In general I have found that working over formalisms is one necessary part of developing understanding for a topic; don't just read it, fight it. Follow enough syllogisms allowed by the syntax and you end up with a decent intuition of how to prove things in a structure; what a structure can do and how to visualise it. — fdrake
Developing such anchors and being able to describe them seems a necessary part of learning mathematics in general; physical or Platonic grounding deflates this idea by replacing our ideas with actuality or actuality with our ideas respectively. In either case, this leaves the stipulated content of the actual to express the conceptual content of mathematics without considering how the practice of mathematics is grounded in people who use mathematics and whether that grounding has any conceptual structure. — fdrake
No, division and multiplication are not at all symmetrical, because you never have a remainder in multiplication. In multiplication, you take a designated number as the "base unit", a designated number of times, and you never end up with a remainder. You have no such "base unit" in division, you have a large unit which you are trying to divide down to determine the base unit, but you often end up with a remainder.
Evidence of this difference is the existence of prime numbers. These are numbers which we cannot produce through multiplication. We can still divide them, knowing there will be a remainder, but that doesn't matter, because there's often a remainder when we divide, even if the dividend is not prime. — Metaphysician Undercover
[The] impossible synthesis of assimilation and an assimilated which maintains its integrity has deep-rooted connections with basic sexual drives. The idea of "carnal possession" offers us the irritating but seductive figure of a body perpetually possessed and perpetually new, on which possession leaves no trace. This is deeply symbolized in the quality of "smooth" or "polished." What is smooth can be taken and felt but remains no less impenetrable, does not give way in the least beneath the appropriative caress -- it is like water. This is the reason why erotic depictions insist on the smooth whiteness of a woman's body. Smooth --it is what reforms itself under the caress, as water reforms itself in its passage over the stone which has pierced it....It is at this point that we encounter the similarity to scientific research: the known object, like the stone in the stomach of the ostrich, is entirely within me, assimilated, transformed into my self, and is entirely me; but at the same time it is impenetrable, untransformable, entirely smooth, with the indifferent nudity of a body that is beloved and caressed in vain. — Sartre
On paper you produce "a representation" of the Euclidean ideals. That representation is something completely different from the square root, which is part of the formula behind the representation which you draw on paper. When I want to lay out a square corner, a right angle, on the ground, I might use a 3,4,5, triangle. In this exercise I am not using a square root at all. I could make this square corner without even knowing the Pythagorean theorem, just knowing the lengths of 3,4,5. But if one side of the right angle is to be 5, and the other side 6, I'll need to know the Pythagorean theorem, and then figure the diagonal as the square root of 61 if I am going to make my right angle. — Metaphysician Undercover
That's not quite right. We, as human beings, cannot necessarily distinguish two distinct things, due to our limited capacities of perception and apprehension. So it's not quite right to say that you can always distinguish a thing from all other things. A thing is distinct from other things, but we cannot necessarily distinguish it as such. And that difference may be a factor in quantum mechanics — Metaphysician Undercover
Right, but to perceive a thing, name it "X", and then claim that it has the "identity" of X, is to use "identity" in a way inconsistent with the law of identity. You are saying that the thing's identity is X, when the law of identity says that a thing's identity is itself, not the name we give it. The law says a thing is the same as itself, not that it is the same as its name. — Metaphysician Undercover
Consider that human beings are sometimes mistaken, so it is incorrect to say "the name is a reference that identifies always the same concrete object". The meaning of the name is dependent on the use, so when someone mistakenly identifies an object as "X", when it isn't the same object which was originally named "X", then the name doesn't always identify the same concrete object. And, there are numerous other types of mistakes and acts of deception which human beings do, which demonstrate that the name really doesn't always identify the same concrete object, even when we believe that it does. — Metaphysician Undercover
Do you recognize that Einstein's relativity is inconsistent with Euclidian geometry? Parallel lines, and right angles do not provide us with spatial representations that are consistent with what we now know about space, when understood as coexisting with time. My claim is that the fact that the square root of two is irrational is an indication that the way we apply numbers toward measuring space is fundamentally flawed. I think we need to start from the bottom and refigure the whole mathematical structure. — Metaphysician Undercover
Consider that any number represents a discrete unit, value, or some such thing, and it's discrete because a different number represents a different value. On the other hand, we always wanted to represent space as continuous, so this presents us with infinite numbers between any two (rational) numbers. This is the same problem Aristotle demonstrated as the difference between being and becoming. If we represent "what is" as a described state, and later "what is" is something different, changed, then we need to account for the change (becoming), which happened between these two states. If we describe another, different state, between these original two, then we have to account for what happens between those states, and so on. If we try to describe change in this way we have an infinite regress, in the very same way that there is an infinite number of numbers between two numbers. — Metaphysician Undercover
If modern (quantum) physics demonstrates to us that spatial existence consists of discrete units, then we ought to rid ourselves of the continuous spatial representations. This will allow compatibility between the number system and the spatial representation. Then we can proceed to analyze the further problem, the change, becoming, which happens between the discrete units of spatial existence; this is the continuity which appears to be incompatible with the numerical system. — Metaphysician Undercover
Right, my argument is that there is no such thing as an abstract object represented by "2" — Metaphysician Undercover
It was clear even then that the real numbers had a certain magnificent unreality or ideality. — mask
When I studied some basic theoretical computer science (Sipser level), I saw the 'finitude' of now relatively innocent computable numbers like pi, — mask
It's basically ridiculous to do philosophy of math without training in math: sex advice from virgins, marital advice from bachelors. — mask
I always follow your posts. You know much more set theory than me, so I learn something. — mask
I don't know if there is a way to express the same theory with similar results on first approximation making use only of mathematics based on integral fields. But even if there is a way, I suspect that it would become an extremely complex theory, impossible to use in practice. — Mephist
I couldn't do calculus integrals for beans, but I took naturally to Zorn's lemma. Did you know that the proposition that every vector space has a basis, is fully equivalent to the axiom of choice? Isn't that wild? — fishfry
It's very doubtful to me that such a structure has an analog in the physical world. And if it did, it would be quite a surprise. — fishfry
Mathematicians have a tongue-in-cheek saying: The imaginary numbers are real; and the real numbers aren't! — fishfry
It's an extremely widely held false belief that pi encodes an infinite amount of information, when it of course does no such thing. Bad teaching of the real numbers in high school is the root cause of this problem. Whether there is a solution that would serve the mathematical kids without totally losing everyone else, I don't know. — fishfry
The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world. — Mephist
f you consider geometric spatial figures as real physical objects, there are a lot of "problems" with them: first of all, they are 2-dimensional (or 1-dimensional, if you don't consider the internal surface), and all real physical objects are 3-dimensional. — Mephist
They are not real objects, and there is no problem with the distinction between finite or infinitesimal distances: it works even if you consider space-time as discrete. In fact, in practice it's very common in GR simulations to approximate space-time as a 4-dimensional discrete grid of points. — Mephist
The main point to keep in mind with physical models is that they don't have to be considered the real thing: they simply have to WORK as the real thing. — Mephist
Now, if you think that the distinction between measures expressed with rational or with real numbers is essential in your theory (represents some important characteristics of the real physical space), I don't see any other way other than making lengths become discrete at the microscopical level. — Mephist
Why should this background of mathematics remain a secret? And is it merely aesthetic in nature (a consideration of mathematical beauty alone)? — fdrake
OK, division and multiplication are not symmetrical for integers, because integers are "quantized": you can't give one candy to three children, because candies are "quantized". But physical space is not quantized, or is it? The mathematical description of continuous measures is not inconsistent: there are several ways to make them at least as consistent as natural numbers are.
So, if integers (quantized) objects exist in nature, why shouldn't continuous objects exist? — Mephist
So is the 3,4,5 triangle really straight or not? I don't understand... — Mephist
OK, so what can I do with identities?
If I cannot refer to them with names, I would say that it's impossible to speak about identities. So, they surely cannot be used in logic arguments. Logic is basically manipulation (operations) of language, isn't it? — Mephist
But Einstein's relativity is based on differential calculus and real numbers. How can it be correct, if the whole system is wrong? — Mephist
OK, continuous change cannot be identified by a finite number of steps. But does this prove that continuous change cannot exist? — Mephist
As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction. — fishfry
Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago. — fishfry
Question: I get that you do not believe in the ontological existence, however you personally define that, of 2–√2. My question is:
Do you believe in the mathematical existence of 2–√2?
If you say yes, then our disagreement is over whether mathematical existence is sufficient for ontological existence.
If you say no, then our disagreement is whether 2–√2 has mathematical existence.
So, do you think 2–√2 has mathematical existence; even though you maintain that's not sufficient justification to grant it ontological existence as you define it? — fishfry
Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence.All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence". — Metaphysician Undercover
In this mode, the stuff of the proof itself is the medium of thought — mask
The mathematical system being employed premises that a symbol represents an object, and that each time the symbol appears within an expression, like an equation, it represents the very same object. Any conclusions produced must uphold this premise. — Metaphysician Undercover
Right, clearly there are "problems" if we represent mathematical figures as real objects. Notice I removed your qualifier, "physical" objects. If we begin with a statement as to the nature of an "object", a definition, such as the law of identity, then we must uphold this definition. If the claim is that a "mathematical object" is fundamentally different from a "physical object", such that the same definition of "object" cannot apply to both, then we need to lay out the principles of this difference so that equivocation can be avoided. — Metaphysician Undercover
The problem is that any such "grid of points" is laid out on a spatial model. If a square is an invalid spatial model, then so is the Cartesian coordinate system Then "space-time" itself is improperly represented. — Metaphysician Undercover
Pragmatism is not the answer, it is the road to deception. Human objectives often stray from the objective of truth. When we replace "the truth" with "they simply have to work", we allow the deception of sophism, because "what it works for" may be something other than leading us toward the truth.. — Metaphysician Undercover
Thirdly, we'd need some principles to relate the continuous to the quantized. For example, to me time appears to be continuous, and space appears to be quantized. If this is the case, then we need different principles for modeling time than we do for space, and some principles to relate these two systems to each other. — Metaphysician Undercover
Thanks for the reference! I took a quick look at the book (just a quick look at the equations, really) and the first think that I thought is: what's the difference? — Mephist
But in this case, the two formulations are completely equivalent (in the sense of equivalence of categories, if you see theories as functors from formal systems to models), and from the point of view of physics the choice between two equivalent representations doesn't make any difference. — Mephist
And probably, after Voevodsky, it doesn't make much difference even from a mathematical point o view. — Mephist
As I said I find it nihilistic because you must then reject all of the modern world that sprung from that basic act of abstraction.
— fishfry
This is not the case. To reject that "the abstraction" exists as an object does not require that I reject abstraction. What I reject is any instance where an abstraction is presented as an object. — Metaphysician Undercover
Now that you mention it, that makes perfect sense relative to your neo-Pythagoreanism. By that I mean that you still profess to be "Shocked, shocked, I tell you!" at the fact that the square root of 2 is irrational. The rest of the world got over that a long time ago.
— fishfry
I'm not "shocked" at the fact that the square root of two is irrational, what shocks me is that the rest of the world got over this. — Metaphysician Undercover
I'd answer this, but I really don't know what you would mean by "mathematical existence". — Metaphysician Undercover
Many things can be expressed mathematically, but what type of existence is that? — Metaphysician Undercover
I suppose the short answer is no. The symbol √2 does not stand for anything with real "existence". — Metaphysician Undercover
I agree that is has a large amount of mathematical significance, and it is quite important mathematically, so the symbol definitely has meaning, but I don't think I'd agree that the symbol stands for anything which has "existence", in any proper sense of the word. — Metaphysician Undercover
All existence is "ontological existence" so it makes no sense to try and separate "mathematical existence" from "ontological existence". — Metaphysician Undercover
I had heard that, but never studied the proof. It is indeed wild. — mask
At my school we only had to learn the set theory that comes with analysis and algebra. — mask
I did look into ordinals on my own. I remain impressed by the usual Von Neumann constuction. I used it in visual art and I also think it has a philosophical relevance. It's a nice analogy for consciousness constantly taking a distance from its history. 'This' moment or configuration is all previous moments or configurations grasped as a unity. It works technically but also aesthetically. — mask
That would be surprising indeed. I think we agree on the gap between math and nature. — mask
As you mention, our measurement devices don't live up to our intuition and/or formalism. I have a soft spot for instrumentalism as an interpretation of physical science. — mask
I haven't heard that one. But I know a graph theory guy who thinks the continuum is a fiction and an analyst who believes reality is actually continuous. Another mathematician I know just dislikes philosophy altogether. — mask
I like philosophy more than math when I'm not occasionally on fire with mathematically inspired, though I have spent weeks at a time in math books, obsessed. (At one point I was working on different models of computation, alternatives to the Turing machine, etc. Fun stuff, especially with a computer at hand.) — mask
Right! Because of the infinite decimal expansion. One of my earliest math teachers had lots of digits of pi up on the wall, wrapping around the room. Some of the problem may be in the teaching, but I've wrestled with student apathy. Math tends to be viewed as boring but useful, the kind of thing that must be endured on the path to riches. Its beauty is admittedly cold, while young people tend to want romance, music, fashion, fame, etc. — mask
Are you saying that classical and constructive physics are equivalent as categories? I'm afraid I don't know exactly how you are categorifying physics — fishfry
Nevertheless, how you stipulate or construct the object lends a particular perspective on what it means; even when all the stipulations or constructions are formally equivalent.[/quotre]
Yes definitely. Each perspective adds to your intuition and understanding of what's going on. A little like being baffled by trig and then baffled by "the square root of -1" in the context of solving quadratics; and then at some point in the future, maybe, you find out that trig and complex numbers are two ways of talking about the same thing; and everything becomes so much more clear.
Learning math is sort of about learning more and more abstract and general viewpoints for the same thing.
— fdrake
I remember studying abstract algebra at university, and seeing the isomorphism theorems for groups, rings and rules for quotient spaces in linear algebra and thinking "this is much the same thing going on, but the structures involved differ quite a lot", one of my friends who had studied some universal algebra informed me that from a certain perspective, they were the same theorem; sub-cases of the isomorphism theorems between the objects in universal algebra. The proofs looked very similar too; and they all resembled the universal algebra version if the memory serves. — fdrake
Regarding that "nevertheless", despite being "the same thing", the understandings consistent with each of them can be quite different. For example, if you "quotient off" the null space of the kernel of a linear transformation from a vector space, you end up with something isomorphic to the image of the linear transformation. It makes sense to visualise this as collapsing every vector in the kernel down to the 0 vector in the space and leaving every other vector (in the space) unchanged. But when you imagine cosets for groups, you don't have recourse to any 0s of another operation to collapse everything down to (the "0" in a group, the identity, can't zero off other elements); so the exercise of visualisation produces a good intuition for quotient vector spaces, the universal algebra theorem works for both cases, but the visualisation does not produce a good intuition for quotient groups. — fdrake
If you want to restore the intuition, you need to move to the more general context of homomorphisms between algebraic structures; in which case the linear maps play the role in vector spaces, and the group homomorphisms play the role in group theory. "mapping to the identity" in the vector space becomes "collapsing to zero" in both contexts. — fdrake
There's a peculiar transformation of intuition that occurs when analogising two structures, and it appears distinct from approaching it from a much more general setting that subsumes them both. — fdrake
Perhaps the same can be said for thinking of real numbers in terms of Dedekind cuts (holes removed in the rationals by describing the holes) or as Cauchy sequences (holes removed in the rationals by describing the gap fillers), or as the unique complete ordered field up to isomorphism. — fdrake
Basically, what I wanted to say is that there is a "trick" in his kind of "constructivist" theory. For example, from page 55:
"As in the classical logic, we can add to intuitionism the axioms of arithmetic or of the set theory, which gives the constructive versions of these logical theories"
All the results are exactly the same, and all theorems are equivalent, only reformulated in a different way (encoding the rules of logic in a different, but equivalent way)
For physics, if the formulas are the same and the method to calculate the results is the same, there's no difference: the difference is only in non-essential mathematical "details" (from a physicist point of view). — Mephist
Well, basically category theory can be used as a foundational theory for physics. It's rather
"fashionable" today, here's an example: https://arxiv.org/abs/0908.2469 — fdrake
One of the advantages is that equivalent formulations of a given theory can be seen as the same theory: pretty much the same of what Vladimir Voevodsky did with homotopy type theory and his univalence axiom ( https://ncatlab.org/nlab/show/univalence+axiom ). — fdrake
The even more interesting thing (that's why I talked about atoms) is that this is true not only for elementary particles as electrons, but even for atoms (of any element), and even for entire molecules, and this has been verified experimentally. Two atoms in the ground state (https://en.wikipedia.org/wiki/Ground_state) are EXACTLY IDENTICAL (as mathematical objects in the mathematical model of QM) if the ground state is not degenerate (https://en.wikipedia.org/wiki/Degenerate_energy_levels). — Mephist
The tricky thing to realize experimentally is to obtain a non-degenerate ground state for a complex object as an atom: very low temperature, external magnetic field, confined position in a very little "box" (usually a laser-generated periodic electromagnetic field). But this is possible, and in this state the whole atom is COMPLETELY DESCRIBED from by one integer number: the energy level.
In this state you can put a bunch of atoms one over the other, if they are bosons (https://en.wikipedia.org/wiki/Bose%E2%80%93Einstein_condensate) and the theory says that you can have N IDENTICAL objects all in the same IDENTICAL place. — Mephist
The result is obtained by purely mathematical considerations on objects made of complex number functions (the states are the eigenvalues of the system's wave function), but the effects predicted using a purely mathematical abstract model generate real physical predictions in the form of measurable quantities. That seems very strange if mathematical objects are only symbols subject to arbitrary rules. In some way, the rules that we invented for the symbols correspond exactly to some of the "rules" of the physical (real) world. — Mephist
Yes, but the indeterminacy is only for the product position * momentum, and not the position alone (for example an electron emitted from the nucleus of an atom has an indeterminacy of initial position of the size of the nucleus from which it was emitted). And the curious thing is that the wave function, if you want the path-integral over the trajectories to be accurate enough, must be described with a much finer granularity of space than the size of the atom. The wave equation works the best if it's defined on the (mathematically imaginary) real numbers (at least for QED). The renormalization of electron's self-energy (https://en.wikipedia.org/wiki/Renormalization) is a mathematical theorem based on a mathematical model where space is the real euclidean space (real in the mathematical sense: vector space defined on real numbers) (I know the objection: it works even on a fine-enough lattice of space-time points, if you make statistics in the right way, but the lattice of positions have to be much smaller of the wavelength of the electron - that for "normal" energies is comparable with the size of an atom). — Mephist
Yes, however in same cases, the system is symmetric enough that you can use analysis to compute the results instead of making simulations, so you can get infinitely precise answers, (such as for example in the case of hydrogen atom's electronic
orbitals) that however you'll be able to verify experimentally only with finite precision. — Mephist
Well, that was a simple example that doesn't have much sense as a real theory of physics (and I absolutely don't believe that it can be a good model of physical space), but it's still a mathematical model suitable to be used to make predictions on the physical space (well, you should say how big are the sticks: surely there are a lot of missing details). However, as a model, you can decide to make it work as you want: in our case, the squares made with sides of one stick can't have a diagonal (so, let's say, nothing can travel along the diagonal trajectory, as in the Manhattan's metrics), and big "squares" can have diagonals but can't have right edges, or straight angles. — Mephist
Yes, but in loop quantum gravity loops are only "topological" loops: they are used to build the metric of space-time, not defined over a given metric space. — Mephist
I do not see how this can be. The constructive real line is not Cauchy-complete. It's only countably infinite. It does not contain any of the noncomputable numbers. It can not possibly be an intuitively satisfying model of a continuum. I'm troubled by this and I'm troubled that the constructivists never seem to be troubled. — fishfry
This is the way I look at mathematical objects in general, and real numbers in particular. They can be physically represented, if they happen to be. But generally, they are specifications more so then anything. As all specifications, they express our epistemic stance towards some object, not the properties of the object per se. Real numbers signify a process that we know how to continue indefinitely, and which we understand converges in the Cauchy sense. Does the limit exist (physically)? Maybe. But even if it doesn't, it still can be reasoned about conceptually. — simeonz
I think Vovoedsky's name gets used way too much in vain in these types of discussions. It's a perfectly commonplace observation that isomorphism can be taken as identity in most contexts. The univalence axiom formalizes it but informally it's part of the folklore or unwritten understandings of math. — fishfry
Ok this is the post I wanted to get to. — fishfry
Ok that's beyond my pay grade, but maybe I can tell you what I know about it. Say you have a hydrogen atom, one proton and one electron, is that right? The electron can be in any one of a finite number of states (is that right?) so if you take two hydrogen atoms with their electrons in the same shell (is that still the right term?) or energy level, they'd be exactly the same. — fishfry
But you know I don't believe that. Because the quarks inside the proton are bouncing around differently in the other atom. Clearly I don't know enough physics. I'll take your word on this stuff. — fishfry
I don't believe you. I do believe that you know a lot more physics than I do. But I don't believe that there is an exact length that can be measured with infinite precision. I'm sorry. I can't follow your argument and it's clearly more sophisticated than my understanding of physics but I can't believe your conclusion. — fishfry
Wait, what? You just agreed with me. The "real physical predictions" are only good to a bunch of decimal places. There are no exact measurements. The theory gives an exact answer of course but you can never measure it. You agree, right? — fishfry
All of this is quite irrelevant to whether we can measure any exact length in the world. Since it's perfectly well known that we can't, it doesn't matter that you have this interesting exposition. There's some QED calculation that's good to 12 decimal digits and that's the best physics prediction that's ever been made, and it's NOT EXACT, it's only 12 decimal digits. Surely you appreciate this point. — fishfry
If that's all you mean, you have gone a long way for a small point. Of course if we have a theory we can solve the equations and get some real number. But we can never measure it exactly; nor can we ever know whether our theory is true of the world or just a better approximation than the last theory we thought was true before we discovered this new one. — fishfry
I'm not following this. — fishfry
Mathematicians and philosophers of mathematics, with the presumed exception of platonists, reject the premiss that all "existence" is ontological existence. Specifically, they acknowledge that mathematical existence does not entail ontological existence. — aletheist
Yeah well, let's say so... The way QM is formulated is: there are "observables" that represent the objects (or better: the results of experiments), and then there are other mathematical "objects" (such as wave functions- https://en.wikipedia.org/wiki/Wave_function) that are not meant to represent something that normally we could call "objects" in physics. — Mephist
Well, the grid of points represents only the topology of space-time, not the metric.
Meaning: there are 4 integer indexes for each point (the grid is 4-dimensional), and then the ordering of the points, only defines which are the pieces of space-time that are adjacent (attached to each-other), not their size or the orientation of the edges. — Mephist
So, basically, every "object" that is composed of well-identifiable parts can be considered to be a natural number, if you specify how to perform the arithmetical operations with the parts. — Mephist
Well, the problem is that what you say doesn't seem to be in any way "compatible" with current physical theories. And current theories are VERY good at predicting the results of a lot of experiments.
To me, it seems VERY VERY unlikely that a simple physical theory based on a simple mathematical model can be compatible with current physics at least in a first approximation. The physical world seems to be much more complex than we are able to imagine... — Mephist
That's what abstraction is! It's giving a name to something immaterial in order to manipulate it. — fishfry
You say you have found a loophole that allows you to accept the world yet reject ... something. The square root of 2. — fishfry
* If we believe in the rationals, we can build a totally ordered field containing the rationals in which there is a square root of 2. — fishfry
Yes I know you already believe that. The question is whether you're willing to believe that it has mathematical existence. You ask me what that is but I've given you many demonstrations of the mathematical existence of 2–√2; as the limit of a sequence, as an extension field of the rationals, as a formal symbol adjoined to the rational numbers. All these things are part of mathematical existence. You will either have to take my word for it, or work with me to work through a proof of the mathematical existence of 2–√2 . — fishfry
You showed me that you have a psychological block in dealing with symbology; leading to a massive area of ignorance of math; leading to making large errors in your philosophy. That's my diagnosis. — fishfry
Like Captain Ahab, who only has fictional existence in a novel. Nevertheless statements about him can can truth values, such as whether he's the captain of the Pequod or the cabin boy. So there's fictional existence. — fishfry
I can indeed specify 2–√2; but when I do so I am merely "expressing my epistemic stance" toward 2–√2; yet not necessarily saying anything about 2–√2 itself, whatever that means. I think this is Metaphysician Undercover's point perhaps.
And then "expressing my epistemic stance" towards a mathematical object is what I mean by endowing that object with mathematical existence. Perhaps this is the distinction being made. — fishfry
Actually, if you analyze this situation closely, "wave functions" are produced from observations, so they are still mathematical representations of the movements of objects. The wave function is a use of mathematics to represent observable objects. There is no such separation between the representation of a physical particle and the wave functions, the wave functions represent the particles. They are of the same category, and I think the physicist treats the particle as a feature of the wave functions. Wave functions are used because such "particles" are known to have imprecise locations which they can only represent as wave functions. With observed occurrences (interactions) the particles are given precise locations. Wave functions represent the existence of particles when they are not being observed. — Metaphysician Undercover
I don't quite get this. "Space-time" here is conceptual only, like "the square" we've been talking about, or, "the circle". Therefore, the positioning of the points is what creates the "object" called space-time, just like we could position points in a Euclidian system, to outline a line, circle, or square. What is at issue, is the nature of the medium which is supposed to be between the points, which accounts for continuity. The continuity might be "the real", what exists independently of our creations of points according to some geometrical principles. — Metaphysician Undercover
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