Tuesday, July 23, 2013

A response to a comment about Platonism

A visitor, Marc Missildine, responded to my recent post about why I suspect Platonism is such an alluring way of thinking about mathematics (Link). I expect I'll need more space than a reply box to reply, and I hope to make the response interesting and poignant, so a new post for it. Thanks, Marc, for your comment!

As is customary for me, I will put Marc's commentary in block quote format in green letters. The full text of Marc's comment is here (Link).
"...and nothing except the unfolding of the cards really happens." So.. nothing but EXPLORATION happens?? 
I prefer not to use such charged language as "nothing but" here. In my experience, particularly in philosophy, "nothing but" statements are usually dubious and frequently wrong.

The problem here with "nothing but" is that eventually we meet the seams of what we can say within the axiomatic system we're exploring, e.g. when we follow the Peano Axioms out and realize that they predict the concept of infinity but do not account for it. At that point, we get to decide upon new axioms about the questions raised, which isn't really exploring anymore, at least not in the sense that it's all already there waiting to be found.
This takes us back to the cave example. We can still explore the unknown (in candyland the unknown is what color the next card is going to be), and can reveal truths about the universal system. 
The point here reveals why the cave example isn't any good. A video-game cave is actually a better example than a real physical one. Why? Because it's as if our axioms define a certain digital cave, but the programmer isn't actually able to close off all of the boundaries within that cave. Sure, the boundaries are relatively hidden in most cases, but like playing a glitchy game, there are places where you can kind of fall off the edge of the map. There is an outside to what the axiomatic system is able to describe in exactly the same way that the unprogrammed "void" is a "place" outside of the intended playable area in a glitched video game.

We try to set up the program to close off all the glitches, but we know because of the Incompleteness Theorems that we cannot actually do that. We'll always be able to find glitches that let us "leave the map." At that point, programmers can decide either not to bother with it or to flesh out areas for some of the more accessible, obvious glitches. Outside the metaphor, this is choosing new axioms that define more (or different) cave space to explore--knowing again because of the Incompleteness Theorems that these attempts will never close off all of the seams.

I'm most uncomfortable with your use of the term "universal system" for this reason. We have axiomatic systems that define truth values for a huge variety of statements, but they do not touch those statements that the axiomatic system cannot assess--which by the Incompleteness Theorems are always there (at least if we choose to avoid logical inconsistencies).
Similar to the child, all of humanity does not know what the cards in the deck of mathematics are, nor do we know how big the deck is. 
It's actually worse than that. The Candyland example really breaks down in a sense, it really just being a metaphor to give a sense of what's going on here, and that's hardly the point. We are actually in a situation where not only do we not know that, we also don't know what kinds of cards we'll decide we want to add to the deck until we face those questions, and we're in a situation in which we know that whatever cards we choose, literally no matter what, we aren't guaranteed to be able to "finish the game."

Don't get too caught up on the metaphor, though. It's just an intuition pump.
Nor do we know who 'shuffled' the cards. 
Whoa. Just whoa. I know where this is going, and whoa. Stop.

First, this is the wrong question to be asking, and second, yes we do. "The cards" in this case, really, are the truth values assigned by the axiomatic system, but those values are determined by the choices of axioms and logical framework, which are ultimately choices we have made and continue to make.
This points to a rules-maker who is outside of the boundaries of humanity who created these simple and universal(transcendental) truths(what you call axioms) such as 1=1. 
No it doesn't. First, I just said why it doesn't. Second, inventing a "rules-maker" is choosing an axiom (meaning you choosing an axiom, you being a person, just like all the other axiom-choosers).

I don't call these things axioms. "Axioms" is the word for these things. I use that word, but it's not the same as your connotative implication. Also, "axioms" aren't "universal(transcendental) truths," they are baldly asserted statements that we agree are sufficiently parsimonious and basic to be getting on with getting started, i.e. to be willing to baldly assert as sufficiently self-evident*. The asterisk here indicates a constant willingness to reassess them at a later date: Cf. the Peano Axioms predicting the axiom of infinity but being unable to handle it and so the Zermelo-Fraenkel axioms of set theory taking over the job.
This points to an ordered universe in which laws and rules are discoverable, because if 1=3 sometimes according to some axioms, and 1=red in other axioms, then the universe loses all meaning.
Incredibly, you missed my entire point, then. It doesn't. It points to an ordered axiomatic system that humans have developed in order to attempt to make sense of the universe we live in. The point isn't that the universe "has meaning," it's that humans, as thinkers, can find meaning in it, which as often as not means defining for ourselves the constructs (languages, as it turns out) in which meaning resides. The central point is that we invent these logical systems and language to attempt to understand what we observe and to communicate those attempts with other people. The logical systems are not the universe itself.

Incidentally, under some definitions, 1 does equal 3 (e.g. in any algebraic system over the field with two elements). "1=red" is just silly--a category error, to let you know.

Again, thank you for your time and your comment.

8 comments:

  1. Given infinite axioms and infinite time, a formal system can be consistent and complete, correct? Gödel's results only apply to axiomatic systems with a finite number of axioms. And so, humans (and computers), being able to construct only finite axiomatic systems, will always encounter the incompleteness and inconsistency of these systems. Is this not where religious faith comes into play? After all, the universe were a complete, consistent, finite system, that would be impossible wouldn't it? Only because of the incompleteness and inconsistency are we allowed subjective statements that cannot be proven objectively true or false and may change over time (they are unprovable outside the system). And if the numbers zero and infinity didn't exist, the universe would be a complete, consistent, finite system, correct? Something like Leibniz's universal God-mind however would be completely compatible with this view.

    Consider these two quotes:

    “It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of spacetime is going to do?" – Richard P. Feynman, The Character of Physical Law, 1965.

    "…[T]here is always, underneath, a reason for the truth, but the reason is understood completely by God, who alone traverses the infinite series in one stroke of mind. (A VI iv 1650/AG 28)" - Leibniz

    Now remember that Leibniz was, after all, derided as an atheist "Lovenix" (believer in nothing) at the end of his life (and he never went to church). His idea of God was a universal, rational, mathematical mind, the ground of all being, that calculates to infinity. If rational unobservables such as zero and infinity exist, this is not a problem for Platonists, yet it is a horrible catastrophic problem for materialists and formalists. On your view, does the fact that axiomatic systems arise (to begin with) have no meaning? In other words, is mathematics entirely devoid of meaning because it is the result of a random universe? Or does the appearance of structures or patterns imply anything greater? Will science ever be able to produce a rational grand unified theory of everything that explains reality from the ground up, or will we be forced to reckon with pure randomness? What explains the "unreasonable effectiveness" of mathematics that makes science so powerful? Is there not some correspondence between mathematics and reality?

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    1. I don't have a lot of patience left with you, as you know. Thus, this will be terse.

      1. "Is this not where religious faith comes into play?" No.

      2. "After all, the universe were a complete, consistent, finite system, that would be impossible wouldn't it?" The universe isn't an axiomatic system. Category error.

      3. "And if the numbers zero and infinity didn't exist, the universe would be a complete, consistent, finite system, correct?" Infinity isn't a number, even if it is a quantity. I doubt this is correct, but I don't actually know what you're talking about.

      4. I don't care about Leibiniz's weird definition of God: a mind "that calculates to infinity." Why should anyone care about this? Honestly. Also, why should anyone call it "God"? What a horrible word for such an idea! It already means other things! How confusing and misleading!

      5. Materialism doesn't depend upon zero being a "rational observable."

      6. "On your view, does the fact that axiomatic systems arise (to begin with) have no meaning?" Axiomatic systems don't "arise," we construct them.

      7. "In other words, is mathematics entirely devoid of meaning because it is the result of a random universe?" See the paragraph where I discuss where the term "meaning" makes sense. Stop there. This question is empty.

      8. "Or does the appearance of structures or patterns imply anything greater?" We've covered this before. Stop wasting my time. Prove that a "pattern" exists outside of a mind to call it a pattern.

      9. "Will science ever be able to produce a rational grand unified theory of everything that explains reality from the ground up, or will we be forced to reckon with pure randomness?" Don't know. Not my field. Doesn't matter for this discussion. Eat your herring alone.

      10. "What explains the "unreasonable effectiveness" of mathematics that makes science so powerful?" We made it to do the job rejecting failures as we went, rather like the "unreasonable effectiveness of our shoulders for throwing objects."

      11. "Is there not some correspondence between mathematics and reality?" In much the same way that there's a correspondence between calling a tree a "tree."

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    2. Hi James, thank you for your response. Firstly, I don't aim to test your patience or risk being banned from your blog. If anything I want to find truth (if there is such a thing, which is an open question). And that cannot happen unless I ask questions, and I would not be asking questions if I already thought I knew the answers. And even if I did, I am not dogmatically attached to my own views. I'm not a fundamentalist Abrahamist, for whom you should really reserve your impatience and anger. I am not here to assert anything based on stupid irrational faith or holy books, I'm just here to ask so-called "meaningless" and harmless "empty" questions that no one need take seriously if they find it distasteful to do so. No need to get so bent out of shape, I really don't understand that. If you don't want to respond to my comments, just don't respond. If you'd like to only waste a little time, you could consider a short answer which points me to some resources, books, websites, etc. that I should consult before asking my next question. I never said I didn't want to learn; it's in fact the complete opposite.

      I suspect from all the facepalming we do to each other's questions and answers that we are simply talking past each other. You merely assert the question of meaning vs. randomness is "empty," and there we seem to have an impassable difference of opinion: merely asserting something is meaningless or empty does not necessarily make it so. I wonder why you seem reluctant to consider the metaphysical implications of your views about whether the universe is ordered or random. I think the universe is based on mind, so the existence of patterns in the universe is, to me, the same as the existence of patterns in minds. The physical laws of the universe, which seem to be mathematical, should be enough "proof" to you to take this idea seriously (inasmuch as any scientist believes in them, so I hope you respond to my further comments below). If nothing else, despite your crankiness it seems you appreciated my past comments enough to make them fodder for your blog posts, so I hope you may at least find that as the minimum (or should I say infinitesimal!) amount of value.

      Lastly I would like to point out that we cannot construct an axiomatic system out of nothing. It must come from something. It's like saying computers compute because we made them, as if we can make anything to do anything! We can't. The laws of nature have to be mathematically compatible in whatever sense you prefer in order for us to contruct axioms in the first place which allow us to, for instance, build computers, otherwise we would only be able to build machines that act randomly with no meaning. You can say "Computers compute because we made them to do that," but you're missing the point! They are made of matter which is no different from all the rest of matter in the universe, but they perform mathematical operations. Why should matter be doing all of this mathematical computation? Why should the laws of physics seem to be mathematical? You can assert that it is a meaningless question or that you don't know, but that doesn't mean that the question is in principle empty or meaningless or that we can never know. If you know of books or resources that address this issue, I would be interested to know of them. Thanks again.

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    3. "I wonder why you seem reluctant to consider the metaphysical implications of your views about whether the universe is ordered or random."

      If that's your question, I don't think you're phrasing it correctly. Also, I don't necessarily think this has to be a dichotomy. Both could be possible. Further, I don't put much stock in speculation on these matters--which is why you see very little of my own speculations on these things, which I have done plenty of.

      "I think the universe is based on mind"

      I don't. Would the universe exist if there were no minds to experience it? Our observations tell us that it did for eons, if we accept realism.

      I suggest that everywhere you talk about axiomatic systems, you replace the term with the word "language," meaning specifically ideas like French and English, and see what you get from them.

      The laws of physics "seem to be mathematical" because we wrote them down in mathematics. In fact, we invented mathematics (see: calculus) to do it in many cases. The harder question of why the universe behaves as it does, fundamentally, is open and likely to be solved by physicists eventually if it can be solved. That mathematics works to describe it is a matter of our design of mathematics to do that very job. That's why I say it's a "meaningless question" to ask why the universe appears mathematical. The universe is how it is; it is not behaving according to mathematics. We wrote mathematics to describe the behaviors we see in the universe. Why do we see those behaviors? Who knows, broadly?

      My contention, very generally, is that you're looking at this issue through the wrong end of the telescope and wondering why everything you see seems to tiny.

      I don't know if any specific books that have been written on this subject. I'm basing my opinion on years of considered study in all of the relevant fields (physics, mathematics, and philosophy).

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  2. (continued)...
    Consider that most physicists consider the fundamental laws of physics to be mathematical, in some sense. See 56:57 of this video: https://www.youtube.com/watch?... Yet Weinberg can't account for what they are! Why not? He says "they're not made of anything" so what are they? "It's the relations [between the particles] that are real" but what does that mean? If you rewind the video to about 39:45 Weinberg says "I suppose you would say that it's real because it exists in minds and the minds are in brains..." Yet Carroll asks, "so Schrödinger's equation is real because people can write it down?" For me, Alex Rosenberg nails it at 40:20 when he says "the biggest problem for me philosophically ... that's 'what is the metaphysics and epistemology of mathematical truths'?" Would you be inclined to say that (regardless of symbols, characters, or semantic terms used) that A always equals A? Or would you argue that A=A is not always true (even regardless of symbols)? Is the universe just random, or are there underlying laws and principles? Is math just a language that humans made up, or do the underlying structures or patterns actually exist in some sense? If so, how and in what sense? If not, isn't it essentially a matter of faith on behalf of scientists like Steven Weinberg (who admits if he didn't think they were real that he in fact made a mistake in his choice of career) who believe in such mathematical "laws" that can not be experimentally detected or scientifically observed? But again if the structures / patterns / laws do exist, would we not have a justification to infer that mathematical objects exist in some non-material sense (in the sense that Weinberg admitted)? In effect, doesn't this spell the end for materialism (naturalism)? How can a scientific law exist without being subject to all the laws of entropy, change, and decay like the rest of matter?

    As you must know, Gödel in his unpublished essays defends Platonism and, in fact, suspected some sort of conspiracy against him because so many people used his work to attempt to prove the opposite of what he believed. I know you remarked that "he was looking for the star on the ground and never found it," but that response doesn't clarify your position in any way.

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    1. Again, terse, almost angry, because you've previously worn my patience with you out completely. It is not yet restored.

      You asked me 13 questions here, some of which are huge unanswered questions. Do you seriously expect me to take time to answer all of them for you? Here's the take home on your first slough of them:

      We don't know the fundamental nature of matter yet. Philosophers will not be the ones to figure it out. Physicists will. I'd bet the farm on that. Leibniz's weird 500-year-old idea of monads is very unlikely to be correct in this regard. I will not speculate about the fundamental nature of matter for you.

      I don't know what to tell you about a physicist, or group of them, being flummoxed over the idea that we think up our mathematics in order to try to describe reality. Ian Stewart in Letters to a Young Mathematician claims that most working mathematicians operate on an unexamined blend of Platonism and Formalism, so why shouldn't we expect that from physicists to even greater degree? So it's likely these fellows are wondering about their unexamined foundations of mathematics. Good for them. Maybe they'll get somewhere with that one day.

      The law of identity: first of all, why are you people so wrapped up on that? Second of all, no, maybe not. The law of identity strikes me as being a foundational proposition of our logical system, which embraces the law of the excluded middle. I don't know if that's the right logical system to use or not and don't pretend to, nor do I base my philosophy on such an uninformed choice!

      Go learn what "random" means before you use that word anymore.

      None of this spells the end of materialism. "Scientific laws" don't exist. They're descriptive tools we've written down to try to make predictions about how nature appears to have always behaved. How absurd so many of your questions are!

      Yes, Gödel was a bit of a fruit-bat. Wait until you discover Tesla considered himself to be in a romantic relationship with a pigeon.

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    2. Thanks again for your response. I hope to aim to restore your patience; we'll see if that happens or not.

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  3. Sorry, that link should be: https://www.youtube.com/watch?v=qeyBqxY3MsQ&list=PLFSiHJ_VzQh8UeyiBMyaC1s16uMQZmt2M

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