The Platonic Mathematical World

[SerScot]

Vraith,

It's still an interesting discussion.

[Vraith]

Vraith,

It's still an interesting discussion.

Oh, hell yeas, or I'd have said "Shut up SS, you're boring me." [at least abstractly/conceptually, by NOT having a concrete, material post that said so].
I'm not sure if that is actually a joke like I think it is, or if I just confused myself.

[Zarathustra]

...the semantic content of mathematics doesn't change because, unlike ordinary language, it has no content except its definition.

No. Not only is this not true, but even if it were true, how could that explain why math doesn't change? If its content is nothing more than our definitions, then why can't we change math by defining it in different ways?

We don't define the relationship between the diameter of a circle and it's circumference; we discovered it. Sure, we say that a certain symbol stands for this relationship, and that's a kind of defining. But the content of the concept exists independently of this act of choosing which symbol we'll use to talk about it. The content of this concept is much more than its definition. It's a geometrical relationship that can be expressed numerically--with digits that go on infinitely. It's a relational fact that never changes. It existed within the framework of geometry long before we discovered it and named it. How can that be a product of our definitions? How can our definitions include an infinite series of digits which none of us has ever seen or known? That's not a definition, that's an infinite process of discovery.

Thus, it's obejctively real in the sense that the meaning of "pi" isn't dependent upon our subjective intentions, connotations, feelings, impressions, etc. It has its own reality separate from our subjectivity.

But I agree with you that "concrete existence" is an incorrect way to describe this objective reality. "Ideal existence" is better.

[Linna Heartbooger]

[Edit: Oh bah... if I'm not going to join the conversation, I must forfeit my right to complain about it.]

[Zarathustra]

rancho.pancho.pagesperso-orange.fr/Abstra.htm

I found this while Googling "ideal reality of numbers Husserl." It's a very dry read, but you can probably skip down to the section "Husserl Reasons his Way into the Realm of the Ideal" and get the gist of it.

In my opinion, you can't do justic to this topic without examing Husserl's contributions to it. His Logical Investigations is the largely considered the nail in the coffin for psychologism (i.e. what some seem to be arguing for here when they dispute the ideal/Platonic reality of numbers).

[Vraith]

...the semantic content of mathematics doesn't change because, unlike ordinary language, it has no content except its definition.

No. Not only is this not true, but even if it were true, how could that explain why math doesn't change? If its content is nothing more than our definitions, then why can't we change math by defining it in different ways?

We don't define the relationship between the diameter of a circle and it's circumference; we discovered it. Sure, we say that a certain symbol stands for this relationship, and that's a kind of defining. But the content of the concept exists independently of this act of choosing which symbol we'll use to talk about it. The content of this concept is much more than its definition. It's a geometrical relationship that can be expressed numerically--with digits that go on infinitely. It's a relational fact that never changes. It existed within the framework of geometry long before we discovered it and named it. How can that be a product of our definitions? How can our definitions include an infinite series of digits which none of us has ever seen or known? That's not a definition, that's an infinite process of discovery.

Thus, it's obejctively real in the sense that the meaning of "pi" isn't dependent upon our subjective intentions, connotations, feelings, impressions, etc. It has its own reality separate from our subjectivity.

But I agree with you that "concrete existence" is an incorrect way to describe this objective reality. "Ideal existence" is better.

I'm trying to reply in a way that isn't repetition/doesn't lead back to stuff we've covered [if not fully exhausted], but points to new territory...because I have a nebulous intuition that there is other stuff that might be interesting...
The pi relationship isn't dependent on our subjectives, true...except insofar as I prefer strawberry/rhubarb to apple, and coconut cream most of all. But what it is and how it works is preceded by/dependent on the definition of a circle...the set of all points on particular plane equidistant from a single point.
Wherever there is a perfect pi, no one can exist to eat it...wherever the eater exists, there is no perfect pi.
Pine trees and decision trees are mutually exclusive.
I try to imagine, while I sit here contemplating how useful and accurate mathematics is, some intelligent rhombus awestruck by the way abstract conceptual mass/energy/dimensions/change [which can't "really" exist there] explains her world.
Every particular horse, I say, is a pale imitation of pure horse-ness.
This pale perfect horse, she says, is a meaningless generalization of the multitudes of real horses in their infinite expressions of truths.
I wonder, if an exception proves the rule, is it really a rule?
Rhombus ponders, if a rule proves an exception, is it really an exception?
Mathematics is just as mystical as God when asking "how, precisely, does it touch/make the world?" One's top down, the other bottom up...but that doesn't mean either actually exists.

Edit to add...posting on top of each other, I'm gonna read that link now.

[SerScot]

Vraith,

If none of the mathematical approximations or abstract concepts of Mathematics can be truely represented in the real world, why are they so useful in science, medicine, enginnering, etc...?

[Vraith]

Vraith,

If none of the mathematical approximations or abstract concepts of Mathematics can be truely represented in the real world, why are they so useful in science, medicine, enginnering, etc...?

But that's just it. They are only truly true in themselves, in the real world they are only approximate; they are represented [approximately] but not present. My post cannot truly represent my thoughts...yet it is useful, is it not? [if it isn't I should just shut the f#^# up...which may be true anyway, people in my "real" life tell me to often enough]

They are useful because things are relational, have some sort of "order,"...any place with order and relation will be describable by systems/logic/math...NOT because the math MAKES it that way, but because the matter/forces within it will not allow it to exist if there is none...at least not for long.

In the end, very simply, and thus approximately, represented but not present, the Ideal is inferior, not superior...because I can define a circle but it cannot define me. I can create as many of them as I like, it cannot even create an impression.
By analogy [again insufficient] it is, as so many things are, at least I say so quite often, much like we said in the army: The map is not the territory.

[SerScot]

Vraith,

But given your contention that they are approximations created by human intellect why don't they change over time as with things created by human intellect change over time? What is different about mathematical concepts as opposed to other human created intellectual concepts?

You haven't been able to explain mathematics amazing constitency and utility to the real world.

[Zarathustra]

I'm trying to reply in a way that isn't repetition/doesn't lead back to stuff we've covered [if not fully exhausted], but points to new territory...because I have a nebulous intuition that there is other stuff that might be interesting...

Yeah, it does seem we're repeating ourselves a bit. But it's better than watching TV, I suppose.

Wherever there is a perfect pi, no one can exist to eat it...wherever the eater exists, there is no perfect pi.

I'm not sure why this matters. I've already agreed that "concrete existence" is the wrong way to describe the independent, objective existence of ideal objects. To require "pi" to be a real pie (even understanding your point metaphorically) is to insist upon a criterion that misses the point. It's as unreasonable as requiring love to have a color, in order to exist. Or to require that electrons make you feel happy.

Every particular horse, I say, is a pale imitation of pure horse-ness.
This pale perfect horse, she says, is a meaningless generalization of the multitudes of real horses in their infinite expressions of truths.

You can't apply the same reasoning to "horseness" as you can to circles. As you have said yourself, we can define a circle precisely. We can't do this with "horseness." The latter is an abstraction, a generalization, from known particulars (real existing objects). The ideal objects within math are purely formal. Sure, we may have abstracted the concept of "circle" from seeing things in nature like the moon and the sun, but we've never seen objects with a 10,000 sides, and yet we can easily conceive them, draw, them, define them, ect. Math isn't merely a generalization from real objects, but its own "realm" in which we discover things we've never guessed or imagined. It's impossible for all these discoveries to be abstractions from known objects, when there are no known objects that fit all these discoveries, and we didn't imagine them until we discovered them.

But the fact that math doesn't derive from abstractions from known objects (in the same way we derive "Horseness" from horses) doesn't mean that math is pure definition or doesn't have a connection to reality. Indeed, the fact that it isn't derived from physical reality--and yet physical reality can be seen to conform to it nonetheless--strengthens the case for math being independent (not derivative) and real.

Mathematics is just as mystical as God when asking "how, precisely, does it touch/make the world?" One's top down, the other bottom up...but that doesn't mean either actually exists.

I think we can agree that the regularity we see in nature actually exists, right? We're not just making it up, or seeing an illusion, like shapes in the clouds. So how do you characterize that regularity? Is it an accidental, approximate correspondance with an imaginary system we invented? (That sounds suspiciously close to shapes in the clouds). Is there nothing we can conclude from this correspondance with math, other than our own peculiar tendency to describe things with numbers?

Sure, our mathematical models only approximately capture this regularity, and thus an argument can be made that the regularity isn't actually "in" the universe in the same way a perfect circle isn't "in" the universe. But a curious thing happens the deeper we look. As our knowledge increases, the match between reality and models increases in accuracy. For instance, Newton's laws were pretty good in predicting the positions of the planets. Tiny inaccuracies in our Newtonian predictions were corrected when Einstein developed a new understanding of nature and a new mathematical formula to describe it. So the "imperfections" weren't inherent in nature; they were a limitation of our understanding of nature. It turns out that nature was a lot more ordered than we first understood.

Why would that be the case? Why does reality look more mathematical the closer we look at it? If reality were merely an imperfect approximation of a formal system which we invented, one might expect for this correspondance to break down the closer we looked. For instance, if you're a good artist with a steady hand, and you draw a circle, it looks pretty good at a distance, but if you grab a magnifying glass it will look less and less perfect the closer you peer. But with reality, the opposite is the case. The more we look at the universe, the more order we discover. In the example above (Newton vs Einstein), it would be like seeing a physical circle (for instance, an orbit of a planet ... ignoring for the moment that orbits are more like ellipses), and then discovering that the "imperfections" in this circle weren't actually random fluctuations of an unsteady hand--as with the case of the artist's circle--but rather that the physical "circle" was actually a regular, ordered shape with 10,000 sides. That's a phenomenon completely different from the situation with the artist's circle. With the artist's circle, the imperfection is real. With the universe, the "imperfection" is actually a level of order we didn't know about yet.



And finally--to go in a completely different direction--if "perfect circles" only exist in our thoughts, and our thoughts are products of an organic brain, then doesn't this mean that there is a sense in which these abstract objects exist in nature? Our brains are definitely in nature. Our thoughts are in some sense "in" our brains--at the very least, the chemical/electrical/neural counterparts of our thoughts are most certainly in our brains. So how it is possible for these fundamentally imperfect processes to produce the conception of perfect ideal objects? If there is a real, causal correspondance between our thoughts and our brains (and there must be, unless you're going to fall back on some kind of God-maintained parallelism between mind and body) then what do we make of the correspondance between thoughts of perfect circles, and the underlying physical processes which produce and/or comprise these thoughts?

[Vraith]

Vraith,

But given your contention that they are approximations created by human intellect why don't they change over time as with things created by human intellect change over time? What is different about mathematical concepts as opposed to other human created intellectual concepts?

You haven't been able to explain mathematics amazing constitency and utility to the real world.

But I am not contending that they are created by the human intellect.
I am saying they are an emergent, though only approximately precise, implication of things existing for any measurable amount of time. A mind is necessary only for their explication. And we do that, fundamentally, by mentally stripping away the apple-content from the statement "One apple." The one is then free to roam [or be pushed] about, entangling and disentangling from a multitude of things.
What I'm contending with is the idea [central to the book that started all this] that the Ideal is somehow transcendental compared to the material, that without the math itself there could be no "real," the Platonic notion that physical existence is cheap imitation/reflection, and therefore an illusion/falsification of the pure, "real" TRUTH.
The amazement at the fact that math can so closely approximate real things, be so useful, imply other real things no one has/had thought of yet ignores the simple truth that the number of things it can ALSO approximate/are non-applicable/imply that are impossible, is infinitely greater in number. And that you need different kinds of mutually exclusive maths to describe different kinds of real things.
It's like being in awe at the number of things you can do with a pebble while completely dismissing the incomprehensibly larger number of things you CAN't do with one, and that you need pebbles with contradictory properties to perform different functions.

[Zarathustra]

But I am not contending that they are created by the human intellect.
I am saying they are an emergent, though only approximately precise, implication of things existing for any measurable amount of time.

See my points above about "horseness." How is an imaginary number or an irrational number an emergent implication of things existing for any measurable amount of time?

A mind is necessary only for their explication. And we do that, fundamentally, by mentally stripping away the apple-content from the statement "One apple."

You really think that all mathematical truths are explicated by stripping away physical content from their meanings? What about all the mathematical truths that are discovered merely by manipulating the formal rules of the system?

What I'm contending with is the idea [central to the book that started all this] that the Ideal is somehow transcendental compared to the material, that without the math itself there could be no "real," the Platonic notion that physical existence is cheap imitation/reflection, and therefore an illusion/falsification of the pure, "real" TRUTH.

Is that really what the book says? A cheap imitation/reflection? I realize this is close to what Plato said, but I don't think you have to maintain that idea to describe numbers and their relationship to the universe as Platonism. I don't consider teh universe to be an illusion of the pure real TRUTH, no more than I'd consider the performance of a song to be the illusion of a written transcription of that song ... or the running of a computer code to be an illusion of that code.

The amazement at the fact that math can so closely approximate real things, be so useful, imply other real things no one has/had thought of yet ignores the simple truth that the number of things it can ALSO approximate/are non-applicable/imply that are impossible, is infinitely greater in number.

Like what? How do you know they are impossible? How do you know there aren't other universes which conform to different mathematical rules?

And that you need different kinds of mutually exclusive maths to describe different kinds of real things.

Mutually exclusive? Again, like what?

It's like being in awe at the number of things you can do with a pebble while completely dismissing the incomprehensibly larger number of things you CAN't do with one, ...

I'm not sure why it matters that math has infinitely more functions than those which we use to describe the universe, no more than it matters that the realm of the Possible is infinitely larger than the realm of the Actual. What matters here is that the realm of the Actual can be completely described using these tools, and there is no part of the physical universe which is non-mathematical.

... and that you need pebbles with contradictory properties to perform different functions.

How can properties be contradictory? Are you saying there are objects which simultaneously exhibit property A and ~A (not A)? Are you thinking of particle/wave duality? That's not exactly the same as contradiction (A/~A), but rather an indication that our ability to comprehend through metaphors is limited. The mathematics of particle/wave duality is perfectly intelligible and noncontradictory.

[Vraith]

But I am not contending that they are created by the human intellect.
I am saying they are an emergent, though only approximately precise, implication of things existing for any measurable amount of time.

See my points above about "horseness." How is an imaginary number or an irrational number an emergent implication of things existing for any measurable amount of time?

A mind is necessary only for their explication. And we do that, fundamentally, by mentally stripping away the apple-content from the statement "One apple."

You really think that all mathematical truths are explicated by stripping away physical content from their meanings? What about all the mathematical truths that are discovered merely by manipulating the formal rules of the system?

The stripping away is first, and emerges from a thing existing...but only needs to be done once, after that it's all a matter of not contradicting one, of consistent and logical process...one implies two, implies things between one and two, some of which are irrational, eventually you end up facing the square root of -1. But all of those operations are internal, may or may not have any relationship with actual things.

What I'm contending with is the idea [central to the book that started all this] that the Ideal is somehow transcendental compared to the material, that without the math itself there could be no "real," the Platonic notion that physical existence is cheap imitation/reflection, and therefore an illusion/falsification of the pure, "real" TRUTH.

Is that really what the book says? A cheap imitation/reflection? I realize this is close to what Plato said, but I don't think you have to maintain that idea to describe numbers and their relationship to the universe as Platonism. I don't consider teh universe to be an illusion of the pure real TRUTH, no more than I'd consider the performance of a song to be the illusion of a written transcription of that song ... or the running of a computer code to be an illusion of that code.

It does say that [well, not cheap...but lesser/inferior] through arguments/events...beings are moving from less ideal universes to more ideal ones, each level the inhabitants are "more ideal,"...I say the end result of that is eventually entering a level where living things can't exist...and never could have/can/will.

The amazement at the fact that math can so closely approximate real things, be so useful, imply other real things no one has/had thought of yet ignores the simple truth that the number of things it can ALSO approximate/are non-applicable/imply that are impossible, is infinitely greater in number.

Like what? How do you know they are impossible? How do you know there aren't other universes which conform to different mathematical rules?

I think it is theoretically possible for SOME different math universes to exist...but, for instance, one in which imaginary numbers are the ordinary everyday...not possible. There simply can't be a place where there are 1i apples.

And that you need different kinds of mutually exclusive maths to describe different kinds of real things.

Mutually exclusive? Again, like what?

Simplest example: you need three different maths [that do have some commonalities, but also contradict each other] to describe triangles in convex, flat, and concave spaces.

It's like being in awe at the number of things you can do with a pebble while completely dismissing the incomprehensibly larger number of things you CAN't do with one, ...

I'm not sure why it matters that math has infinitely more functions than those which we use to describe the universe, no more than it matters that the realm of the Possible is infinitely larger than the realm of the Actual. What matters here is that the realm of the Actual can be completely described using these tools, and there is no part of the physical universe which is non-mathematical.

I'm not worried about the realm of the Possible...it's the realm of the impossible, that which can exist nowhere except in the math, and that no part of the universe is mathematical...it IS, and is roughly describable mathematically.

... and that you need pebbles with contradictory properties to perform different functions.

How can properties be contradictory? Are you saying there are objects which simultaneously exhibit property A and ~A (not A)? Are you thinking of particle/wave duality? That's not exactly the same as contradiction (A/~A), but rather an indication that our ability to comprehend through metaphors is limited. The mathematics of particle/wave duality is perfectly intelligible and noncontradictory.

I was being sloppy, but meaning the first [a/~a].

[Avatar]

Doesn't math describe a perceived relationship between things? If our perception of that relationship is wrong, then our description must be wrong too.

SS says it's consistent...but is that strictly true? What did it look like before we realised, for example, the particle/wave duality of light?

Didn't the math change with our perception?

--A

[SerScot]

Avatar,

Doesn't math describe a perceived relationship between things? If our perception of that relationship is wrong, then our description must be wrong too.

SS says it's consistent...but is that strictly true? What did it look like before we realised, for example, the particle/wave duality of light?

Didn't the math change with our perception?

--A

That's my point. Perceptions change over time but the way mathematical concepts and relationships interact do not change. If Mathematics is created by the human mind and has no concrete existence of its own for us to discover I contend that it would alter over time.

[Zarathustra]

Doesn't math describe a perceived relationship between things? If our perception of that relationship is wrong, then our description must be wrong too.

Yes, perception is involved. But unless you're a solipsist/nihilist/skeptic, then you must admit the possibility that our perceptions can actually discern real relationships. Sure, there can be error. Senses are fallible. But that's why we have the scientific method, to minimize that potential for error. After 100s or 1000s of scientists check the same data and results, after they perform 1000s of experiments to verify their conclusions, how can you continue to say that the relationships we're seeing between math and objects is merely one of perception? It's not just in our heads ... otherwise our hypotheses wouldn't turn out to be correct and our technology wouldn't work. If mathematical relationships between gravity, mass, friction, etc. weren't real, then when we used math to do things like design buildings, our buildings would fall down. Our computers wouldn't work. Our space probes wouldn't reach their targets. Our predictions wouldn't be proven true. At some point you have to admit there's a real relationship between math and reality, because if these patterns are all in our heads, you'd have to resort to magic or ludicrous amounts of coincidence to account for why it works so well in the real world.

But let's say you're right. Even if mathematical relationships were merely in our heads, why should our perceptions be ordered in this way? Why would evolution shape our sensory organs such that they gathered data in ways allowed us to do things like invent--and apply--calculus in our daily lives? There is absolutly nothing in our survival contexts that would give natural selection a reason to select for this peculiar perceptual trait, nothing about having mathematically ordered perceptions that would confer survival advantages upon our ancestors ... especially if that perceptual ability doesn't connect with some real aspects of the world.

You might as well say there are no such thing as "one" apple or "two" apples, if you're going to say that mathematical relationships are merely in our heads.

Or maybe I'm misunderstanding your point. Maybe you're not saying the relationships aren't real, but instead that we can never know for sure if we're using the correct mathematical description? If so, I'd reply that we can know we have the "correct" description in as much as it works. If we use a particular mathematical formula to send a probe to Jupiter, and the probe goes to Jupiter, then we know that the description we're using is at least that accurate. Could it be more accurate? Of course. In fact there is an anamoly in the Viking probes' motion that scientists still can't account for. But to dismiss the overwhelming mathematical nature of the way matter moves through space merely because we could be more accurate is just nit picking, in my opinion. It's missing the point. It's like finding a pot of gold the size of a mountain and complaining that it could be a few inches higher.

[Fist and Faith]

Also, Av, as somebody's sig says, if you can't tell the difference, what difference does it make?

[Avatar]

Or maybe I'm misunderstanding your point. Maybe you're not saying the relationships aren't real, but instead that we can never know for sure if we're using the correct mathematical description?

That's closer to what I was getting at. Didn't the "nature" of math change when we discovered light's particle/wave duality?

Our perception changed, therefore we had to change (or "find" or "make") the math to describe this new perception.

I'm not disputing the existence of those relationships. But the relationships exist anyway. They exist independently of our perception. The math is just a way to describe our perception. New perceptions require new or better or more accurate descriptions. Therefore the descriptions are not consistent over time.

--A

[Zarathustra]

I don't think our math changed when we realized that light behaves as both a particle and a wave. We just realized that our metaphor for light was inadequate. It is neither a wave nor a particle. Under some circumstances, it exhibits wave-like behavior, and under other circumstances, it exhibits particle-like behavior. Thus, neither of these conceptual models can completely capture its nature ... but the math certainly can.

Nothing changed about math after this discovery. We already had math that described either wave behavior or particle behavior. We just didn't realize that both of these mathematical "toolkits" could be used on the same phenomenon.

While it's true that the descriptions we use aren't consistent over time, it's also true that our descriptions are continuously getting more accurate. This doesn't put the math in doubt--in fact it makes it more reliable over time.

[Avatar]

I'm not saying it's in doubt. It's flexibility is one of the good things about it. It allows it to change to fit changing understandings. But all that tells me is that it's not concrete. This description is good enough until we realise we were missing something, and then we change the description.

Math isn't a law, it's a model.

--A