You clearly do not understand, if you think that I accept the Fourier transform. I accept it as an example of an unresolved problem. — Metaphysician Undercover

I am surprised you've heard of it. Yet you don't agree that 2 + 2 and 4 refer to the same thing. In my prior conversations with you, you've convinced me that you utterly reject symbolic mathematical formalisms. And without those, there certainly aren't any convergent infinite trigonometric series. Those are very abstract gadgets. There's a mismatch in your level of discourse.

A couple of months ago you refused to except the notion of the finite field extension $\mathbb Q[\sqrt 2]$, the field of rational numbers adjoined with a symbolic square root of 2. That is a much simpler construction that Fourier series. It's illogical for you to complain about the one and then casually invoke the other to make some point.

And when that unresolved problem is united with the bad metaphysics of special relativity, the result is the uncertainty principle of quantum mechanics. — Metaphysician Undercover

But special relativity and physics in general need not be about metaphysics. "Shut up and compute," which they say about QM but which might just as well apply to relativity, special and general. Do theories explain the world, or just describe it's approximate behavior in our laboratory experiments?

We keep coming back to the same point. Nobody is making metaphysical claims except you. I agree that SOME scientists think their theories are True with a capital T, but I don't. You're fighting against someone's opinion that isn't mine.

Relativist seemed to be arguing that a metaphysician is better trained to do metaphysics than a physicist, yet there is some metaphysics, such as the metaphysics of time, which a physicist is better trained to do. — Metaphysician Undercover

I can't comment on these inside baseball conversations with other posters. I rarely even read my own posts, let alone anyone else's.

However, the uncertainty principle is clear evidence that physicists should leave the metaphysics of time in the hands of metaphysicians. — Metaphysician Undercover

Ok, fine. I stipulate that. Science isn't metaphysics, science is not ontology. What of it? I've been conceding you this point for days. You won't even acknowledge that I've said that, you just keep coming back with arguments as if I haven't said it.

Thanks for offering your take on this. I think this is exactly where the unresolved problem lies. It appears like the size of a chosen base unit might be completely arbitrarily decided upon. — Metaphysician Undercover

I was describing how Cantor stumbled upon the transfinite ordinals while studying the zeros of trigonometric polynomials. Not sure how this applies.

Yet the possible divisions are not arbitrary because divisibility is dependent on the size of the proposed base unit. Take 440 HZ as the baseline, for example. From this baseline, one octave (as a unit) upward brings us to 880HZ, and one octave downward brings us to 220HZ. So the higher octave consists of 440 HZ, and has different divisibility properties from the lower octave which consists of 220 HZ. This results in a complexity of problems in music. — Metaphysician Undercover

Sounds a little handwavy to me.

I ask you directly: Do you understand that I make no ontological or metaphysical claims for science? I make only the claim of accurate predictions to the limits of the experiments that we can do and the observations we can make.

You know, like when Eddington came back from photographing the 1919 eclipse and proved that the observations were consistent with Einstein's special theory of relativity and not Newton's theory of gravity. That's science. If you want to claim that this doesn't mean it's "true" in some metaphysical or absolute sense, I totally agree with you. So what's your point?

My understanding is that, for example, the Planck length is the length at which our current theories of physics break down and may no longer be applied. So that we can't sensibly speak of what might be happening below that scale. We can't say that reality is continuous or discrete; only that our current theories only allow us to measure to a discrete limit. — fishfry

But I am still confused about your conclusion. You're saying that a situation is ontological if there's no knowledge I could have that would settle the matter. Whereas I seem to mean something different. There's what we can know, and there's what really is. Two different things. — fishfry

I think I break the terms up differently from you.

To my understanding, you treat all that a mathematical model of something says as epistemic. Because a mathematical model is knowledge of a thing, what it predicts about that thing is knowledge of a thing. I agree with that. And I'm inclined to take another step; if a mathematical model of something is good, I'll accept what it concludes as if it were the thing. I treat good mathematical models as representational knowledge; and they represent the thing. Part of representation to me is being able to stand in for the thing when considering it.

So when the theory says; "it doesn't matter how much you sample about (blah), the variance of (blah) has a lower bound", I tend to treat it as being about (blah), rather than about our knowledge of (blah).

With statistics and averages, I'm less inclined to do this. A lot of randomness in statistical models is epistemic, and thus it can in principle be reduced by sampling. I'm happy treating that as as a fact about how the model's estimates relates to the sample, not a fact about the model the sample is being used to estimate things from. EG; samples of heights of people in Wales having a variance that can be arbitrarily reduced (in principle) by more sampling (epistemic) vs sampling from a signal in time space and that sampling strategy inducing a lower bound on the error in the frequency space regardless of the specifics of the sampling strategy (aleatoric).

But with aleatoric randomness, I'll treat it as a fact about the model. It doesn't matter what samples you do to estimate stuff from the model, the thing is gonna be random. I don't have a "physical intuition" or a "physical meaning" associated with these uncertainty principles when predicated of signals; as if I knew what they implied about reality. Some people seem to do this, interpreting an uncertainty principle about the modelled thing (blah) as meaning (blah) really is a distribution. Maybe it is, it seems useful to think that, but all I wanted to do here by calling it "aleatorically random" was to say that the uncertainty associated with an uncertainty principle is a model property rather than a sample property; it's about the (representation of) the thing, rather than about samples about the (representation of) the thing. In other words, its application is invariant of sampling, so it's about the model.

If you wanna call it "epistemic" because it's some content of the model, and don't want to let the model "stand in" for the thing like I'm inclined to, that's fine with me.

I am surprised you've heard of it. Yet you don't agree that 2 + 2 and 4 refer to the same thing. In my prior conversations with you, you've convinced me that you utterly reject symbolic mathematical formalisms. And without those, there certainly aren't any convergent infinite trigonometric series. Those are very abstract gadgets. There's a mismatch in your level of discourse. — fishfry

There's no mismatch in my discourse, you simply refuse to try and understand what I'm saying. I believe that two plus two equals four. I do not believe that "two plus two" and "four" refer to the same thing. Since you think that they refer to the same thing, you and I give "2+2=4" different meaning. We simply interpret this phrase differently. It is an ontological difference. So I reject some conclusions of mathematical formalism as unsound, based in unsound premises. This does not exclude me from taking a look at some of these unsound conclusions. Comparing unsound conclusions with what is really the case helps in the effort to produce better premises.

We keep coming back to the same point. Nobody is making metaphysical claims except you. I agree that SOME scientists think their theories are True with a capital T, but I don't. You're fighting against someone's opinion that isn't mine. — fishfry

The premises, axioms, theories, are metaphysical claims. whether you recognize this or not. I know we disagree on this, and you think that such premises might be based in something called "pure mathematics". but I explained to you in the other thread why this is an unsound principle itself. There is no such thing as "pure mathematics" in an absolute sense. Mathematics is ultimately guided by utility, and even those who might seem to be engaged in pure math are doing what they are doing (choosing whichever problems they choose to be working on instead of working on other problems) for a reason, so utility cannot be removed from mathematics.

Ok, fine. I stipulate that. Science isn't metaphysics, science is not ontology. What of it? I've been conceding you this point for days. You won't even acknowledge that I've said that, you just keep coming back with arguments as if I haven't said it. — fishfry

Do you recognize that scientists, in their scientific endeavours, regularly employ metaphysical principles?

Do you understand that I make no ontological or metaphysical claims for science? — fishfry

In saying that "2+2" and "4" refer to the very same thing, you make a metaphysical (ontological) claim.

I do not believe that "two plus two" and "four" refer to the same thing. — Metaphysician Undercover

Kant said the same thing :) Synthetic vs analytic

Do you recognize that scientists, in their scientific endeavours, regularly employ metaphysical principles? — Metaphysician Undercover

They most certainly do

saying that "2+2" and "4" refer to the very same thing, you make a metaphysical (ontological) claim. — Metaphysician Undercover

No, that's a psychological claim

I'm the rhetorical foil of last resort? I'm not sure how to take that — fishfry

No way! I just tagged you because you were "openminded and pluralistic about math" in the past with me

If Descartes's vortices are coming back that would be great news. Have you a specific link please? — fishfry

I only read it in the Stanford Encyclopedia of Philosophy on Descartes Physics. It said people are having renewed interest in it. I watched a video once that showed how his definitions of forces and reactions couldn't work on a billiard table. The guy said he had another video on Descartes "flawed" optics, but I couldn't find it and even the first video isn't around anymore. Sad

No, that's a psychological claim — Gregory

Any claim, which anyone makes, says something about one's mind, so they can all be said to be psychological claims. So that's really irrelevant. But to say that "4" refers to a thing, and that "2+2" refers to the very same thing, rather than that "4" and "2+2" have meaning (in which case one might see that the meaning of each differs), is to make an ontological claim.

Mathematics is ultimately guided by utility, and even those who might seem to be engaged in pure math are doing what they are doing (choosing whichever problems they choose to be working on instead of working on other problems) for a reason, so utility cannot be removed from mathematics. — Metaphysician Undercover

This is true only if "utility" includes fascination with exploring a subject, finding what's behind the next intellectual door, where an investigation might lead, etc. That's been my motivation for many years.

This is true only if "utility" includes fascination with exploring a subject, finding what's behind the next intellectual door, where an investigation might lead, etc. That's been my motivation for many years. — jgill

Yes, utility includes that, because there's always a reason why one explores one subject rather than another. I study for the very same reason, to find what's behind the next intellectual door, but I don't deny that there's always a reason why I head in one direction rather than another.

2+2 is two numbers making a process. One might think of this process when they think of four. The whole subject is stupid and Kant should not have made it an issue

No way! I just tagged you because you were "openminded and pluralistic about math" in the past with me — Gregory

Yes thank you. People often don't like math because it's taught in a dogmatic way. But mathematicians themselves are very openminded, at least after a few decades when the new ideas start to sink in!

I only read it in the Stanford Encyclopedia of Philosophy on Descartes Physics. It said people are having renewed interest in it. I watched a video once that showed how his definitions of forces and reactions couldn't work on a billiard table. The guy said he had another video on Descartes "flawed" optics, but I couldn't find it and even the first video isn't around anymore. Sad — Gregory

I'm not actually familiar with any of the details. I perused the SEP article but didn't see where it said people have renewed interest. May have missed it.

at least after a few decades when the new ideas start to sink in! — fishfry

Lol, true. It actually says in the first paragraph of the Stanford article "It is this unique amalgam of both old and new concepts of the physical world that may account for the current revival of scholarly interest in Descartes’ physics."

Lol, true. It actually says in the first paragraph of the Stanford article "It is this unique amalgam of both old and new concepts of the physical world that may account for the current revival of scholarly interest in Descartes’ physics." — Gregory

Oh from https://plato.stanford.edu/entries/descartes-physics/, different article than I was looking at.

So there is scholarly interest meaning historians of science? Or actual scientists? I'd be surprised if the latter.

Sean Carroll explicitly says this. He says the world is a brute fact of quantum fluctuation, using Russell's old phrase btw — Gregory

I responded to this yesterday but this afternoon I happened to come across this article by philosopher of science Massimo Pigliucci. He's debunking and ripping to shreds some bit of pseudoscientific woo from a guy named Klee Irwin, whose videos I've seen and who definitely strikes me as a crank. Pigliucci begins:

Physicists seem to be on a roll these days. Unfortunately, I’m not talking about a string of new discoveries about the fundamental nature of reality, but of a panoply of speculative notions ranging from the plausible but empirically untestable (and therefore non-scientific), such as Sean Carroll’s marketing of the many-worlds interpretation of quantum mechanics, to sheer nonsense on stilts, like the idea that is the subject of this essay.

Thought that was interesting in the context of our conversation.

https://medium.com/science-and-philosophy/the-universe-simulates-itself-into-existence-and-other-nonsense-from-modern-physics-32e958b690b

There's no mismatch in my discourse, you simply refuse to try and understand what I'm saying. — Metaphysician Undercover

On the contrary. I've made a concerted effort to engage with your ideas. I just can't discern any.

I believe that two plus two equals four. I do not believe that "two plus two" and "four" refer to the same thing. — Metaphysician Undercover

Well then you have no idea what a number is, what a representation is, and you're just flat out wrong both philosophically and mathematically.

Since you think that they refer to the same thing, you and I give "2+2=4" different meaning. We simply interpret this phrase differently. — Metaphysician Undercover

I could see that ... except that in all the times we've talked, I've never been able to understand what meanings you assign to the symbol "2 + 2" and "4". You have not succeeded in making yourself clear.

That you attribute this to my own lack of effort in trying to understand you reflects on your own lack of self-awareness regarding the incoherence of your position, whatever it is.

It is an ontological difference. So I reject some conclusions of mathematical formalism as unsound, based in unsound premises. — Metaphysician Undercover

You haven't made a coherent case that I could even agree or disagree with. You've made no case at all.

This does not exclude me from taking a look at some of these unsound conclusions. Comparing unsound conclusions with what is really the case helps in the effort to produce better premises. — Metaphysician Undercover

I say again: If you don't know that 2 + 2 and 4 refer to the same abstract entity, then you are in no position to have an opinion on Fourier series, which are far more sophisticated mathematical objects.

The premises, axioms, theories, are metaphysical claims. whether you recognize this or not. I know we disagree on this, and you think that such premises might be based in something called "pure mathematics". but I explained to you in the other thread why this is an unsound principle itself. There is no such thing as "pure mathematics" in an absolute sense. Mathematics is ultimately guided by utility, and even those who might seem to be engaged in pure math are doing what they are doing (choosing whichever problems they choose to be working on instead of working on other problems) for a reason, so utility cannot be removed from mathematics. — Metaphysician Undercover

You feel the same way about chess? Yet another of my arguments that you never bother to engage with.

Do you recognize that scientists, in their scientific endeavours, regularly employ metaphysical principles? — Metaphysician Undercover

Some do. Some just "shut up and compute," which is a well-known saying in QM. And Newton said, "I frame no hypotheses," and explicitly rejected metaphysical considerations in his theory of gravity.

In saying that "2+2" and "4" refer to the very same thing, you make a metaphysical (ontological) claim. — Metaphysician Undercover

No, I'm making a claim of Peano arithmetic, a purely syntactic system. I presented you a carefully crafted proof that 2 + 2 and 4 are the same thing. You pointedly ignored my argument and wouldn't even engage with my having presented it. You didn't just say, "Oh yeah well I have it on good authority that Giuseppe Peano cheated at cribbage." You just ignored what I wrote entirely and you continue to ignore it to this day.

You only showed your own ignorance and terror of symbolic reasoning. Yet you persist in caring about it. You had a bad experience with a math teacher along the way and it's scarred you for life.

I read half the Stanford article on Descartes Physics. This below is the important stuff so far. I have three more sections I will read tomorrow (I got tired tonight):

"Descartes explicitly deems motion to be a 'mode' of extension". author

"Foremost among the achievements of Descartes’ physics are the three laws of nature (which, essentially, are laws of bodily motion). Newton’s own laws of motion would be modeled on this Cartesian breakthrough" author

Descartes "incorrectly regards (uniform, non-accelerating) motion and rest as different bodily states, whereas modern theory dictates that they are the same state." author

"Descartes insists that the quantity conserved in collisions equals the combined sum of the products of size and speed of each impacting body." author

"a body which is at rest puts up more resistance to high speed than to low speed; and this increases in proportion to the differences in the speeds." Descartes

"change is always the least that may occur." Descartes. Therefore the author says "a body’s determination is apparently linked to its magnitude of speed."

"In his Optics, published in 1637, Descartes’ derivation of his law of refraction seemingly endorses this interpretation of determinations. If a ball is propelled downwards from left to right at a 45 degree angle, and then pierces a thin linen sheet, it will continue to move to the right after piercing the sheet but now at an angle nearly parallel with the horizon. Descartes reasons that this modification of direction (from the 45 degree angle to a smaller angle) is the net result of a reduction in the ball’s downward determination through collision with the sheet, 'while the one [determination] which was making the ball tend to the right must always remain the same as it was, because the sheet offers no opposition at all to the determination in this direction'"

Well i mustered the stamina to read the rest of the article. I don't feel like the author really gets Descartes (he is too into modern physics for that). Descartes was the first philosopher I got into, the first I really liked, as an adult. For him, there are spiritual things, extension (matter), and forces which live in extension. Here are the rest of the important stuff from the article:

"In order to better grasp the specific role of Cartesian force, it would be useful to closely examine his theory of centrifugal effects... Besides straight-line motion, Descartes’ second law also mentions the 'center-fleeing' (centrifugal) tendencies of circularly moving material bodies: 'all movement is, of itself, along straight lines; and consequently, bodies which are moving in a circle always tends to move away from the center of the circle which they are describing' (Pr II 39). At first glance, the second law might seem to correspond to the modern scientific dissection of centrifugal force: specifically, the centrifugal effects experienced by a body moving in a circular path, such as a stone in a sling, are a normal consequence of the body’s tendency to depart the circle along a straight tangential path. Yet, as stated in his second law, Descartes contends (wrongly) that the body tends to follow a STRAIGHT line away from the center of its circular trajectory. That is, the force exerted by the rotating stone, as manifest in the outward 'pull' on the impeding sling, is a result of a striving towards straight line inertial motion directed radially outward from the center of the circle, rather than a striving towards straight line motion aimed along the circle’s TANGENT." (my emphasis, so you can see where modern theory disagrees with Descartes)

The article also says: "If, for example, God removed the matter within a vessel (such that nothing remained), then the sides of the vessel would immediately become contiguous (but not through motion)". It is not through motion because Descartes had what he called the “first preparation for motion”. This is the center of his vortex theory of the universe, which for him acts as a unified mechanical whole. He didn't believe time existed outside us, so there are simply the turnings of the wheels, going back to the first motions. He didn't believe you can prove there is a God from nature. Nature simply works like a clock, and as for clocks (which he loved), they don't delineate anything that is outside our own brains

No, I'm making a claim of Peano arithmetic, a purely syntactic system. — fishfry

This is exactly my point, such axioms are based in ontological principles, they are not "pure mathematics. You can insist that there is no ontology to them all that you want, refusing to consider the evidence, in denial, that's a matter of your own free choice.

You pointedly ignored my argument and wouldn't even engage with my having presented it. — fishfry

Actually I demonstrated your faulty interpretation of the premise of extensionality. That two symbols refer to something of "equal" value is not sufficient for the conclusion that they refer to "the same" thing. Being the same implies being equal, but being equal does not imply being the same. You commit a fallacy of conversion.

You are the one in denial, insisting that mathematical axioms are exempt from judgements of true and false, being "pure mathematics", and absolutely abstract, refusing to accept the truth in this matter.

This is exactly my point, such axioms are based in ontological principles, they are not "pure mathematics. You can insist that there is no ontology to them all that you want, refusing to consider the evidence, in denial, that's a matter of your own free choice. — Metaphysician Undercover

PA is a formal symbolic system no different in principle than the game of chess.

Do you think chess has ontological significance? Yes or no? If no, then why do you think PA does?

Note that as usual I ask you direct, probing questions and you'll respond by changing the subject. I dare you to prove me wrong.

Actually I demonstrated your faulty interpretation of the premise of extensionality. That two symbols refer to something of "equal" value is not sufficient for the conclusion that they refer to "the same" thing. Being the same implies being equal, but being equal does not imply being the same. You commit a fallacy of conversion. — Metaphysician Undercover

Perhaps we're done for now. I tire of this game.

You are the one in denial, insisting that mathematical axioms are exempt from judgements of true and false, being "pure mathematics", and absolutely abstract, refusing to accept the truth in this matter. — Metaphysician Undercover

There is no middle 'e' in judgment. Jus' sayin' but nevermind . Axioms are formal statements, strings of symbols that are well-formed according to specific syntactic rules.

You can have different axiom systems such as Euclidean and non-Euclidean geometry, or Abelian and nonabelian group theory, that are mutually inconsistent yet both interesting and valid; and both applicable in their respective domains.

Therefore there can be no "truth" in axioms; only logical consistency and interestingness. You don't want to understand that, knock yourself out.

I'm taking a break from our chats but perhaps we'll meet again down the road.

It isn't an epistemological limit. — fdrake

Hi, this remark has been on my mind. It's totally counter to everything I think I know, so I wanted to make sure I understand you. This is in reference to whether we can say that Heisenbergian uncertainty is epistemological or ontological, with my strongly taking the position of the former.

I did ask you about this the other day when I replied to your post ... if I did. I remember replying but who knows. In any case, can you explain this more please? I've told a lot of people online that it's epistemological, so if I'm wrong I want to find out.

Being the same implies being equal, but being equal does not imply being the same. — Metaphysician Undercover

Depends on the contexts of usage. My friend and I are equal (in the eyes of the law), but we are not the same (in the eyes of the law).

This is distressing. I'm beginning to agree with MU . . . :worry:

Did you see my reply here? Link.

I remember a physics prof in university spending maybe 15 minutes scolding me when I said something like "the uncertainty principle says we can't know...", they responded "The uncertainty principle has nothing to do with how much we know about particles, it's not about our knowledge of the particles, it's about the particles" - not that exactly since it was a lot of years ago now, but that was definitely the gist. They were pretty mad at the suggestion it was epistemic, and their research was quantum theory, so I trust 'em.

Edit: I have another story like that which is pretty funny. We had an analysis lecturer that was extremely eccentric, and one of the masters theses they were willing to supervise was on space filling curves. They handily included a "picture of a space filling curve in a subset of the plane", which was just a completely black square. I asked another prof if the eccentric prof actually wrote out code to draw the space filling curve, since it was the kind of thing he'd do if he could. The other prof got pretty angry and said "Computers can't do that, it's noncomputable, the construction relies upon the axiom of choice!".

This is distressing. I'm beginning to agree with MU . . . — jgill

It's not that bad! There are lots of equivalence relations!

PA is a formal symbolic system no different in principle than the game of chess. — fishfry

That's not true, we went through this in the other thread. The application of the rules of mathematics always has a different purpose from the application of the rules of chess. If we want to apply mathematics toward understanding the universe, we need truth (in the sense of correspondence) in the axioms. Playing chess has a purpose of competing with another person for supremacy, within extremely limited conditions. If the rules of the game are designed to fulfill the purpose of the game, the people who have composed the rules must have kept that purpose in mind when creating the game. If not, there would be incoherence within the rules of the game, having been designed for different purposes.

If utility is removed from mathematical axioms, and they are composed simply for aesthetic beauty, then there will be random difference between one set of axioms and another, and real application would not be practical. This is not what we have in mathematics. Therefore we can conclude that mathematical axioms are not composed for aesthetic beauty.

If the purpose of mathematical axioms is simply utility, in an unconditional sense, then different mathematical systems will be inconsistent with others, depending on the purpose (the game) they are designed for. This means incoherency within the mathematical rules as a whole. This is what do we have in mathematics, as you yourself admit.

To produce consistency within the rules of mathematics as a whole, there needs to be one principle of utility, one purpose for which all the axioms are designed. Since mathematics is most widely applied in sciences we can look at the purpose of the axioms within science to get an idea of what that fundamental purpose might be. There are two distinct purposes which appear to me, one is the understanding of the universe, as mentioned above, and the other is for the prediction of events.

These two purposes are distinct. The former involves the bivalent logic of truth and falsity, while the latter involves probability. There is a fundamental incompatibility between these two, expressed in the law of excluded middle. When the axioms of one meet with the axioms of the other, paradox appears, as demonstrated by Zeno. This is because the logic of being (what is and is not) is inconsistent with the logic of becoming (what will be). The arduous task of the ontologist (metaphysician) is to determine the principles by which the two might be related to each other, to establish compatibility between them.

Note that as usual I ask you direct, probing questions and you'll respond by changing the subject. — fishfry

I'm changing the subject because your analogy has been demonstrated as completely insufficient and not irrelevant. Playing chess has a completely different purpose from applying mathematics.

There is no middle 'e' in judgment. Jus' sayin' but nevermind . Axioms are formal statements, strings of symbols that are well-formed according to specific syntactic rules. — fishfry

An axiom is a proposition which may or may not be accepted. That is the nature of an axiom. There are various reason why one might accept or reject a proposition. To say that an axiom must follow "specific syntactic rules" is one of these reasons, but clearly there are others.

Therefore there can be no "truth" in axioms; only logical consistency and interestingness. — fishfry

Obviously you have considered the acceptance and rejection of axioms from a very narrow perspective, without observance of the many real factors involved in this process.

Depends on the contexts of usage. — jgill

The attempt to unnecessarily restrict usage is fraught with problems. We are creatures of habit, and if habitual interpretation is different from the one imposed by a logical rule, the habit will often impose itself into interpretation of the logical conclusion in the form of equivocation.

So for example, we can define "equal" as "same", such that within this logical circle all uses of "equal" mean nothing other than "same", and all uses of "same" mean nothing other than "equal", but there would be absolutely no point to this. We could just use one of those words without losing anything. The only reason to use both, is if they have different intension. But that difference in intension requires that we determine the relationship between the two before we proceed with any logic procedure.

If in common usage, "equal" is the broader category, such that not all cases of being "equal" mean "the same", and all cases of being "the same" mean being "equal", then we ought to adhere to this in our logical definitions and proceedings to avoid possible confusion and equivocation. So there is a fundamental principle which we habitually recognize, and this is that two distinct things might be equal, but only one thing is the same as itself (law of identity). This distinguishes the uniqueness of a "thing" from any other similar or equal thing. But we commonly use "same" to refer to two distinct things which are of the same class, type, or category, and this is a completely different meaning of "same". If we equate "equal" with "same", we appeal to this other meaning of "same", which allows that two distinct things are the same. But then we are in violation of the law of identity.

It's not that bad! — fdrake

If agreeing with me was really that bad, I'd be afraid to leave my house.

I can concede that "2+2" and "4" are equal but not the same. They do, however, represent the same Platonic ideal. Not being a philosopher, this is as deep as I dare go into the subject. And being a (non-foundational) mathematician this is as deep as I need go. :cool:

How can they represent the same Platonic ideal when "+" represents an ideal in itself, which is part of "2+2", but not part of "4"?

How can they represent the same Platonic ideal when "+" represents an ideal in itself, which is part of "2+2", but not part of "4"? — Metaphysician Undercover

Irrelevant. 4-ness is the ideal in discussion. :roll:

I can concede that "2+2" and "4" are equal but not the same. — jgill

They are exactly the same set. If you have some mathematical framework in which 2 + 2 and 4 do not represent the exact same abstract mathematical object, I would appreciate your filling in the details. Even though your area of specialization was far removed from undergrad set theory, I'm sure you must have had some glancing acquaintance with that material at some time in the past.

↪jgill
How can they represent the same Platonic ideal when "+" represents an ideal in itself, which is part of "2+2", but not part of "4"? — Metaphysician Undercover

Ah! My friend @MU I believe I have achieved a glimmer of understanding your position. Let me see if I can say this back to you.

* First, there's what Plato said. Nevermind that he may or may not have been right about the ultimate nature of things. He's got a lot of mindshare over the millennia. But still, he's just a person who wrote down some thoughts in a context very different from ours. So you are saying that according to Plato things are such and so; but that's not necessarily the case.

* If we accept Plato for sake of discussion; then there's an ideal or a class or a category of thought called "plus" and another one called "2". And when you combine 2 + 2 to get 4, you are stating a mathematical equality but not a metaphysical one; because the left side of the equation 2 + 2 = 4 denotes the combination of two ideals and the thing on the right is only one ideal.

Well maybe. I think that point's a stretch. Plato could be wrong. But more to the point, 4 already includes within itself the possibility of being partitioned into 2 + 2. or 1 + 1 + 2, or 1 + 1 + 1 + 1. This is in fact the mathematical subject of partitions. It's what Ramanujan was working on inThe Man Who Knew Infinity. IMO doing a good job of explaining the partition function to a general Hollywood audience is one of the greatest math feats in cinematic history.

Point being that if 4 is an "ideal" or whatever you call it by itself, it ALREADY CONTAINS the possibility of all its positive integer partitions.

Truly, 2 + 2 and 4 are the same Platonic object. I don't find your argument convincing for this reason:

Sure, 2 + 2 expresses the fact that 2 and + can be combined to make 4. But 4 already expresses the fact that 4 can be represented as 2 + 2. Partitions are a natural and built-in aspect of a number.

Am I at least representing your position correctly?

If you have some mathematical framework in which 2 + 2 and 4 do not represent the exact same abstract mathematical object, I would appreciate your filling in the details. — fishfry

I'm speaking of the two symbols. 2+2 and 4 are not the same symbols. And I don't appreciate your snide remark. Of course they represent the same mathematical object. I'm not that far gone! They are "equal" in the sense they represent the math object, but are not the same symbols. :angry:

Did you see my reply here? Link. — fdrake

Thank you. I missed that. I'll reply separately.

I remember a physics prof in university spending maybe 15 minutes scolding me when I said something like "the uncertainty principle says we can't know...", they responded "The uncertainty principle has nothing to do with how much we know about particles, it's not about our knowledge of the particles, it's about the particles" - not that exactly since it was a lot of years ago now, but that was definitely the gist. They were pretty mad at the suggestion it was epistemic, and their research was quantum theory, so I trust 'em. — fdrake

My own sense is that the very last people in the world who have a clue about philosophical issues are the physicists. I've personally seen big time, well-known celebrity physicists, make elementary errors talking about infinity, or whether what they study is real. Most physicists, or at least many, actually think their theories are True in some absolute sense. Or more commonly, they don't even think about it at all. They just "shut up and calculate," which is very wise advice.

When it comes to metaphysics. physicists are the last people I'd listen to; and celebrity physicists the least of all :-)

Edit: I have another story like that which is pretty funny. We had an analysis lecturer that was extremely eccentric, and one of the masters theses they were willing to supervise was on space filling curves. They handily included a "picture of a space filling curve in a subset of the plane", which was just a completely black square. I asked another prof if the eccentric prof actually wrote out code to draw the space filling curve, since it was the kind of thing he'd do if he could. The other prof got pretty angry and said "Computers can't do that, it's noncomputable, the construction relies upon the axiom of choice!". — fdrake

I'm not sure what was funny about that except that it's perfectly computable and doesn't require choice at all. Did I understand that and/or get the math right? And on a practical level we could input the resolution of the printer or display device, and calculate exactly how many iterations of the curve would show up as solid black. And it would of course be a finite number, so definitely computable and not needing any mathematical foundations beyond counting to a large but finite number. That's way less than the Peano axioms. An ultrafinitist, someone who doesn't believe in the infinitude of sufficiently large sets, would be able to compute the space filling curve to the point that it appeared black on the display. I'd be willing to guess you don't need that many iterations. Your eye couldn't make out the lines, it would all black pretty soon.

Irrelevant. 4-ness is the ideal in discussion. — jgill

You've forgotten about summation. It might be the case that "4" represents 4-ness, but "2+2" represents a particular instance of the general rule of summation, not 4-ness.

Well maybe. I think that point's a stretch. Plato could be wrong. But more to the point, 4 already includes within itself the possibility of being partitioned into 2 + 2. or 1 + 1 + 2, or 1 + 1 + 1 + 1. This is in fact the mathematical subject of partitions. It's what Ramanujan was working on inThe Man Who Knew Infinity. IMO doing a good job of explaining the partition function to a general Hollywood audience is one of the greatest math feats in cinematic history.

Point being that if 4 is an "ideal" or whatever you call it by itself, it ALREADY CONTAINS the possibility of all its positive integer partitions.

Truly, 2 + 2 and 4 are the same Platonic object. I don't find your argument convincing for this reason:

Sure, 2 + 2 expresses the fact that 2 and + can be combined to make 4. But 4 already expresses the fact that 4 can be represented as 2 + 2. Partitions are a natural and built-in aspect of a number.

Am I at least representing your position correctly? — fishfry

I think you almost understand, but not quite. The symbol "4" represents a particular unity, if we adhere to Platonic idealism. Four is an object, a number. It may be the case that this number could be constructed through the summation process of 1+1+1+1, but this process is not the same thing as the object itself. Processes are not objects, they are activities which objects are engaged in. A cause is not the same as the effect. So it is ontologically incorrect to say that this process which creates the object "4" is the same thing as the object "4".

Evidence of this fact is that the object "4" may be created in an infinity of other ways, "6-2" for example. So, all these logical possibilities which are inherent within "4", cannot be the same as "4" because each one is itself a different process. The fact that a different process can be utilized to make an object indicates that the process is not the same thing as the object.

When it comes to metaphysics. physicists are the last people I'd listen to; and celebrity physicists the least of all :-) — fishfry

Eh, if it was someone else I wouldn't've trusted it. The guy lectured in quantum physics and philosophy of quantum physics. I'm sure that it can be doubted.

Most physicists, or at least many, actually think their theories are True in some absolute sense. — fishfry

I think they operate quite rightly, provisionally treating the theory as if it is the thing is part of how it works I think. If the discussion we'd have is "what properties of a model can be treated as standing in for a property or behaviour of the thing", that'd be quite different from "are all models merely epistemic" - the first would actually be about the uncertainty principle, the second is a much broader realism vs anti-realism of scientific content debate. If you and I have to go through the latter to get to the former, that's fine with me, both are interesting.