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Ok, continuing what I was saying in part 3 regarding mathematical realism. Let me explain the way in which I currently view mathematics. As I say, this is what I consider a very difficult problem, so my views are still unfolding and may be refined in the future. But I think I've made a good amount of progress within the past few years so I'd like to summarize it.

Before getting to my views, let me mention that there is another book that I've read a portion of within the past couple years which has helped inform my opinions on these issues... The Emperor's New Mind by Roger Penrose. While Penrose is fairly well respected as a mathematical physicist, he has a rather unique view of mathematics, and is known for having very bizarre and non-traditional views both on that, the mind, and on quantum computing and quantum gravity. His view is that there are 3 different worlds: the physical, the mental, and the mathematical. The mental world for him serves as a bridge or a gateway between the physical world and the mathematical world. He sees mathematicians essentially as shamans, people who can reach out with their mind and contact another world, bringing back news from it to the rest of us who live mostly in the physical world. Needless to say, nearly everyone else in physics thinks he's crazy. That said, he's not even nearly as crazy as Godel was. Godel believed essentially the same thing about the mathematician's mind being in direct contact with the objects of mathematics, but he took it even further and also believed in God (he attempted to prove God's existence at one point) and other supernatural phenomena like psychics and contacting ghosts of past ancestors. He was also a hypochondriac and suffered from anxiety and paranoid delusions, especially throughout the latter portions of his life. Penrose, as far as I know, doesn't have any mental illnesses and his views are by comparison fairly sane.

My view is rather different from Penrose, Godel, and Plato. My view instead is most similar to the views of two other mathematical realists: David Deutsch and Max Tegmark (both outspoken advocates of the Many Worlds Interpretation). In fact, it's maybe worth my calling the former 3 Platonists and the latter 2 mathematical realists as there seems to be a big distinction here. I know a good bit about Deutsch's take on this and a fair bit about Tegmark's take, but I will nevertheless not try to speak for either of them and instead simply explain how I see the matter.

The main difference between the Platonists and I is that I'm thoroughly an empiricist. Godel and the others believed that mathematical truths could be justfied by "mathematical intuition" which results from a mystical connection between the mind and mathematics. None of that fits into my view at all. There is a longstanding tradition to treat mathematical truths as "a proiri" and "necessary" in a similar way that tautologies in logic are treated. I think this is a mistake. First of all, I don't believe there is a such thing as a priori truth. All truths must be justified empirically and so ultimately there is no truth which is not a posteriori in some sense. The so-called "truths" of logic I see as language conventions which are set up that way for convenience, not because they represent any actual truths about the world. But this does not make sense when you're dealing with actual truths about the mathematical objects themselves. The main thing that I think we need to do more of is to separate two historically conflated ideas from each other: Mathematics meaning a system of formal proof, and Mathematics meaning the structures which mathematicians study. But I don't think they are the same thing. Or if they are, it's not in an obvious way. My preference is for reserving the term Mathematics or "the mathematical world" or "mathematical objects" or "mathematical structures" for the things that mathematicians study, and to refer to the other thing as Formal Systems. There may be arguments for using other language here, but this is the convention I'll adhere to for the rest of the post.

Another difference between Penrose's view and mine is that I believe in one world whereas he believes in three. I acknowledge that you can approximately speak of a mathematical world, a physical world, and mental worlds (there are a lot of separate ones since there are a lot of different minds), but a part of my reductionist beliefs/suspicions is that each one reduces to the last. (This part I would thank Tegmark for.) Just as the mental worlds that constitute what people call "phenomenology" or "subjecthood" are illusions that ultimately reduce to the physical world, I suspect that our physical world is ultimately an illusion and reduces to a portion of the mathematical world. In other words, the property of physicality is an emergent one which certain mathematical structures acquire when they get complex enough. Once those physical objects combine and get even more complex, they start to acquire mental properties as well (they become conscious). Other types of worlds which can emerge once things get really complex is virtual worlds. For this you need a whole network of minds combining, for instance those connected through what we call the Internet, all participating in the same virtual world. In the future, I think virtual worlds will have the most practical importance, but the mathematical world is the root of everything, serving as the foundation of reality, supporting all of the other worlds on its back. The mathematical world is what I'd call the "real world"... the only one that is strictly speaking, not an illusion. The rest can all be deconstructed and shown to fall apart into nothingness, but mathematics is eternal and everlasting, and the whole of it will always remain a step beyond human comprehension. It's something we can't reach out and touch or feel or see, but it's always there behind our world, generating it and sustaining it.

So as an empiricist, how do I know that mathematical structures exist? What experience have I had that gives me evidence they exist? Well, to start out very simply, consider the first mathematical objects which were discovered, the natural numbers. Note that these were discovered long before formal systems were discovered. Before anyone had any notion of proof or even logic, we knew about the natural numbers. The classic experiment you can perform involves 4 apples. Take two of them in your left hand, and two of them in your right hand, and now bring your hands together. Now count them... how many do you have? You've demonstrated empirically that 2+2=4. This was known before there was any such notion of proof. You can do the same physical experiment anywhere you like and any time you like, and you'll still see that it's true. Fast forward all the way to modern theoretical physics. We've discovered empirically that our microscopic world is very precisely described by an SU(3)xSU(2)xU(1) quantum gauge field theory in the spontaneously broken Higgs phase. This is a particular mathematical structure, and trust me... it's quite a bit more complex than the structure of arithmetic of natural numbers. Between them is a whole continuum of complexity building up, and centuries of history as we did more and more complicated experiments and uncovered more and more abstract mathematical structures in the world. The thing that gets really erie, which is something which was pointed out by Wigner at some point (google "Unreasonable Effectiveness of Mathematics in the Natural Sciences")... is that by the time we uncovered this particular SU(3)xSU(2)xU(1) structure (known as the Standard Model of particle physics) it had become commonplace for mathematicians to discover new mathematical structures on their own before we do the official experiments which "discover" them! What's going on here? Rationalists (or even weaker empiricists than myself) would say that the mathematicians used their own "a priori" methods to discover the truth before it was discovered empirically. One striking example of this is the discovery of non-Euclidean geometry before we realized that the physical spacetime we live in is actually described by non-Euclidean geometry. A second thing that happens is theoretical physicists take current mathematical structures we know describe the world, and extend them in consistent ways that are motivated by solving problems with the current theory... and they end up predicting new discoveries before they happen. An example of this is Dirac's prediction of antiparticles from pure mathematics and theory. He predicted that an antiparticle must exist (in order to avoid backwards causality and the kind of grandfather paradoxes associated with time travel) for every particle in the universe... just by figuring out what the only new mathematical ingredient could be in our world that would give just the right mathematical cancellation to ensure causality. About a year after his prediction, the first antiparticle was discovered in the laboratory. I take stories like this (and many others) as compelling empirical evidence that the methods we use in mathematics, whatever you may want to call them, allow us to gain real knowledge and insight about the world. It's not just random guessing.

So while some would say that the discovery Dirac made was justified "a priori" before it was later justified "a posteriori", I would disagree. If this were the first time anyone had thought of using math to try to predict the real world, then there would be absolutely no reason to think it would succeed... there would be no justification to think that it would work or to think that what was claimed was true. The reason we believe math works is because it is justified a posteriori in the same way that the existence of the natural numbers is justified a posteriori because it predicts the right number of apples when you put your hands together.

So here comes the really interesting part. . . the way in which mathematicians study mathematics is to use formal systems of axioms in order to "prove" statements about them. They set up the axioms and then it's just a matter of turning the crank. And if they've set up the axioms properly, it appears to work. By "work" I mean that if the axioms they use define a structure that we've already discovered directly through sensory experience with the physical world (such as the natural numbers) then they always predict the right future sensory experiences when dealing with that same structure. After working with these formal systems a lot, what it looks like is that these axioms are a very reliable tool for studying these mathematical structures. Again, note that the only reason we know we can rely on it is because it has been shown to work empirically. Not because some foolish rationalist has "proven" it will work. Ok, so we trust that certain mathematical structures exist, and we trust that the axioms are a tool to investigate these structures. What else do we know empirically? Well, we also know that if we take one of the axioms and modify it but keep the rest the same... there's a very good chance that we'll end up with another structure that exists somewhere in the physical world. Sometimes you get one that we already know exists, but the most interesting case is when you get one that is later discovered to exist. That's where we get empirical evidence that even something more exciting is true! From cases like this (such as the example of non-Euclidean geometry) I believe that what the empirical evidence is screaming at us is this: our physical world (the world we experience with the senses) is built out of mathematical structures, and those structures can be generated by building blocks themselves, and the building blocks are in some sense the axioms. By recombining the blocks we've already discovered to exist in one form, we can create a new form which also exists. We've seen this happen again and again. So we have reason to believe (or at least evidence which should make us suspect) that no matter how we recombine these blocks we will end up with other things which also exist.

The situation with mathematics is analogous to finding a toy house made out of legos. You take apart the house and build a toy car. Your friend says "that's nice, but you can only build a house or a car... you couldn't, for instance, build a boat!" And maybe after only two trials, you would believe him... until you tried it and were able to build a boat. Then you build a tractor, and a limosine, and a truck, and etc. etc. Every time your friend says "but we only have evidence that you can build A, B, C, and D... we have no evidence that you could build E!" But I would argue that this is not the case. I would argue that you have good evidence to believe you can put the blocks together however you want and you'll get a new structure which exists every time. This is my argument for how we know that other mathematical structures exist besides the ones which we observe directly in the physical world. And I hope I've explained clearly why I see the justification for that knowledge as entirely empirical... it's not something that follows from pure reason. And it's not something that requires you to believe in a priori synthetic truths, or any kind of a priori truth for that mater! I remain an empiricist, even though I believe we have evidence that there are other universes besides ours and that "all" mathematical structures exist (where the "all" is a bit fuzzy and I will try to clarify in part 5).

I still haven't gotten to Turing Machines. Turing Machines are the link between formal systems and mathematics... or one of the links, anyway. I think this provides further evidence that what I'm saying is true and we should take it seriously. Other things I need to talk about in part 5 are clarification on which sorts of mathematical structures exist, how to deal with such a huge ensemble, what implications this has for the possibility that we're living in a simulation, and connecting it back to part 2 where I mention the one way in which I could see Copenhagen maybe actually working. Whew! Thanks for reading, if you got this far. I've been saving all this stuff up for a couple years, wanting to write it down. A lot of it I would have to thank David Deutsch for, as well as Max Tegmark for other parts. Both of them influenced my thinking on these issues greatly.

Comments

( 26 comments — Leave a comment )
fallen_x_ashes
Jan. 26th, 2008 08:59 am (UTC)
The mathematical world is what I'd call the "real world"... the only one that is strictly speaking, not an illusion. The rest can all be deconstructed and shown to fall apart into nothingness, but mathematics is eternal and everlasting, and the whole of it will always remain a step beyond human comprehension.

Not true. As an example, please show me an object in the real world that has a circumference of exactly 2pi-r. :-p

The various branches of math can't even agree on what the nature of the integers are. Math is pretty much the ultimate Platonic Device you can think of.
fallen_x_ashes
Jan. 26th, 2008 09:03 am (UTC)
And here's a book to add to your reading list.
Morris Kline


Edited at 2008-01-26 09:04 am (UTC)
spoonless
Jan. 26th, 2008 08:01 pm (UTC)
Re: And here's a book to add to your reading list.
Looks like a good book. I added it to my wishlist! thanks
spoonless
Jan. 26th, 2008 08:06 pm (UTC)

please show me an object in the real world that has a circumference of exactly 2pi-r

Our universe is a much more beautiful and complicated mathematical structure than a circle. If it were just a circle, it would be pretty boring. Nevertheless, it is just as "perfect" as a circle, as far as we can measure. Obviously, you cannot measure anything exactly as that would require an infinite amount of time (and in our universe, because of the way it's set up it would also require an infinitely large measuring device).

Incidentally, it is true that some of the features of our universe have been measured out to 12 decimal places and agree exactly with the pure mathematical structure we hypothesize it to be, to as far as we can measure.
dankamongmen
Jan. 26th, 2008 08:05 pm (UTC)
good article. i'm going to have to get you to review my book when it's done, i think?
spoonless
Jan. 26th, 2008 08:29 pm (UTC)
Thanks. What's it on?
darius
Jan. 27th, 2008 02:00 am (UTC)
Just want to say I'm reading and think this is good stuff, though I haven't the attention to spare to comment on it.
killtacular
Jan. 29th, 2008 11:42 pm (UTC)
Interesting stuff, some comments :

In other words, the property of physicality is an emergent one which certain mathematical structures acquire when they get complex enough.

Could you elaborate on that? It seems that many properties of physical things are just completely different than properties of mathematical structures. So how, exactly, does physicality emerge from appropriately complex mathematical objects. I can see how this might happen for certain sorts of complex physical things, but mathematical structures and physical objects just seem too different. So how does this happen?

You've demonstrated empirically that 2+2=4. This was known before there was any such notion of proof. You can do the same physical experiment anywhere you like and any time you like, and you'll still see that it's true.

This may be trivial, but the answer seems dependent in some ways on the fact that we have 10 fingers on our hands. If, say, we had four fingers (and so used a base-4 system instead) we would get 2+2=10. In some ways they are equivalent, but there is still something there I think ...

I believe that what the empirical evidence is screaming at us is this: our physical world (the world we experience with the senses) is built out of mathematical structures, and those structures can be generated by building blocks themselves, and the building blocks are in some sense the axioms. By recombining the blocks we've already discovered to exist in one form, we can create a new form which also exists. We've seen this happen again and again. So we have reason to believe (or at least evidence which should make us suspect) that no matter how we recombine these blocks we will end up with other things which also exist.

I'm not sure about this. Why not say that we've discovered that the universe can be described by certain mathematical structures, rather than these structures exist and the universe is built up out of them? I think all our evidence is consistent with this view which is ontologically more parsiminous than your view, which sanctions the "existence" of all types of mathematical structures independent of ourselves, or of logic, or the axioms we use to define them, or whatever.

Finally, for someone who takes a radically different opinion than you do, but who does it quite well and shows how (at least some) science can be done without mathematics, I'd recommend picking up Hartry Field's Science Without Numbers, just to see what you think.
spoonless
Feb. 3rd, 2008 03:05 am (UTC)
oh, and thanks for the book recommendation... I should probably read something like that to hear that side of it. I can't really tell what the book is going to be like though, just from the one review that's there.
killtacular
Feb. 5th, 2008 08:31 am (UTC)
It's pretty cool. If you have access to JSTOR, there is a review by Kenneth Manders from a 1984 issue of the Journal of Symbolic Logic that might be illuminating. Field basically argues that mathematical entities are eliminable from our scientific theories in favor of purely physical relations, and illustrates how this can be done for (classical) gravitational theory. It is an incomplete project, and maybe ultimately unsuccessful, but if you want to hear the opposite side its probably one of the best places to look.
easwaran
Feb. 7th, 2008 09:51 pm (UTC)
Field is great. In this book he gives a formulation of Newtonian gravitational theory where the only entities quantified over are physical objects, and the only relations and properties they have are physical ones, like spatial betweenness, and betweenness in mass. Then he shows that we can add a separate realm of entities ("real numbers") and add certain relations between physical objects and the numbers (like "O is located at x-coordinate N" and "O has mass N") and add the standard phrasing of Newtonian gravity, and he shows that anything expressible in the purely physical language that can be proven in this new theory can already be proven in the purely physical theory. Thus, we can treat the math as a "useful fiction" that simplifies our physical calculations, while admitting that the math is not indispensable, and therefore we have no good reason to think the mathematical entities exist.

Regardless of what one thinks of the metaphysical project, I think the mathematical project of extending Hilbert's geometry to a purely intrinsic characterization of Newtonian gravity is neat.
spoonless
Feb. 3rd, 2008 03:17 am (UTC)

It seems that many properties of physical things are just completely different than properties of mathematical structures.

People made similar arguments for why mental properties had to be irreducible. Some of them still even do. How could mental properties emerge from physical properties if they seem fundamentally different?

Well, in both cases I think people are just underestimating the power of emergence/reductionism. When things get really complex, they can look very different to humans.
A house doesn't resemble a pile of bricks very much, nor does a swimming pool resemble a swarm of threaded strings of H2O molecules slithering over each other, or a model of a single molecule that you might build out of sticks and balls.


[Specifically here, I'm thinking of the property of "wetness". Someone skeptical of science or reductionism might argue that you could never get something with the property of wetness just by combining molecules that by themselves don't have anything like this property. They might say you need something fundamentally different... a new ingredient. But in fact, it's just the molecules there... and when a huge number of them swarm together we use different words to talk about it, and our mental picture of what it's like becomes rather different.]

The biggest part of what we think of as physicality is something whose emergence is I think already understood. The main property that physical things seem to have is their rigidness... the ability for us to reach out and touch something and press on it, and feel it pressing back on us. We already understand that this is not due to any intrinsic "rigidness" of the parts making up the matter... it's just due on one level to the repulsive electrical forces between our hand and the electron clouds of the atoms on the outside of the "object". And on a deeper level, it's due to the Pauli exclusion principle (two fermions can't go into the same state, so they tend to spread out) which in turn just comes from the statistics of anticommuting mathematical structures.

The fundamental structures in quantum field theory (or string theory) that everything else is presumably built out of don't take up any space... like lines or points in mathematics, they lack dimensionality until there is a combination of several of them and certain "characteristic length scales" develop which makes them stay a certain distance apart and appear to "take up space".

If, say, we had four fingers (and so used a base-4 system instead) we would get 2+2=10

10 in base 4 represents the same thing as 4 in base 10. Two different languages, describing the same truth. The structure of arithmetic on the integers and the facts about what happens when you put apples together are both independent of the language you use to describe them.

Why not say that we've discovered that the universe can be described by certain mathematical structures

To me, the word described here is wrong because it implies that humans are responsible for making up the structures that are being called "merely descriptions". I think we have pretty solid evidence that we're not just making up these structures... we're discovering them... or at least part of them, and they would still exist and have the same structural relations even if humans had never discovered them. I see this part of it as fairly firmly established. The part that I'm arguing that I think is weaker is the part where I conjecture that the physical world is a subset of the mathematical world. As you point out, it seems incredible, and almost impossible, that these two could be the same thing. So maybe they aren't... maybe mathematics needs to be instantiated in some way in order to exist "physically". But even if that were true I would still believe that mathematical objects exist... it would just be that they lack certain physical properties.
killtacular
Feb. 5th, 2008 08:41 am (UTC)
People made similar arguments for why mental properties had to be irreducible. Some of them still even do. How could mental properties emerge from physical properties if they seem fundamentally different?

True, and the charge of a "category mistake" does need to be made with caution. But, for instance, mathematical entities are causally inert. No mathematical entity has any causal influence on anything. We appeal to them in our descriptions of causality, but the number "4" doesn't cause anything to happen. But physical entities do have causal influence, indeed, that might be a fundamental property that makes something a physical object : it can cause something else to happen, or not, or whatever. I guess my problem is I don't see how the complexity of mathematical structures gives rise to causality, or how causal influence can emerge from their complexity.

wo different languages, describing the same truth. The structure of arithmetic on the integers and the facts about what happens when you put apples together are both independent of the language you use to describe them.

You are right, and I don't think this is anything like a real objection. I think it might be illustrative, though, that at least some mathematical reasoning is a result of contingent facts about us, rather than about any platonic realm of "numbers" (which, although you don't want to go there, is where I think any strong realist view of mathematics inevitably leads).

I think we have pretty solid evidence that we're not just making up these structures... we're discovering them... or at least part of them, and they would still exist and have the same structural relations even if humans had never discovered them

Oh sure. But I'm not convinced that what we are discovering is mathematical structures, rather than lawlike regularities that we describe with our mathematical language. There is definitely something out there about the real world that we are discovering. But we don't ever discover that there exists a "set," or a "real number" or anything like that. We discover that we can make very good predictions, and we use mathematics to make those predictions. But that doesn't mean we are discovering any mathematical entities. Or, at least in my most nominalistic moods, thats how I see it.
easwaran
Feb. 7th, 2008 09:54 pm (UTC)
Is it really clear that mathematical entities don't have causal powers? Maybe we just don't understand the relevant notion of causation. They certainly don't transfer energy, or do many of the other things we associate with causation, and it's hard to conceive of relevant counterfactuals involving them, but neither of those is an especially good theory of causation.

Ben Callard, the husband of one of the Berkeley grad students, has a paper in last summer's Philosophia Mathematica (or maybe it was in the fall?) arguing that mathematical entities do have causal powers. I'm not convinced, but it's an interesting point.
killtacular
Feb. 7th, 2008 11:13 pm (UTC)
Hmm, ya, I've just kind of bought the Benacerraf line without really thinking through it, I guess. I certainly have a hard time seeing how they could have causal powers, and at the least they seem to fail to fall under many of the paradigm cases of causation. But there may be a different way to understand causation for abstract mathematical objects. And thanks for the article recommendation, I'll have to check it out next time I'm at the library.
spoonless
Feb. 8th, 2008 09:01 am (UTC)
regarding causal powers, see my comment to killtacular. Could be there are more sophisticated definitions for "causal powers" than I'm thinking of, but I would definitely regard such things as illusions as far as I understand them. I think it's been a long time since I've believed in "cause and effect" except in a sort of useful fiction sense.
spoonless
Feb. 8th, 2008 08:51 am (UTC)

No mathematical entity has any causal influence on anything.

I don't think I believe in "causal influence". There was a time when science described the world in terms of cause and effect, but I think that time has long past and there is no more use any more for such things, except to describe how the world appears to us. The way modern physicists describe the world is by figuring out the mathematical relationships between things in a holistic way, not by breaking things into different specific events and saying "event A causes event B". It could be that I'm just not familiar enough with the idea of "causal powers" but they seem like something that is purely an illusion... and an illusion whose origin is already understood. Most of the illusion I think comes from time... we experience time as if it is something we are "moving forward in" even though in some sense it is just as static as any of the other dimensions... our lives are finite worldlines in spacetime that follow a specific trajectory. Not things that are caused or unfold "as things happen".
spoonless
Feb. 8th, 2008 08:57 am (UTC)

But I'm not convinced that what we are discovering is mathematical structures, rather than lawlike regularities that we describe with our mathematical language.

Right. But this is just an issue of language. Do we call those "lawlike regularities" mathematics or do we call them physics? Or do we call them something else? I've come up with the best system I can for naming these different things given what I currently know. It could change in the future.
easwaran
Feb. 7th, 2008 09:41 pm (UTC)
I've got a bunch of things here that I want to comment on - hopefully I'll think of them all.

In your previous post, I almost mentioned Penrose as someone who misuses Godel's theorem - The Emperor's New Mind is I believe the locus classicus of his misuse. He seems to assume without any good argument that humans really can recognize our own formal system and recognize its Godel sentence as true. Hilary Putnam recently gave a talk at Berkeley (and I think elsewhere) where he came up with a better interpretation of the same argument - not that humans aren't Turing machines, but rather that we can't know which Turing machine governs our knowledge.

Coincidentally, I just started reading The Road to Reality yesterday on the plane back from NYC. I was quite struck by the metaphor of the three worlds, but I didn't realize how seriously he might actually take the existence of these three separate worlds. I was thinking of it more as the claim that all mental phenomena are physical, all physical phenomena work mathematically, and all mathematical phenomena are understandable by the mental. This doesn't commit one to anything about whether or not these three realms are at all separate or the same, and if they're the same, then which is fundamental (which probably doesn't actually make sense to ask in the end, if they're the same). I probably disagree most with the third claim - there's no good evidence that all mathematical phenomena are understandable, and in fact we know there's all sorts of non-computable, non-definable, and non-constructive stuff that goes on all over the place in mathematics, so we seem to have good reason to think some of this stuff can't really be grasped (like a particular non-measurable set).

What I recall about David Deutsch is that he seems to think that the physical world is fundamental and that mathematics is essentially a physical phenomenon (proofs are concrete inscriptions, and so on).

I'll put the rest in another comment, because I might be running out of space.
easwaran
Feb. 7th, 2008 09:46 pm (UTC)
I'm generally inclined to go along with you and say that there's no distinction between a priori and a posteriori knowledge. However, there are some important objections to this claim. One is that in order for empirical knowledge to be possible, it looks like there has to be some way for us to recognize which evidence supports which conclusions. In fact, we might even have to be able to know this before having the evidence, in order for it to justify things properly. But then these facts about the relation between evidence and conclusions would be ones that we could know a priori.

Also, you seem willing to concede that logical truths are just truths by convention. But why don't you concede that about mathematical truths? Mathematical terms might just be conventional abbreviations for "whatever things happen to satisfy axiom system S" (it seems that terms like "group" and "Hilbert space" and the like really do work like this), and then truths about these things just are logical truths about these conventions.

But I think Quine has some serious arguments against the possibility of truth by convention - I'll have to read that paper again and see what I think about it.
spoonless
Feb. 8th, 2008 08:40 am (UTC)

in order for empirical knowledge to be possible, it looks like there has to be some way for us to recognize which evidence supports which conclusions.

I don't have a good "knock down" answer to this at the moment. Just a strong feeling that it's not really a problem.

you seem willing to concede that logical truths are just truths by convention. But why don't you concede that about mathematical truths?

Because I see mathematics as distinct from semantics. This is what I meant by "we should separate formal systems from mathematics more". Maybe the thing I'm calling mathematics shouldn't really be called mathematics... maybe it should be called something else... but for now it's the word I use to refer to the actual things that I believe mathematicians are studying. Not the axioms and proofs, but the topological spaces, categories, uncountable sets, groups, varieties, schemes, braids, sheaves, etc. themselves. I think that you can use logic to study these things just like you can use logic to study the sensory world (for instance, you can deduce that Socrates is mortal if you know that Socrates is a man and that all men are mortal... might come in handy if you get into a brawl with Socrates).

I think that there is a part of the world that transcends logic. When I say "the world" I mean both the thing Penrose would call the mathematical world, and the thing he would call the physical world... since I see them as the same thing. It could be that a better term for what I call mathematics would just be "the physical world". I've thought about calling it that, but I feel like it's not really the right term since most of the things in it are not "physical" in the usual sense of the word... ie, they are not tangible, you can't touch them, listen to them, eat them, etc.



Edited at 2008-02-08 08:42 am (UTC)
easwaran
Feb. 7th, 2008 09:56 pm (UTC)
Oh, also:

We've discovered empirically that our microscopic world is very precisely described by an SU(3)xSU(2)xU(1) quantum gauge field theory in the spontaneously broken Higgs phase. This is a particular mathematical structure, and trust me... it's quite a bit more complex than the structure of arithmetic of natural numbers.

Is this really at all more complex than the structure of arithmetic of natural numbers? Godel's completeness theorem can be proved in an arithmetical form, so that every consistent theory has a model in the natural numbers. So in some sense, arithmetic of the natural numbers is as complex a structure as there can be.

Also, I think I see how I managed to get more comments than anyone else in your journal...
spoonless
Feb. 8th, 2008 08:02 am (UTC)
hmm... you may have a good point, I don't know enough about this to say. I don't fully understand what the consequences of the completeness theorem are or what it applies to... but my first objection would be that I've heard it only applies to first order logic. When you say "every consistent theory has a model in the natural numbers" you mean any theory based on first order logic, right? What if you want to make statements about countable versus uncountable sets or something? Quantum gauge field theories have not been axiomitized yet, because all sorts of weird infinities show up... so I think we don't know yet what sorts of axioms will be required to describe them. But like I say, I don't know too much about this stuff so maybe what I'm saying doesn't make sense.

Also, I think there is still a sense in which it may be "more complex" than the natural numbers, even if you can represent it by a set of axioms that has some direct correspondence with the natural numbers.
easwaran
Feb. 8th, 2008 05:51 pm (UTC)
Yeah, any theory expressible in first-order logic. The natural numbers obviously don't contain any actually uncountable structures (and even including sets of naturals, you don't get anything bigger than the continuum - that's still enough to do quite a lot though, including talking about analytic subsets of the reals and the like). However, there is for instance a model of ZFC that is countable and contained entirely within the natural numbers. According to this structure, some sets are countable and others are uncountable. However, this is an internal distinction between countable and uncountable, which doesn't necessarily line up with the external distinction that we make. (This result is called Skolem's Paradox.)

You may well be right that there's an important sense in which this other structure is more complex than the naturals, but I was just pointing out that the naturals are actually already hideously complex, so there may not be a sense in which anything definable can be more complex.
(Anonymous)
Apr. 5th, 2008 10:57 pm (UTC)
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Feb. 12th, 2011 04:44 am (UTC)
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