 3 exists? Of course the positive integers have a pretty good claim on existence because we can so easily instantiate the smaller ones. So how about $2^{\text{googolplex}}$? That's a finite positive integer that could in theory be instantiated with rocks or atoms, but there aren't that many distinct physical objects in the multiverse. So make an argument, say something interesting about this. Forget sqrt(2). Do you think that extremely large finite positive integers exist?

https://en.wikipedia.org/wiki/Googolplex

My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind? — GrandMinnow

Professionals, yes. Non-professionals (forum participants, for example) a lot weaker.

today’s generation of mathematics undergraduates who are studying mathematics using theorem provers from the outset. — sime

What's a "theorem prover"? Computer program? A tutor?

My rough impression is that professionals in the field of philosophy of mathematics usually do know about mathematics. Which philosophers in, say, the last 85 years do you have in mind? — GrandMinnow

Mostly the ones here. The famous philosophers of math know a lot of math, like Putnam and Quine and so forth. I may have overstepped a bit. Having just stuck my contrary toe in a political thread I think I've had enough for tonight. You know when you say something contrary to the dominant narrative, people don't just respond rationally. They come with insults and non sequiturs. It's like the 1950's all over again and I just came out as a Commie.

What's a "theorem prover"? Computer program? A tutor? — jgill

More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving.

Lean with Mathlib, to my understanding is a state of the art approach to logic and mathematics programming that embodies the above principles , an approach which together with advances in deep learning theorem proving will undoubtedly revolutionise mathematics education , automation and research. Remarkably, the mathematics library of lean now codes the proofs of a lot of undergrad level mathematics.

https://leanprover.github.io/live/latest/

But why the square root of 2? How about the number 3? That has no more claim on existence than sqrt(2). — fishfry

This is doubtful, and seems to contradict the rest of your statement. If we're talking mathematical existence, I do not think that natural numbers have more claim to existence than irrational numbers. In fact, I think the opposite is more likely the case. "The natural numbers" were in use prior to the Pythagoreans who are supposed to have demonstrated the "existence" of irrationals. So it was only by the work of the Pythagoreans that "existence" was assigned to numbers, and existence was stipulated in order to provide reality for the irrationals. Prior to this the natural numbers were in use without any assumptions that numbers exist, so the naturals are lacking in the claim of existence. There is no need for them to be stipulated as existing.

If we're talking "existence" in the philosophical sense, we'd have to first agree as to what existence means before we might judge whether one type of number has more of a claim to existence than another. If we do not find a definition of "existence" which allows that numbers exist in the first place, then the suggestion is meaningless.

Clearly numbers don't have the same claim to existence as rocks or fish. — fishfry

Why not? I don't see the reason for approaching the question with such a bias. It will only make a true answer more difficult to find. Plato demonstrated the pitfalls of this bias thousands of years ago. It is a mistake to assign a higher degree of being to something apprehended through the senses over something apprehended directly with the mind.

This thread hasn't even begun to touch on the subtleties of that subject. I've seen no decent arguments one way or the other. And if that's what the OP really cared about, they'd have asked if 3 exists. Once you bring in sqrt(2) you are talking about mathematical existence. — fishfry

This again shows some sort of bias toward natural numbers over irrational numbers. If "3" represents a number, and "sqrt(2)" represents a number, then why assume that the question is better asked of "3" than of "sqrt(2)"? That's just admitting that "3" is in some sense a better representation of a number than "sqrt(2)", and in doing this you undermine the concept of "mathematical existence". If some numbers have a better, or more valid "mathematical existence" than others, then there must be ambiguity within the concept which could allow for equivocation.

It's "above their pay grade" as Obama would have said. So make an argument. Do you think 3 exists? Of course the positive integers have a pretty good claim on existence because we can so easily instantiate the smaller ones. So how about 2googolplex2googolplex? That's a finite positive integer that could in theory be instantiated with rocks or atoms, but there aren't that many distinct physical objects in the multiverse. So make an argument, say something interesting about this. Forget sqrt(2). Do you think that extremely large finite positive integers exist? — fishfry

As I said, there's really no point in making an argument as to the existence or nonexistence of something unless we have a workable definition of "existence". That's why the thread really won't get anywhere because all the members in this forum have wide ranging biases about what constitutes "existence", and a relatively small number of them have any formal training in this subject, so it will end up with people arguing to support their own prejudices.

I would be inclined to define "existing" as having either temporal or spatial extension.

More generally, it is a dependently typed logical programming language, with clause resolution and other rules of logical inference, together with SAT solvers, methods of analytic tableaux and heuristics for automated or interactive theorem proving — sime

Thanks for the information. From my perspective the project sounds dreadful, but for coming generations it may become standard. It's given me a moment to reflect on a theorem I am putting together and proving at present. The intuition and imagination I have used to both design the theorem, then prove it, presumably in the future could be generated in some computerized fashion. But of course I don't see how. :chin:

As for simply formatting the proof of a theorem in a computer language I might be able to do that in a version of BASIC. But why would I?

the square root operation is closed over real numbers — TheMadFool

I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. I don't deny the value of irrationals or question the centuries of work on calculus, I'm just questioning whether we actually know that irrationals are numbers.

The op is asking a philosophical question about the existence of certain mathematical objects, not whether those mathematical objects have mathematical existence. — Metaphysician Undercover

A few have pointed to my original question so let me try to clarify. My view of existence encompasses mathematical existence. While I have doubts about the existence of irrational numbers, I don't doubt the value of our descriptions of irrational 'numbers' (e.g. Dedekind Cuts, Cauchy Sequences, infinite series, and so on). I know this seems like a contradictory statement since conventionally they're all equivalent. But I see an infinite series as an algorithm (described completely with finite characters) which if executed to completion would output an irrational number (described completely with infinitely many characters). Since that output could never be generated, the output (the number) cannot exist but the algorithm certainly can, for example on my laptop. This is why I think the number 3 can exist but not the 'number' sqrt(2). We never actually work with irrational 'numbers', we only work with their algorithms or rational number approximations. So why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with?

I think one issue is that some non-computable irrational 'numbers' don't have algorithms...

I can accept that the square root operation is closed over the 'reals', but that doesn't mean it's closed over the real numbers. — Ryan O'Connor

'The reals' means 'the real numbers'.

We construct the set of real numbers. It doesn't make sense to debate whether a real number is a number. Mathematics doesn't have a universal definition of 'number'. Mathematics doesn't really involve questions of what is or is not a number. Instead there are many different number systems. Each number system has its carrier set. And we may ask whether certain objects or are or not in the carrier set of a given number system. The carrier set of the real number system is the set of real numbers (sometimes just called 'the reals). Every real number is a number in the sense that it is in the carrier set of the real numbers system.

A version of this information was provided to you earlier in this thread.

Cauchy Sequences — Ryan O'Connor

Equivalence classes of Cauchy sequences. This has been mentioned to you previously in this thread

why do we even need to assume that irrational 'numbers' exist? Why not assume that irrationals are the algorithms that we actually work with? — Ryan O'Connor

(1) We don't assume they exist. We prove they exist.

(2) I imagine that instead of constructing the real numbers, we could instead take the carrier set to be the set of algorithms. But limiting to only computable reals makes calculus a lot more complicated. Since calculus works with the irrationals included, and since it is more complicated for the calculus to regard the irrationals as algorithms and not as real numbers, we must weigh the advantages and disadvantages of both approaches. In particular, if we ask 'What is the length of the diagonal of a unit square?' we may answer 'sqrt(2)' rather than have to say 'Well, I can't tell you except there is an algorithm that computes successive approximations. Tell me what degree of accuracy would your like your approximation to be, and I'll tell you the answer to that degree of accuracy.' And I would say, 'Thanks, but I got a more succinct answer from the mathematician who said it is sqrt(2).'

And, by the way, if you show me an algorithm, then I may ask, 'What does it compute?' And how would you say what it computes without already having in mind that it computes approximations of ... wait for it ... sqrt(2). So, to get me sold on your algorithm, you would already have to presuppose that there is a thing that it approximates - and that thing is ... wait for it .. sqrt(2).

(3) There are proposed systems in constructivism, computationalism, and predicativism that may very well satisfy your desiderata. You only need to first inform yourself of the basics of the subject.

Most of this also has been mentioned to you previously in this thread.

algorithm (described completely with finite characters) which if executed to completion — Ryan O'Connor

Algorithms that execute to completion do so in a finite number of steps. As far as I know, what you may have in mind is not an algorithm but rather it is a supertask. I am not well informed about supertasks, so I don't know whether there is a rigorous mathematical definition of the notion or whether the notion is purely philosophical.