Over at The Economist, we have this brief interview with Edward Frenkel, a mathematician at Berkeley. We met Frenkel in this post from last week. Frenkel has a new book out called Love and Math, one copy of which is currently residing on my Kindle.
One part of the interview caught my attention:
Symmetry exists without human beings to observe it. Does maths exist without human beings to observe it, like gravity? Or have we made it up in order to understand the physical world?
I argue, as others have done before me, that mathematical concepts and ideas exist objectively, outside of the physical world and outside of the world of consciousness. We mathematicians discover them and are able to connect to this hidden reality through our consciousness. If Leo Tolstoy had not lived we would never have known Anna Karenina. There is no reason to believe that another author would have written that same novel. However, if Pythagoras had not lived, someone else would have discovered exactly the same Pythagoras theorem. Moreover, that theorem means the same to us today as it meant to Pythagoras 2,500 years ago.
This is a common view among mathematicians, but it doesn't make much sense to me. I don't know what it means to say that mathematical concepts exist outside of the physical world and outside of the world of consciousness.
To make it specific, consider the notion of a continuous function. The idea of a continuous function is an abstraction developed by mathematicians because it is useful for modelling a great many processes in the physical world. In what sense would this abstraction continue to exist if humanity suddenly winked out of existence? We could imagine that the physical processes themselves would continue to exist. But what could it mean to say the abstract model itself would still exist?
Of course, when you undertake mathematical research it certainly feels as though you are making discoveries about objects that exist independently of anyone's ideas about them. That's because you are. But you're not so much making discoveries about the objects themselves, but about the logical relationships among the concepts you have chosen to define. It is like chess. The rules of chess are invented. But then you discover that a consequence of those rules is that a king, knight and bishop can force checkmate against a lone king while a king and two knights cannot. The rules of chess are so rich that even after centuries of analysis people continue to play the game and to make new discoveries about it. And so it is with the objects mathematicians choose to define and study.
So it seems to me at any rate. But Frenkel takes a different view:
So it’s not subject to culture?
This is the special quality of mathematics. It means the same today as it will a thousand years from now. Our perception of the physical world can be distorted. We can disagree on many different things, but mathematics is something we all agree on.
The only reason the theory means the same is that it describes the reality of the physical world, so mathematics must need the physical world.
Not always. Euclidian geometry deals with flat spaces, such as the three-dimensional flat space. For millennia people thought we inhabited a flat, three-dimensional world. It was only after Einstein that we realised we lived in a curved space and that light doesn’t travel in a straight line but bends around a star. Pythagoras theorem is about geometric shapes in an idealised space, a flat Euclidian plane which, in fact, is not found in the real world. The real world is curved. When Pythagoras discovered his theorem there were, of course, inferences from physical reality, and a lot of mathematics is drawn from our experience in the physical world, but our imagination is limited and a lot of mathematics is actually discovered within the narrative of a hidden mathematical world. If you look at recent discoveries, they have no a priori bearing in physical reality at all.
The naive interpretation that mathematics comes from physical reality just doesn’t work. The other interpretation that mathematics is a product of the human mind also has serious issues, because it seems clear that some of these concepts transcend any specific individual.
Take Evariste Galois, who was killed in a duel at the age of 20. He came up with a beautiful theory on symmetry called Galois theory. His contemporaries didn’t get it but this theory now forms the core of modern mathematics. But what if the work had been burned? Would we never have known Galois theory? No. Someone else would have discovered it because it is inevitable.
I don't see how that paragraph about Euclidean geometry justifies Frenkel's claim that mathematics doesn't come from physical reality. It is certainly true that Euclidean geometry is not adequate as a description of the large-scale structure of the universe. But the fact remains that it is a wonderfully accurate model for the world of our day-to-day experience. We hardly needed modern physics to tell us that Euclidean geometry is about an idealized space that does not exist in the physical world. That fact does not weaken in the slightest the suspicion that mathematical objects come from physical reality.
Indeed, the whole point of developing abstract models is to strip away the extraneous details of the real world so that we can focus solely on what's important. That our abstract constructions do not actually exist in the real world is precisely the point.
As for Galois theory, of course someone else would have discovered it had Galois never been born. But that's because the logical relationships among the objects Galois studied were out there waiting to be discovered, as per my earlier chess analogy. But all of the abstract objects Galois studied, like fields, groups and polynomials, ultimately owe their existence to abstractions drawn from physical reality.
I cannot think of a single example of a mathematical object which doesn't owe its existence ultimately to some real-world consideration. Of course, the mathematical world is vast and I have only studied a small part of it, so I'm certainly willing to consider possible counterexamples I have overlooked. But I'd like to know what specific examples Frenkel has in mind. I guess I should read his book!
Time to wrap this up, so here's one last excerpt:
Because it is simply true?
Yes. It’s a difficult philosophical question, to which we still don’t have the answer, but it’s an important question to be aware of. It’s not the same as the mathematics we use to calculate a tip—it goes to the heart of reality and of consciousness. It is all around us, with smart phones and computers and GPS devices and the algorithms that control our lives. The Amazon recommendations we are offered are based on very sophisticated algorithms, which analyse our past purchases, correlating us with other users. Mathematics is invading our world more and more and it communicates timeless, persistent and necessary truths which transcend time and space. The Langlands Programme should be as familiar to us as the theory of relativity.
I've gotten a little disenchanted with talk of “the heart of reality and consciousness.” It's too grandiose for my taste. I think it's enough just to say that certain abstract models help us order our experiences and certain other models are not helpful. To go beyond that seems, well, unhelpful.
And that's how I view talk about the existence of mathematical objects. It's not helpful. What matters about mathematical objects is what you can do with them, not the sense in which they can be said to exist.
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I don't agree that math exists "outside" the physical world, it's more like the scaffolding to it, the smallest pieces, the structure required for the assembly of it - at least that's what I understood from Garret Lisi's work: it's likely everything is made up of one tumbling equation.
I don't think it is outside of consciousness either because I'm an adept of "biocentrism" and I believe consciousness comes *before* the physical world, but that's bearing on religion... so I'll stop here. But I do believe it.
Good argument!
I entirely agree with your conclusion, Jason.
I think Frankel is mainly trying to express the idea that there's a sense in which mathematics is objective, i.e. observer-independent. I would say he's right about that, but a clearer way to express it is to say that mathematical facts are true in an observer-independent sense. Focusing on "objects" and "worlds" (as mathematical Platonists do) is unhelpful. I'm not sure I would say that mathematical Platonists are necessarily wrong, but at least they have an unfortunate and potentially misleading way of expressing themselves.
"I cannot think of a single example of a mathematical object which doesn’t owe its existence ultimately to some real-world consideration."
Umm... you've just slipped into talking about the existence of mathematical objects (which you later say is not helpful)! But I think your meaning here is reasonably clear, because you're using the expression in relation to the history of (human) mathematics, so it's apparent (I think) that you're referring to human use of such objects. Context is all-important. Still, I would replace the word "object" with "concept" here.
It's useful to have some word to refer collectively to such things as numbers, sets and functions, and the word "objects" serves that purpose. My word "things" is just another word for "objects". Also, it's perfectly meaningful for mathematicians to talk about existence within the context set by their axiomatic systems, e.g. to say "there exist n prime numbers between A and B". But mathematical Platonists steps outside these contexts and use the terms "object" and "exist" in a new context where there's no established usage that makes sense of them.
Part of the problem, I think, is that the grammar in which we speak about mathematics is very similar to the grammar in which speak about the real world. Both use nouns and verbs, for example. And once we assign a concept (e.g. "number") the grammatical role of a noun, we reify it, and start to treat it as rather similar to real-world things. But mathematics isn't really about objects. It's about relationships, in a sense. (I think Hume referred to mathematical/logical statements as "relations of ideas".) We can't separate the "objects" from the axiomatic systems in which they occur. A "line" in Euclidean geometry is not quite the same thing as a "line" in a non-Euclidean geometry. And none of these is quite the same as the word "line" as we use it in talking about the real world. But it's difficult to avoid conflating these subtly different meanings of "line".
Anyway, returning to the substance of your sentence. It's certainly possible to create axiomatic systems without any thought as to whether they have real-world applications, and I'd be surprised if no mathematician has ever done that, though you should know better than me. What about complex numbers? Was the first person to think of the idea trying to solve a real-world problem from the start? Or did he just think it was a neat idea, and came up with real-world applications later?
Oops. I meant "Frenkel".
I usually call myself a fictionalist about mathematics. I had somebody argue that I'm wrong, that I should be a platonist in order to explain why mathematics is useful.
So I thought about platonism. But platonism itself is surely just a made up fiction.
Jason, This question does have a certain air of academic pointlessness to it. I don’t see how it matters much, and why not just say some maths are “out there” and some are pure abstractions, and some—meh—we don’t know.
I guess I don’t see a reason to care much.
sean s.
As far as I know this is an unresolved question.
First of all: what do we mean with "exist"? Take for instance gravity. In the classical meaning of the word it is a force. Do forces exist? It's not that hard to answer this question with no: we do not observe forces, we only observe their consequences (changing form, direction of movement or velocity). Clearly Frenkel thinks forces do exist anyway. It's peculiar though that he doesn't argue why; it's not surprising that he doesn't account for the fact that the word "force" has received a different meaning at least twice in human history. Aristoteles of Stagyra meant not the same as Isaac Newton. In Modern Physics the meaning has changed so much that physicists prefer "interaction".
So my second question is: how can we still maintain that gravity (a special case of force or interaction) exists?
My third question is: what's the relevance of this question? What we do know is that force and interaction belong to meaningful concepts. Why is this not enough for mathematicians like Frenkel? What philosophical problem is Frenkel solving by stating that "symmetry exists independen of the physical world"? And where exactly does he recognize that math is a language, with grammar and all?
http://en.wikipedia.org/wiki/Language_of_mathematics#The_grammar_of_mat…
So I suspect Frenkel is saying something like "the word table existed before the first human being constructed one". What meaning does that even have?
Maybe. Since mathematics is used as a tool to understand the physical world, then in a sense that makes it like engineering a measuring instrument. Creating calculus is somewhat analogous to building a better yard stick.
Now, I would never say that learning how to build a better yard stick is "only" making discoveries about yard sticks. Building such a thing is likely to result in a lot of important peripheral knowledge (about sticks, and measurement, and what sort of yard stick to use when, and, and, and...). Since the human endeavor of building the better yard stick probably helps you to build a better skyscraper or cannon or microscope, then it seems a bit mean and miserly to say that your yard-stick-building "only" taught you about yard sticks. In a very narrow sense of teaching and understanding that may be true, but I'm fine with using a broader sense of those terms too. Mathematical discoveries teach us things about the objective world, because they contribute directly and participate heavily in our gaining an increased understandig of that world.
I don’t even know what existence is. I like the remark of Sidney Morgenbesser, who said, in the midst of a philosophers’ discussion of existence, “Does my lap exist?”
This seems to be a timeless question. Personally, I think mathematical concepts do model real world patterns, at least to some degree of approximation.
Of course, you could say that since mathematics doesn't exactly match real world patterns, then it's just a tool invented by humans. But you could say that about *anything*. A tree exists, except that our conception of a tree is just a mental model we use to describe a complex evolving pattern of elementary particles. Therefore, we could say that the tree concept is just a tool for humans. From a certain point of view, that view would be true. But I'm not sure how useful that truth is.
Dear Jason -- Thanks for your post. The ontological nature of mathematics is a deep philosophical question which is far from being settled. I discuss it in the last chapter of my book "Love and Math", and I also recommend Roger Penrose's book "The Road to Reality" and Mark Balaguer's book "Platonism and anti-Platonism in Mathematics" (Mark Balaguer also has a nice short survey article on these matters, available here: http://www.calstatela.edu/faculty/mbalagu/papers/Realism%20and%20Anti-R…).
I don't want to get into too much detail here, but let me just say this: one can debate whether mathematics is the product of human mind or part of some Platonic reality. Either point of view leads one to some very tough questions, which so far have not been answered in my opinion. So the jury is still out on this question, as far as I am concerned (my conjecture is that at least some part of mathematics exists outside of human consciousness, but I can't prove it). However, there is no question that a huge chunk of modern mathematics has no bearing whatsoever on the physical world around us. I'll give you just a few examples: p-adic numbers, étale cohomology of algebraic varieties over finite fields, and infinite-dimensional Banach spaces. If someone can point them to me in the physical world around us, I would be interested to see that. I don't see them around us, and moreover, if you look at the history of mathematics, these objects were not discovered by mathematicians through the study of the physical world. And yet, to a mathematician they are just as real as stars and atoms. So there are a lot of things to ponder here. "There are more things in heaven and earth, Horatio, Than are dreamt of in your philosophy."
See also another interview I gave recently on related topics: http://gildedbirds.net/2013/12/02/edward-frenkel/
WIth best regards,
Edward Frenkel
Hi Edward. Thank you for the kind reply to my post. I'm looking forward to reading your book, and now that our semester has ended I should actually have time to do that!
I would like to reply briefly to your remarks about mathematics and the physical world, since it is possible we are speaking at cross-purposes. I think a lot depends on what you mean when you say that certain mathematical objects have “no bearing whatsoever” on the physical world.” That's pretty strong. The claim I was making was that the objects mathematicians study, no matter how abstruse, ultimately owe their existence to some real-world consideration.
Take p-adic numbers for example. Certainly the rational numbers arise from real-word considerations, as does the idea of embedding the rationals within some larger field. The p-adic numbers then arise from investigating what happens when we fiddle with the definition of “distance” in this environment. It is not that the p-adic numbers themselves have some obvious real-world applications, but that they arise naturally from a consideration of things that do have real-world significance.
I would say the same about infinite dimensional Banach spaces and etale cohomology. A Banach space is a special kind of vector space, which is a notion that arises from analytic geometry, which plainly has real-world applications. Cohomology arises from homology, which arises from natural questions about the topological properties of actual surfaces.
Mathematicians certainly study a great many highly abstract things that are quite far removed from their historical foundations and real-world applications. But we don't just produce abstractions from nothing. So I don't think p-adic numbers and the like are evidence for an independently existing mathematical realm that we gain access to through our consciousness. I think instead they are evidence for the power of human curiosity and imagination.
Certainly these objects feel real to us when we study them. Chess pieces feel real to serious chess players, but I wouldn't say they exist within their own mind-independent realm. (Are there chess Platonists?) Fictional characters seem real as well, and we routinely discuss them as if they had an existence outside the stories in which they appear. But here, again. it would just seem weird, at least to me, to say they exist in their own Platonic realm.
In short, I would simply return to what I said in the post. I just don't understand what it means to talk about existence in this context, and I don't find the idea helpful. But there is certainly no shortage of people who disagree with me, so maybe I need to give Platonism another look.
This is the most pertinent question you could have asked. I'm currently finishing my PhD in music composition, and we have lots of similar discussions about Platonism and musical structures. I think about it almost every day.
One way of looking at these things is to say that once a set of axioms is invented, all of the theorems derivable from those axioms are invented right along with the axioms as a logical necessity. However, given a rich enough set of axioms, it takes tons of work to "find" those theorems -- one hears about new "discoveries" in some mathematical field or other. This process often feels almost like a kind of archaeology or other applied science because the results can be so unexpected in what seemed to be fallow ground at the outset of the investigation.
But -- "given a rich enough set of axioms" is a pretty big "given." Finding a desirable set of axioms is kind of empirical, in that it can be seen as an iterative honing based on the fruitfulness of the theorems they deliver. Go back to chess -- the rules of chess are amazing, but I doubt anyone had the foresight to simply sit down one day and invent the rules in a few minutes. The size of the board, the opening positions, the relatively immobile king, the variations on knights' moves, castling, etc. are all parts of the rules, but those rules likely had to be tempered by play with older, less satisfactory rules. In some sense, the balance of the rules as we know them had to be "discovered," and in that sense it might make sense to speak of chess Platonists, but I think that goes a bit too far. It's not that these rules were "out there waiting to be discovered," but that the inventors of chess were reasoning "forward" and deductively with game play according to the agreed upon rules, and "backward" and inductively about the fecundity of the rules themselves. A good painter requires similar multilevel thinking -- he has to negotiate foreground, background, detail, and form all at once, and make adjustments to each as he proceeds.
By the way, it's totally possible to be a Platonist with respect to Anna Karenina: It's one of the books in the Library of Babel. It's just not nearly as obvious a "discovery" in that library as the Pythagorean Theorem is in the Book Of All Theorems, so of course it's exceedingly unlikely that another author would have written it.
In what sense do mathematical objects exist? In what sense does the color blue exist? We've entered the realm of very meta-physical philosophy.
I second the motion: we are all pleased that Jason again has the time and motivation to write more here.
As a mathematical platonist (but maybe more emotionally than rationally!) I of course strongly disagree with him on the issue here. And I don't feel totally naive, given that some serious contributors to human knowledge advocate mathematical platonism, even though they don't entirely agree with each other on all details. Examples are Penrose above; David Deutsch, the inventor, if there is one, of quantum computers; Max Tegmark, a physicist at MIT; and now Edward Frenkel. And I'm not trying to give an argument 'from authority'---many serious intellectuals are on the opposite side. Some here seem to think that even discussing the question is a waste of time, despite 2300 years (at least) of these discussions.
Tegmark's position (from a few years ago anyway) has not appeared here yet. It would, in a sense, reverse things: the entire natural world, in its guise as part of his Multiverse IV, is asserted to BE (not just be approximated by or described by) a mathematical system. If the arguments for this were accepted, one would have to solipsistically deny the existence of any physical world outside one's own mind to deny the existence of that one particular mathematical system, whatever it might turn out to be, were humans able to find it. (Read his stuff for details, but here a system is 'up to isomorphism' so to speak, and I'm not as confident as he seems to be that the idea of such a system is in final form at all; and Tegmark would assert that the previous follows logically from accepting the existence of a "Final Theory" as Stephen Weinberg puts it; and finally he presents the possibility of cosmological observations which could falsify his claim, so it's not just philosophical puffery). It does seem to finesse away the 'problem' of the "unreasonable effectiveness of mathematics", given the reasonableness of good approximations to a system being themselves mathematical systems.
If some abstract system is 'our' physical reality, it would be hardly surprising that every mathematical system 'physically' exists, and there's really no difference between abstract existence and physical existence. So Tegmark's is a pretty strong version of mathematical platonism. It would be interesting to hear if anyone here can see some fatal flaw in this line of argument; and also if anyone here actually accepts that a final theory exists, and yet our physical reality is somehow not a mathematical object.
Please don't conflate the Quantum Field Theory as it exists in, say, Frank Wilszek's mind, with some specific mathematical system of QFT like the Standard Model, which would be to us a very good approximation to part of the final theory, if it exists.
Where would anti-platonists draw the line dividing existence from non-existence?---at their own body?---at its cells?---at the molecules from which the cells are constructed?---at the quarks 'in' the protons in the nuclei in the atoms in those molecules (to cut it short)?---at the field whose quantization produces those quarks? Those fields in the Standard Model are pretty much mathematical objects, as far as I can see. In Alaska and Iceland, you can often more-or-less see the electromagnetic field some nights. Does something not existing admit a quantization which produces objects which do exist but are not mathematical? Are the mathematical theories which explain the world better and better somehow not approximating any fixed mathematical object, but just going on forever getting closer and closer to something which somehow is not itself mathematical?
Jason's contention that all mathematical knowledge of humans can be traced back to physical observation seems to be quite compatible with this radical platonism, despite his different fundamental conclusion.
Sorry, "Wilczek" it should have seen spelt (or spelled, for USians, I guess!)
To me, mathematics is a hybrid discipline - it is part engineering (mathematicians build mathematical structures), part science (math studies the behaviors and relationships of those very structures); so, although it may be inspired by various aspects of reality, math in itself is not about reality. It just so happens that we focus on mathematical constructs that are most useful when building models that help us understand the world we live in, models we can test against. So, to me, mathematical objects are artificial constructs. However, this is not a bad thing; it is what makes them so powerful - to insist on the idea that they are necessarily manifestations of universal laws or hidden truths is, IMO, to impose artificial limitations on the field.
Many interesting and thought-provoking comments here, and thanks again, Jason, for initiating this discussion. I think at the very least we should all agree that the issues discussed are complex, and one should not jump to conclusions.
I want to comment on something Jason wrote above:
"The claim I was making was that the objects mathematicians study, no matter how abstruse, ultimately owe their existence to some real-world consideration."
I am not convinced that this necessarily means that mathematical objects EXISTin physical reality. The fact that the p-adic numbers are RELATED to the rational numbers (which of course have manifestations in physical reality) does not mean that p-adic numbers EXIST in physical reality. In my view, there an essential difference between the two notions. What's more, mathematicians were not led to the p-adic numbers because they tried to understand something physical reality. So I don't believe that p-adic numbers exist in physical reality (even if they are connected to it). And ditto for the other examples I mentioned above (étale cohomology and Banach spaces; the list can go on and on). But then, if we accept that p-adic numbers do exist, then where do they exist? (Balaguer's article linked to in my comment above gives a good summary of possible answers to this question.)
Here's another example of the same phenomenon: I would argue that Bernhard Riemann (following in Gauss' footsteps) discovered what we now call Riemannian geometry not because he tried to understand something about the physical world. He was trying to describe curved spaces intrinsically, without reference to an ambient flat (Euclidean) space in which they are embedded. Why? Surely, everyone "knew" at the time that curved shapes "can only occur in the physical world inside a flat space", so this must have seemed to others as an abstract exercise that is irrelevant to physical reality -- or to our consciousness, for that matter: Kant, for example, wrote that Euclidean geometry "an aspect of the human mind, the expression of the way the mind confronted the sensuous world of shapes.".
But they turned out to be wrong! Einstein's relativity theory (which was based on Riemann's work) showed that the space we inhabit is curved, and this curved shape is NOT embedded into anything. And to even begin talking about this, one really needed a theory describing such spaces without reference to the landscape into which they are embedded-- that is, precisely the theory that Riemann had developed!
However, I believe that Riemann was NOT motivated by the desire to describe physical reality. In retrospect, he did -- but we only found out about this more than 50 years after his work was finished. I think a much more reasonable explanation is that Riemann was discovering something within some "mathematical reality", but his theory then turned out to be applicable to physical reality (and also connected to our consciousness in some way).
I think this is a very good example of a subtle interplay between three worlds: physical world, mental world (consciousness), and the world of mathematical ideas (one can call it "Platonic world", but that's just a matter of terminology). So there is this triangle, which Roger Penrose (and others) have been talking about for many years. We all agree, I think, that these three worlds exist in some form, and are intricately intertwined with each other. The question is how they are connected. Is any of these worlds subsumed by the two others?
There are many possible answers to this question. My view is that although there are big overlaps between these three worlds, each of them has a certain autonomy, and cannot be reduced to the other two.
Thank you for your comments professor Frenkel. As a philosopher I don't work in the area of mathematics but find the issues being discussed interesting.
Edward--
I can certainly second your remark that these are difficult and complex issues! Since I think it is likely that I will be devoting future blog posts to these questions, and since I think these blog conversations have a tendency to go on forever if folks aren't careful, I will reply briefly and then let that be the last word from my side. For now!
In his short book Mathematics: A Very Short Introduction, Timothy Gowers writes, “[T]here certainly are philosophers who take seriously the question of whether numbers exist, and this distinguishes them from mathematicians, who either find it obvious that numbers exist or do not understand what is being asked.”
I agree that mathematical objects do not exist in the physical world, but we don't need anything as fancy as p-adic numbers to make that point. The whole numbers also do not exist in the physical world. They do exist in a different sense, though, and it is the same sense as the black king in chess exists (to borrow another example from Gowers). The black king exists for a group of people who agree to the rules of chess. It exists in the sense that it has a role to play in the game. And that is roughly what I think of mathematical objects as well.
Another example might be the Supreme Court. It exists, surely, but in what sense? It certainly does not exist in the physical world, in the sense that there is nothing you can point to and say, “That is the Supreme Court.” The Supreme Court is not the ensemble of justices, nor is it the building in which the justices meet, nor is it the paragraphs in the Constitution that created it. Rather, it has a sort of social existence. It exists in the sense that some group of people just agree to acknowledge its authority and procedures..
Likewise for mathematical objects. In the context of modern abstract algebra I know what it means to say that finite simply groups exist. But if you ask whether they existed prior to the advent of modern algebra I no longer understand what you are asking. Did the Supreme Court exist prior to the ratification of the US Constitution? Were the nation's founders gaining insight into some mind-independent realm of legal institutions?
As it happens, I discussed my views on these issues in more detail in this post from last year.
I would also certainly agree that mathematicians routinely study things with no real-world application in mind, and then discover later that their work has applications after all. If it were the case that these mathematicians were studying abstractions they simply created from whole cloth, then I would think there's a basis for believing in an independent realm of mathematical existence that interacts in some way with the physical world. But that's why I emphasize the fact that even the most abstract mathematical objects owe their existence ultimately to real-world considerations. Realizing this creates a far-different impression, I think. Now it seems far less like we are gaining insight into some mind-independent mathematical realm, and far more like we are finding that a tool invented for one purpose turns out to be useful for some other purpose that no one had considered when the tool was first invented. Screwdrivers were invented to drive screws, but they also turn out to be useful for prying the lids off of paint cans. Do we need a philosophy of screwdrivers to explain that?
That's how it seems to me, at any rate. But I'll leave it there for now. Thanks again for your comments. They provide plenty to mull over, as I'm sure your book will as well.
I think mathematical objects exist in a very real sense: The universe itself is a mathematical object.
Let me make this more explicit:
I think mathematical existence is the only form of existence there is. An object exists if and only if it is logically consistent.
That is why the universe exists.
Your two comments are not saying the same, W. In your second one you point out that logical consistence is a necessary condition for something to exist. I agree - but it leaves the option open that there are more necessary conditions.
Hence your first comment needs more back up.
Quoting Jason, where he has some positivity towards (1) but apparently much less towards (2):
(1) "..an independent realm of mathematical existence that interacts in some way with the physical world..." and
(2) "... some mind-independent mathematical realm..."
To reinforce my earlier response, (2) bothers me no more than having some mind-independent physical realm. With something like the Higgs particle, sensory evidence for the latter seems pretty remote, and existence equally related to Hilbert's 'existence=consistency'.
And sympathy for Tegmark's view would say that the two realms are actually the same thing. And that would then make the problem of the "...in some way.." in (1) basically become a non-problem.
To say that it is obvious that abstract and physical objects are different may someday become no more convincing than to say that the earth is flat, or to say that living objects obviously consist of different materials than non-living objects.
I think it's a bit of a meaningless discussion. When mathematicians use the word exists, they use it in a completely different context/meaning than when I say the apple in my fruit bowl exists (and will cease to exist after i've had lunch.)
When you say a continuous function f : A -> B exists, you mean to say it's possible to assign an element of B to each element of A in such a way that ...
This is "existence" in a completely different sense than the existence of an apple, and while people may disagree whether the function exists, I doubt anybody (except some really weird philosophers) will doubt my apple exists.
I think almost everyone will agree that maths is objective, the remaining discussion is one of semantics, what does mathematical existence really mean, and when can we use the word?
Rebutting an argument by calling the arguer "weird" fails to convince at least a few people. Near the beginning of Daniel Dennett's book on consciousness can be found something more convincing, I think, if it is solipsism and 'brain in a vat' that you are thinking of.
Mathematical existence and physical existence are somewhat different if and only if mathematical objects and physical objects are different. Most would say they are, but not all: (pardon the broken record, but) the Annals of Physics doesn't publish 51-page papers just to pad the CVs of the weird. In any case, no padding is needed. Look for vol. 270 (1998). pp.1-51.
Dr. Frenkel:
I am not sure what that third world adds to your example, or what prevents a 'proliferation of worlds' if we accept it. After all, language also contributed to the understanding of Riemannian geometry, so does that mean there's a language world? And if Reimann's dreams helped him come up with it, does that mean there is also a dreaming world along with the mathematical world, language world, physical world, and mental world?
It seems to me that it is much more consistent with observation, and philosophically conservative, to say there is only one world. The other "worlds" are not literal worlds. They are useful linguistic shortcuts for describing the many various ways that we - things in that world - interact with and interpret other things in that world.
Peter Hoffman @25:
The notion doesn't bother me, but IMO there's no evidence for that hypothesis. The notion of there being unicorns doesn't bother me either, but that doesn't mean I believe they exist.
W@23:
Okay, but which logic?
I accept that the world is probably consistent with some logic. That's really just equivalent to saying that the world runs by some set of rules. But I think you're jumping the gun when you assume it must run by human standard two-value logic.
I am not a philosopher; I work in IT supporting an engineering product database. In this database, we have part numbers. Each number represents a product design, each of which corresponds to one or more actual instances of that product (since we tend not to design things unless we have a need to produce them). The product instances indubitably exist. The product drawings also exist (we can point to one on the computer or on paper and say, "that thing"). To keep track of products and product drawings in the database, we use a system of taxonomy familiar to any student of biology, in which we assign each product or drawing a set of attributes (not necessarily the same set in each case, but a set that together with the values for the attributes uniquely describes the item), and then we classify each item into groupings based on how many of which attributes and attribute values they share. An example: A screw may have the attributes "Material", "Length" and "Thread Type", and those attributes might have the attribute values, respectively, of "Brass", "1 inch", and "UNF".
Brass is a demonstrable instance of a type of metal. An inch is "this much". UNF is something that I can show you based on screws cut with that thread. All these things still have existences that I can show you mostly without having to resort to what we think of as "math" (though brass has to have some ratio of zinc and copper, there is no standard ratio unless we specify one; UNF is determined by mathematical measurements to a standard of precision, but I can still teach you to recognize it from existing examples; an inch is "that distance").
But when we get to the attributes themselves, things get harder. Do the attributes "Material", "Length", and "Thread" have existences? Length is the easiest to discuss. We can point to a literal infinity of instances of length. We can define length in an infinity of ways. We know what length is; a rough definition is that it's the shortest distance between a defined point A and a defined point B on the object, that is usually the longest such definable distance that pertains to the object. But does "length" itself, without reference to the object, actually have an existence?
That's where we get to the current discussion about whether mathematical existence is a thing, and whether math can exist without reference to reality. I think math has "dependent existence"; that is, it is a group or category of things we can know about things that can possess attributes (have "independent existence"). Could we know what roundness is if we were unaware of the existence of round things? Only if we defined roundness mathematically. Once armed with that definition, we could go looking for instances of it. Does the definition exist? Well, the definition can itself have attributes (it is so many words long in English, it is expressed using the following symbols in the most typical sort of notation, when graphed it produces a certain shape, etc.), so it does exist. But the attribute "roundness" itself has no independent existence.
Undoubtedly I've "rediscovered" something very trivial and banal in philosophy, but I rarely see this distinction drawn between things that have at best "dependent" or "attribute" existence, and things that have "independent" existence.
A clear statement of the difference between "attributes" and "independent" existents can be found in Bertrand Russell's little book on philosophy (ch. 9). In this chapter he makes the distinction you are referring to in the course of discussing how to talk about attributes (he calls them "universals"). Of note for the current discussion is that he defends a platonist view in this regard.
http://www.ditext.com/russell/rus9.html
Thanks very much, couchloc. Off to read!
OK. Taking Russell's "whiteness" from that chapter, it appears to correspond to the attribute value "white" where the attribute is "color". "Whiteness", I think, has what I'm characterizing as "independent" existence. You can say things about "white"; you can point comparatively at visibly white things to get an idea about what "white" is (provided you can see); "white" can have attributes of its own. I'm going one step back from that. You can't point comparatively at visible things possessing "color" and contrastingly at visible things that lack "color"; all visible things have a color, so the distinction becomes uselessly one between visible and not visible. We need something else here and I think that thing is math, perhaps because math is the same sort of thing as the thing we're describing.
I don't want to annoy anyone by being obtuse; I'm kind of feeling my way around here. I did try to avoid the "color" example initially because I'm aware that the actual experience of color (qualia?) is shown to differ slightly from person to person. But maybe that's helpful. We can have different ideas about what "white" is. But can we have different ideas about what "color" is? I don't think so and perhaps the distinction is useful.
"Sweetness" is a useful test for what you're talking about, which verges on Aristotelian essentialism. A ripe peach has the attribute "sweetness," but only insofar as it's in relation to something that has evolved the ability to distinguish "sweet" sugars from other flavors. Seals and cats do not have sweetness receptors -- they have vestigial genes for them, but since they have evolved to be carnivores rather than omnivores, they no longer need to seek out sweet foods. So what to make of this? Are peaches really sweet, intrinsically, essentially, deep down? In one sense, yes -- they create "sweet perceptions" (which exist as events) in creatures who are able to perceive the sugar content as sweet, so they are sweet in virtue of that ability. In another sense, no -- peaches have sugar content which is real, but not intrinsic sweetness, because many creatures would not taste them as sweet; they are not endowed with some kind of universal "sweetness essence."
So for color, it's just as useful to be clear about whether you are talking about light wavelengths, or the ability to distinguish them plus the actual perception events. Neither of those "is" color, but "color" is a useful shorthand for both.
Russell's view is that there are individual objects that exist and which have properties or universals. A white sheet of paper has the universal of whiteness. There is no difference with respect to existence for him between what you're calling the "attribute value" of whiteness and the attribute "color", since these are both kinds of universals. The universal "color" is merely a second-order universal of the universal whiteness. (I don't think it matters that you can't see "color" directly.) These are all universals from his perspective. And then the rest of his article explains that he thinks universals have a kind of "being" which is different from individual objects but nevertheless is legitimate. So his view overlaps with the platonist view under discussion about whether there are mathematical entities in this respect--there is a kind of "being" which is legitimate but which is different from that of physical objects we interact with. So I see Russell as trying to suggest that the notion of existing is a broader notion.
To respond to Eric re my earlier:
Surely you do not believe in the existence of unicorns because a good deal of observation with the potential to observe them has yielded nothing, so that abundance of lack of evidence for existence allows most of us to think of it as evidence for non-existence, and to assign a probability to unicorn existence an extremely small number [How about 10 to the -(10 to the 100000)?] That's the best one can do, and IMO the same applies to existence of any kind of deity that has any interaction with the physical world.
However, as I mentioned earlier, there is in at least Tegmark's opinion, the possibility eventually of evidence to falsify his hypothesis. In any case, I doubt you could provide me with any history of people making a plausible effort to find evidence of some kind for a mind-independent mathematical reality, but failing, and for that reason concluding that it surely cannot exist.
So the two cases seem very different, and however amusing the non-analogy might be, it does nothing to increase our knowledge.
I'll also take the liberty to answer the other two points Eric makes. Frenkel is busy on much more important stuff than me; but note that maybe a reply by him would be quite different.
But first the logic one, rather straightforward I think. The notion of consistency is entirely syntactic. The notion of 2-valued versus many-valued logics is entirely semantic. So the doubt you attempt to cast here is simply a confusion IMO. I've at least once before here strongly advocated that people try to learn their logic from a more mathematically-minded logician so that this confusion is less likely. Unfortunately logic courses in many philosophy departments are sadly lacking in anything really important, wasting time before students have really grasped much, by jumping into much less important 'relevant' and 'modal' and 'many-valued' and even 'intuitionistic' logics, etc.., quite apart from wasting time on Wittgenstein. That is not to say these are all a waste of time eventually. (Though I personally think so-called relevant logic is quite silly mostly and completely irrelevant to anything important or interesting at all! We had a 'battle' about that on this blog several months ago.)
On Penrose's triangle (mental-mathematical-material), I think his main point has been completely missed in Eric's response. And that point is the apparent contradiction in maintaining: that mathematics is entirely mental; that the mental is just the human brain, or alien brain, or maybe eventually machine intelligence if it ever happens, and so is material; and finally also maintaining that the material is mathematical, as quantum field theory e.g. seems to be saying. Oddly enough, on Tegmark's homepage, you will also find a triple-authored paper on exactly that, with he and the two others each presenting quite conflicting views on the correct way to 'cut' the triangle and avoid any contradiction.
Its analogous because in both cases the burden of proof (or at least rational belief) is on the hypothesis-makers to go get that evidence, and in both cases, the hypothesis-makers have not done that.
Look, there are an infinite number of possible entities, worlds, and platonic object out there. Until a defender of any one of them comes forward with evidence for their preferred entity (or world, or object), none of them have good justification for belief.
Now I'm perfectly willing to agree that if you construct a test for some entity, run it, and get a negative result, that leaves your hypothesis worse off than one that hasn't been tested. But even the one that hasn't been tested is generally not justified. If you support the notion of mathematical objects and think you have a testable hypothesis about them, YOU go test it. It's not up to test it for you and its certainly not up to us to give your idea credence until someone proves it wrong.
I agree, but what you're saying here seems to me very similar to what Jason said originally. I.e. that math is about (the interrelationships between) math symbols. Saying consistency is entirely (logically) syntactical is not saying it's real, it's saying that consistency is a notion that is entirely about the relationships between symbols in a logical system. Did you mean to use some word or concept other than 'syntactic?' Because that word doesn't support the notion of real mathematical objects, which you seem to be defending.
I am not aware that QFT says that the material is mathematical. Can you cite a QFT physics paper which shows that the math indicates that the material is mathematical?
As far as I can tell, QFT equations no more imply that the material is mathematical than earlier Newtonian equations implied it. In both cases you have a set of equations that seems to do a great job of modeling how reality functions. Okay. And??? In what way does this imply that the material is really mathematical?
Answers to Eric's response to mine to him:
On your unicorn's versus the mind-independent math object's existence, my point was that it was no analogy at all, reason as given. It is the many people who accept non-existence of unicorns who provide good 'reverse' evidence. The analogous people who have cocksureness about the non-existence of the math objects have no real reverse evidence to offer. But neither do the others even attempt that kind of direct empirical evidence, platonists for whom I have more sympathy (but probably more emotionally than rationally). The arguments between them, at least the serious thinkers, are quite different, and so your analogy is entertaining but toothless. It seems to me however quite striking that Tegmark's far-from-popular form of strong platonism may very well offer the possibility of that sort of falsification effort, as delineated in the paper referred to.
Pardon for above just rephrasing things once again in the hope of getting through. The challenge to platonists Eric gives is a red herring with respect to his purported analogy.
On the logic, nobody here is "Saying consistency is …real." It is the mathematical system with the property of consistency which was at least for a time Hilbert's idea of 'being real' or 'existing'. I don't think I said or implied anything that you there seem to think I said, in your conflation of the reality of 'consistency', whatever that reality could possibly mean, with what Hilbert apparently maintained.
As to the last point, an example of very recent vintage would be the existence of the Higgs field. It would be very embarrassing to many of the present Nobel committee and a few thousand other physicists if you were able to establish non-existence convincingly, so I'm confident that is not your position. The Higgs field is to me a function on space-time and so a mathematical object. Presumably it is something different to you, and telling me what it is would help clarify our differences.
I'll freely admit, of course, that one is certainly not dealing with a "final theory", as came up previously, and so the Higgs field and much else will likely be emergent objects if we are smart enough to eventually come up with that theory, which of course may not exist. But the example remains. I have not seen any argument casting doubt on the existence of a final theory (including those based on hash-ups of Godel incompleteness) which is any more than the emotionality that I accused myself of both here and earlier with respect to being a platonist. But I do count the extraordinary accuracy QED as a kind of evidence in the other direction, favouring somewhat the existence of such a theory, quite apart from whether we can find that theory ultimately. And so far here, no one has come up with the asked-for argument to dispute Tegmark's claim that the existence of a final theory implies his strong platonism in the form of the reality of at least one mathematical system.
couchloc, that makes a certain kind of satisfying sense to me, thanks for explaining that about the second-order universals. I think I'm distinguishing between the first and second order, then.
Another Matt, I think your "sweetness" is to "flavor" as "whiteness" is to "color". You say a lot about sugar and its intrinsic property of sweetness, but I was reminded this morning (as I made my breakfast with a non-sugar sweetener) about the substance called miraculin. Miraculin works directly on the body to make it taste things as sweet that are not sweet. Sweetness, then, is a perception, but "flavor" is the real concept and something like couchloc's second-order universal. If I understand correctly.
If we could find that altering our perceptions altered how our minds process mathematical objects and relations, then they might tell us something useful, but as far as I know that is not the case. Right?
I think Matt's point is that there is something that is related to "sweetness" which is in the object. If the peach doesn't have a certain structure, then we won't taste it as being sweet. But there is also an aspect of the sweetness that concerns the individual as well. You have to have the right sort of receptors to experience the peach as tasting sweet. For others with different kinds of receptors it may taste different or not sweet at all. In this respect, there is an aspect to how we think of "sweetness" that makes it an observer-relative property. This doesn't seem to be the case with mathematical truths. There is a large degree of agreement that mathematical truths are true for everyone in the same way throughout the universe, and in this sense they are often seen as objective. So these seem like different sorts of things.
Peter Hoffman:
No, we do not decide unicorn-existence based on a hand count. We ask unicorn-believers for observable, reproducible evidence consistent with their hypothesis, and when they can't produce that evidence, we (tenatively) dismiss it until they can. And the same goes for mathematical objects. Or literary objects. Or spirits. Or platonic forms.
I'm not cocksure; I'm happy to tell you that my rejection of your hypothesis is tentative and subject to revision should you demonstrate evidence of these entities. The sticking point seems to be that you think people ought to believe in them before you produce such evidence while I think people ought to believe in them only after you produce such evidence.
The map is not the territory. The mathematics of the Higgs field describes how it (the field) acts. It made testable predictions, etc... But its still just a description of what we observe in nature. Its a map. Its accuracy and so on doesn't make the mathematical description of the Higgs field a real object any more than F=ma is a real object.
Maybe we should start there, at simpler example of the same class of thing. Do you think the mathematical expression F=ma is a real object in and of itself, or just a description of how other real objects act?
peter hoffman,
I make no comment on the relative merits of mathematical platonism, but I will comment on the notion of QFT implying that the physical is really mathematical. That notion really depends on how one distinguishes between a purely mathematical object and a physical one. If we maintain that the distinction is that physical objects are those things which interact with other physical objects (and I recognize that such a definition is certainly a bit circular in nature) then the Higgs field is most certainly a physical object; it imbues mass in other physical objects with which it interacts. We could define physical objects as those which contain energy (always remembering that mass is simply a form of energy). By that definition, the Higgs field is certainly a physical object; the Higgs particle was actually detected and its mass was measured. We could define physical objects as those composed of particles. However the real implication of QFT is that the notion of particle is not a particularly useful one. Particles are just excitations of the appropriate field. At a fundamental level, everything is composed of fields.
If you can come up with a different definition of "physical object" than I have that renders the Higgs field non-physical, then please share and we can discuss that definition. I certainly cannot think of a reasonable definition of "physical object" that makes the Higgs field non-physical
As mentioned eric above, QFT provides a mathematical description of the Higgs field. It also provides a mathematical description of the electromagnetic field, the strong field, and other fields. General relativity also provides a description of the gravitational field as a function on spacetime. Does that render any or all of these fields non-physical or purely mathematical?
Responses to Eric and to Sean T:
Sorry if this is getting tedious to some readers, but being once again misunderstood on the 'unicorn etc. failed but commonly asserted analogy', and also with positions imputed to me which I have not asserted, I must try again.
Eric wrote "No, we do not decide unicorn-existence based on a hand count."
That is not at all what I wrote, and is more-or-less meaningless to me.
" We ask unicorn-believers for observable, reproducible evidence consistent with their hypothesis, and when they can’t produce that evidence, we (tenatively) dismiss it until they can."
That is just about exactly what I explained, both times. I am sorry if you did not realize that my use of "reverse" in front of "evidence" last time was my attempt to get you to understand the statement about 'absence of evidence for existence being regarded as evidence for absence of unicorns' from my response before that. And that is just what you are also saying.
" And the same goes for mathematical objects."
And that is where you are wrong, as explained in both the previous responses by me. Once again, there is not only no record of math platonists attempting to empirically 'find' mathematical objects 'somewhere', there is also no evidence of serious anti-platonists doing the same, failing, and from that believing they have somehow established non-existence. The argument between the two has always taken a different form, and your highly unoriginal, but still somewhat amusing, claimed analogy simply is not one at all. And it is only the feebleness of that attempted analogy to which I am objecting; read what I wrote.
"I’m not cocksure. I’m happy to tell you that my rejection of your hypothesis is tentative.."
I did not say you are. But many anti-platonists certainly are. Examples from Coyne's non-blog, which many here read, of flippant anti-platonism are easy to find, and I make a point of responding to a couple of fellows there who are quite credible on many other points, one a trumpet player, the other a physicist or physics technician from northern climes. (They have plenty of opportunities there to defend their positions, but should be nameless here for those who don't bother with Coyne, since the two may not read this.) Both there and here, my point, that Tegmark's hypothesis (not mine) has not had any convincing refutation, has itself not had any convincing refutation. I am sorry if you took what I said as necessarily applying to you, and thank you for the clarification. The use of the non-analogy above as an attempt at one more regurgitation of humour maybe got me to desire that clarification from you.
"… you think people ought to believe in them before you produce such evidence.."
Again you are putting words into my mouth.
Next, I had said "As to the last point, an example of very recent vintage would be the existence of the Higgs field. ..", and you responded to that:
"The map is not the territory. The mathematics of the Higgs field describes how it (the field) acts. It made testable predictions, etc… But its still just a description of what we observe in nature. Its a map. Its accuracy and so on doesn’t make the mathematical description of the Higgs field a real object any more than F=ma is a real object. Maybe we should start there, at simpler example of the same class of thing. Do you think the mathematical expression F=ma is a real object in and of itself, or just a description of how other real objects act? "
Just below, Sean T. questions my 'views' on the same remarks, and I will only say to Eric's just above that it again exhibits a considerable confusion, this time between a PROPOSITION such as "F=ma", and OBJECTS, such as the "F" the "m" and the "a". Maybe some of the remarks below will help on what Eric asserts just before his F=ma.
Now to quote Sean:
"…If we maintain that the distinction is that physical objects are those things which interact with other physical objects (and I recognize that such a definition is certainly a bit circular in nature)…"
The "..a bit.." could be the understatement of the year. I deliberately chose Higgs rather than other fields in the Standard Model of the particle physicists partly because of hoping for some kind of definition or description of the term 'physical object' which did not suffer from naive philosophical statements from 90 years ago by the logical positivists. As a joke, I could say that your attempted first definition is a bit of evidence in favour of Tegmark, since mathematical objects certainly interact with each other, so maybe they are the same thing as physical objects. But I am not serious there, only trying to point out that your statement says nothing at all.
"…If you can come up with a different definition of “physical object” than I have that renders the Higgs field non-physical, then please share and we can discuss that definition. I certainly cannot think of a reasonable definition of “physical object” that makes the Higgs field non-physical…"
Again, words are being put into my mouth. I agree 100% that it is a physical object. The question here, which I thought was made clear, is the question of whether it is a mathematical object. The misunderstandings here and above really make me wonder how many people have made even the minimal effort needed to download that Tegmark paper I referred to explicitly from his web site and read at least the introduction.
Sean continues:"As mentioned eric above, QFT provides a mathematical description of the Higgs field. It also provides a mathematical description of the electromagnetic field, the strong field, and other fields. General relativity also provides a description of the gravitational field as a function on spacetime. Does that render any or all of these fields non-physical or purely mathematical?"
' No and yes in that order' seems to be what a convinced Tegmarkian would reply. But I get the impression you think it is only one question, not two. But that depends on you giving something much clearer which distinguishes between what you seem to claim are the distinct sorts of objects, the physical and the mathematical, and then using that to make clear the distinction between 'field' as one type of object and 'field' as the other type, and finally making it into one question by saying which type of field you are asking about. It is exactly that which no one seems to be able to do, for those who think Tegmarkians are wrong (and so, in particular, Tegmarkians have no credible argument for the mind-independent existence of mathematical objects).
Finally, between those quotes Sean speaks about energy as possibly part of a definition of physical object. I find nothing there to object to. However let me point to Matt Strassler's blog, easily found, very informative. In particular, it is easy to find his as usual very clear explanation of what particle physicists mean by that word 'energy'. This is not a trivial or obvious matter at all. What they mean is the concept obtained from the proof of a theorem of mathematics. That theorem is the well known one of Emmy Noether, explaining how each symmetry produces a conserved 'quantity'. And it is time symmetry which, by following her 'recipe', produces what modern particle physicists take as the definition of energy, as most of you know. I'll leave it at that, as a point to ponder for those who would prefer mathematics to merely be a language for the description of the behaviour of the physical.
I once had an argument with a Thomist about essences. I suggested that perhaps the only things which can be said to have essences are the particles and fields of QM (about which I am not an expert!), if those do not turn out to be reducible. My reasoning was that since they are (so far) not reducible in the way that something like a "lump of iron" is, the only description possible for them is a reliable set of behaviors, which requires math to model (or perhaps they are a manifestation of something mathematical). He countered that it's impossible for something to be just a behavior -- it has to be made of something to have a behavior in the first place. In other words, there can't be "form" without "content."
This is an epistemological question, though -- seeing the irreducible parts as "pure form" only speaks to what we can know about them given the evidence. I could be wrong, but it seems like something we can afford to be agnostic about and ignore the ontological problems altogether because there probably isn't evidence that would count either way.
peter,
It appears that I have been guilty of strawmanning your argument. If I am understanding you correctly, your argument is that objects that are physical are also mathematical, right? If that's the case, then in order to proceed further, I think we need to establish exactly what that entails. What exactly does it mean for an entity to be both a mathematical and a physical entity?
To be even more precise, what property or properties would an object that is both a mathematical and a physical object possess that is/are different from the properties possessed by objects that are solely physical? If we can't find such properties, then what's the point of claiming that physical objects are mathematical objects as well? Calling such objects both physical and mathematical at the same time would add nothing to the description of such objects unless there actually is some difference such as I've asked us to try to identify.
Personally, that's my main objection to this idea. I am at a loss to come up with such distinguishing properties. Obviously, my incapability of coming up with such properties does not mean that there are none, so I don't reject the idea out of hand. However, if there are no such distinguishing properties, then why bother even arguing the notion?
Sean,
I've been putting out what I understand to be Tegmark's position, and indicated a lot of sympathy for it, but feel not that competent to be an ardent advocate. One negative thing I indicated earlier is some 'nervousness' about how clear and final the notion of a mathematical system is at this point, though modern logic has brought that to a pretty clear thing in this past century.
Anyway, assuming Tegmark's position, there would be no difference at all between the physical and the mathematical. Every consistent mathematical system exists, there is no difference between physical and mathematical existence, and some systems, such as 'ours' (though we don't know what it is, and perhaps never will) contains something he calls self aware subsystems, such as our system and us. And in such systems, the SAS's would 'experience' this mathematical existence as somehow physical. Really I think his paper, and in a few weeks, his book, are the right place for details, and maybe corrections to what I just said. One caveat which has already come into this relates to 'computable versus arbitrary' systems.
Your point about properties which would distinguish between physical objects, depending on whether they are mathematical or not, is perhaps interesting, but would be purely theoretical to Tegmarkians, I think, because the 'nots' would be non-existent to them. Certainly this kind of question about platonism seems unlikely to have much effect on science itself. But he claims his hypothesis is scientific because of the cosmological observations he claims would eventually be capable of falsifying it. And, in any case, anyone who worries about the "unreasonable effectiveness of mathematics" would have a ready-made solution to that problem by becoming a Tegmarkian. And to the extent our earlier discussion is relevant about definitions of physical objects versus mathematical objects, defining the latter is much easier it seems, and the Tegmarkian has no need for the former. And finally, coming back to Jason's essay here, the existence of mind-independent mathematical objects has the obvious answer yes for the Tegmarkian (unless she/he combines it somehow with solipsism, which wouldn't appeal to many and has way less to recommend it as a discussion topic!)
peter,
Fair enough, but asserting that all objects are mathematical is something that requires evidence to back it up. Just saying that since "all objects are mathematical" there is no need to define what properties a non-mathematical object would have is absurd.
Consider, you can pick an adjective at random and claim "all objects are x" where x is your random adjective. Since all objects are x, it is then unnecessary to distinguish between objects that are x and those that are not. We need not define what properties an object must have to be x, and how those properties would be different if the object is non-x.
Let's try it out: I claim all objects are religious. We "know" what if means for an object to be religious. There's no reason to define what a non-religious object would look like or what its properties would be, since no such object exists.
Obviously, if I were to propose the previous paragraph as a serious philosophical argument, it would be torn apart. What it implies is that the mere assertion of a universal statement is sufficient to guarantee the truth of that universal. Ie, since no counterexample to the universal exists, there is no reason to consider what properties such a counterexample would have. Since the universal holds true for all instances, there's no reason to actually test whether or not the universal is true for a given instance.
Surely you can see the glaring circularity in this argument. In order to remove this circularity, we need to find precisely what I had in mind in my earlier post, namely a way to determine for a given object whether or not that object is mathematical. This need not be a practical test, some definition of what properties would in theory distinguish mathematical objects from non-mathematical ones would suffice. Without this, the argument reduces to "since there are no objects that are non-mathematical, everything is mathematical." There is no reason, other than Tegmark's assertion, to believe this argument absent a definition of "mathematical" and "non-mathematical".
In What Sense Do Mathematical Objects Exist? – EvolutionBlog is the air jordan retro 11 cool grey coming to abmerica http://www.couponmom.com/couponmom/index.php?module=content&brand=breds…
Hi Sean et al,
I said that a Tegmarkian (within his or her system--not said explicitly---sorry) would not need two distinct definitions. I certainly did not mean to imply that to arrive at virtual certainty of possessing the truth, the Tegmarkian could just ignore the question.
But the same need for explanation would just as well apply to the anti-Tegmarkian. And despite the common prejudice in its favour, I am surmising that neither you nor anyone else has even the slightest 'proof' or 'experiment' or 'logical derivation' or 'observation' or whatever else you might prefer, which is convincing in the direction that Tegmark's hypothesis is false. And at least he has suggested quite specific observations and statistical calculation which could falsify his hypothesis.
I think the preferable thing is to actually read his Annals paper and other stuff, easily obtained via his website (which IIRC also includes a CV somewhere with 10s if not 100s of technical papers in top notch astrophysics etc journals, in case the moron is listening from the other thread who vituperates the implication of some kind of need to get published as the motivation for this hypothesizing!). And so to then address what his position actually is. From what I say above, it appears his suggestions about some kind of convincing refutation or proof go farther than others, particularly those who insist that the common prejudice, with no proof apparently, must prevail until proved false (and also those who say they do not understand the hypothesis, but seem not to notice that it follows that they do not understand its negation, the common prejudice!)
It appears to be a much more symmetrical situation, and there is still some vagueness even about the meaningfulness of some of these questions. But at least his observational suggestion would also address that, far more than people who simply bleat about their "proof by incredulity" (e.g. 'how could my fingernail be mathematical?' ---I realize that is not you, and I'm mixing this and that other thread, the quality of whose commentators have made me decide not to wade in there myself).
Furthermore, if you read up on Tegmark, I think you will come to agree that the question 'Is the universe a mathematical system?" is somewhat better to address, than the closely related "Are all physical objects mathematical objects?".
peter,
I think maybe we're reaching a bit of common ground here. My main point is that, so far as I can tell, nobody has ever really presented a definition of what it means to say "everything is mathematical". Given that, it's really difficult to see how one would go about determining whether or not that claim has merit. I can certainly agree with you that there's been no real argument, evidence, etc. (beyond the argument from incredulity, which is certainly weak) that Tegmark's hypothesis is false. I just think the reason is that the notion of something being "mathematical" has not been well defined.
Sean,
Yes, somewhat common.
However, in the natural human tendency to try to get in the last word, I would say yes to your second sentence , but clearly the same would necessarily apply to its negation: "nobody has ever really presented a definition of what it means to say...NOT ...everything is mathematical”. Nor has anyone clarified the stronger 'No physical object at all is mathematical', or better, 'No mathematical system has physical existence', those two certainly being the common prejudice, largely of people who have never learned anything outside cookbook arithmetic or maybe cookbook calculus.
They seem to misunderstand Wigner's point about his "unreasonable effectiveness of mathematics". It is not really that after onerous and difficult calculation one sees an extraordinary level of agreement between mathematical theory and experiment, e.g. on the magnetic anomaly of the electron. Rather it is the extraordinary, almost entirely unmotivated by physics, prior invention by mathematicians of systems such as Hilbert space (quantum mechanics after that) Riemannian geometry (General Relativity after that), Clifford algebras and modules (Dirac equation and positron prediction afterwards), grouos and representation theory (symmetry, conserved physical quantities, Wigner's own work on the Poincare group), plus lots of stuff subsequently, each time anticipating rather major physics. One needs more than adeptness at cookbook 'mathematics' to really see the point here. (I realize that teachers like to call the arithmetic they teach in kindergarten by the name mathematics, but inflation applies to more than just cosmology.)
And your last sentence I would disagree with based on the big advances in logic and foundations in the past 100 years. Tegmark explains himself at length on this. But, as I said much earlier, the notion of really what a mathematical system is may very well widen in time, and it is not entirely uncontroversial now. On the other hand, again to quote you with one minor change to go to the opposite: "the notion of something being.. PHYSICAL.. has not been well defined".
In What Sense Do Mathematical Objects Exist? – EvolutionBlog air jordan 11 grey white cool grey http://www.couponmom.com/couponmom/index.php?module=content&brand=jorda…
Apart from smugness, does this assertion have anything to back it up? Is "mathematics" defined as only cutting-edge work in… mathematics? (Hmm, that doesn't work.)
lenoxus,
You, and anyone, can use the word 'mathematics' for anything you wish to.
To back up the claimed word inflation, I will bet that the vast preponderance of elementary school teachers 60 years ago used the word 'arithmetic' as the name of the subject, and exactly the same today, except with 'mathematics' substituted.
Whether you wish to call that inflation or not is up to you. And the so-called new atheists today are called "smug", and many other ad hominem words, by the religiosos who have no other argument to present in defending their ignorance. So I do not mind being in a similar situation, however much less important this matter is.
My claim that putting out a severely incorrect version of Wigner's observations, in ignorance of what he really wrote, is largely due to lack of education in theoretical mathematics, is itself purely conjectural on my part. But any other reason for that misrepresentation of Wigner, which may be seen in responses to that more recent blog about Tegmark's hypothesis, would seem to be a much more serious accusation against those responders.
But maybe you have read Wigner and can think of an innocent reason, other than the innocence of mathematics ignorance on the part of amatuer philosophers.
Sorry, "amateur"---it went by slippery fingers without proofreading.
I retract any inadvertent venom. I suppose you're right that it may be better to call it arithmetic rather than imply something more comprehensive than that, as long as it is acknowledged that arithmetic is part of math.
A defensible division of mathematics as a pedagogical subject is along the lines of theoretical study of mathematical systems versus practical study of computational algorithms, and recently maybe a third called experimental math. Probably my use of "cookbook" is a bit harsh, but it's very far from original.