I don’t trust rocks. Rocks keep fooling me. They sit there looking all solid until you examine them more carefully and find out they’re mostly empty space, with a smattering of charged particles here and there. Then you look a little deeper and find out those charged particles are nothing like they first appeared. They don’t even have locations. Rocks, and their constituents, are nothing at all like they first present themselves. But at least they’re real. I think.
Now here’s what genuinely baffles me: Apparently there are people in this world (and even, occasionally, in the comments section of this blog) who haven’t the slightest doubt about the existence of rocks, galaxies, squirrels, and the rest of the physical universe, but who suddenly turn into hardcore skeptics re the existence of mathematical objects like the natural numbers. (Many of these people, I suspect, are in fact affecting skepticism because of a badly mistaken belief that it makes them look sophisticated. But that’s speculation on my part, so let’s put it aside and take their positions at face value.) I just don’t get this. Why on earth would, say, a scientist, commit to the belief that there’s a physical universe out there but not a mathematical one, when we know that our perceptions of the physical universe demand constant revision, whereas our perceptions of the mathematical universe are largely eternal. My conception of the natural numbers is very close to Euclid’s; my conception of an atom bears almost no resemblance to Demosthenes’s.
Every field of inquiry needs a starting point. I venture to guess that most of the natural-number skeptics go through life without seriously doubting the existence of conscious minds other than their own. They can’t avoid that assumption, because it explains so much. So does the existence of the natural numbers. We prove a theorem that says that every number is the sum of four squares. We pick a number (say 1350591), we try to write it as a sum of four squares, and lo and behold, we succeed! (In this case, 1350591=4312+2262+10552+272.) In other words, arithmetic appears to be consistent. But to explain this consistency, you pretty much have to believe that arithmetic is about something real — just as, to explain the observed behavior of other people, you pretty much have to believe that there are minds inside those heads. (I’d argue that the case for the natural numbers is stronger than the case for other minds, but the cases are similar in structure.)
Now here’s where I expect Bob Murphy to jump in and say: Aha! Yes! And that’s also the nature of the case for God! I agree and disagree. I certainly agree that any system of beliefs needs a starting point (e.g. “I think therefore I am”), and that my belief in the natural numbers is a starting point — it’s not deduced from anything — just like Bob’s belief in God. But here’s the thing: Bob also believes in the natural numbers, because he is sane, and because almost no sane person has ever managed to disbelieve in them. They explain too many things that can’t be explained without them. Whereas it’s easy (trust me!) not to believe in God, because God is quite unnecessary for understanding the Universe. Bob, I anticipate, will say that he needs God to explain the existence of such an intricate structure as the natural numbers. I say quite the opposite: The existence of the natural numbers explains the existence of everything else. Once you’ve got that degree of complexity, you’ve got structures within structures within structures, and one of those structures is our physical Universe. (If that sounds like gibberish, I hope it’s only because you’re not yet read The Big Questions, that you will rush out and buy a copy, and that all will then be clear.)
To put this another way: Bob thinks I should believe in God because that would explain why the natural numbers exist. I believe that the skeptics who show up on my comment pages every now and then should believe in the natural numbers for a great variety of reasons, one of which is that it would explain why the Universe exists. One difference betwen me and Bob is that I think the natural numbers can’t help existing, and therefore need no explanation. But the Universe — now that needs an explanation. And because the natural numbers, which I’m already forced to believe in for other reasons, provide that explanation, I have no need of Bob’s God.
I’ve rambled a bit from my main question, so let me restate it: Why and how would anyone be willing to accept the existence of things like physical objects and other minds, while doubting the existence of the natural numbers? Why would you direct a more skeptical gaze at the one thing that hasn’t spent the past several millenia presenting false appearances?
| —-Inspired by some recent back-and-forth with Bob Murphy. For further reading, see some of my own past posts. |
“when we know that our perceptions of the physical universe demand constant revision, whereas our perceptions of the mathematical universe are largely eternal”
Ha! You’ve obviously never tried to teach addition to my five-year-old son! His perceptions of the mathematical universe are lucky if they last a day!
I propose that rocks are more intuitive than 1,2,3,4,5,6,7,8,9,10…. The latter must be taught, and there are cultures who get by without it. The former seems to come to our intuition far more easily.
I propose also that our intuition (generally or individually) is a very poor basis for judging which of rocks or natural numbers is more fundamentally real.
Re: Empty space
The very idea of empty space is meaningless. As fundamental particles are either points or smears all space is empty, in one sense. In another sense all space is full of fields and virtual particles.
Re: Your Axioms
“and that my belief in the natural numbers is a starting point”
Nonsense! Its deduced from the way your brain works and mentally divides our world into discrete entities. You then just extend the small-ish counting numbers to the natural numbers.
Steve,
I think I’ve asked this tangential question before, but don’t recall if you picked up on it. What’s your take on moral realism, where statements about right and wrong are true or false in the same way that mathematical statements are true or false?
I think it is contradictory to even _argue_ that the universe requires an explanation. Why? Because any such explanation could only be made in terms of rules that do not hold in this universe, but only outside of it. But when you argue something, you are committing yourself to the logical rules that hold in this universe.
Limited rationality has implications in two (at least) areas: in science, we can’t each do all the experiments and deductions; we have to take others’ results and experiences on faith to at least a certain extent. In ethics, there are some rules that operate for the good of the tribe in ways that appear to be against the interests of the individual.
Thus we’ve evolved mental structures that make us susceptible to taking things on faith, especially when taught them at a young age by authority figures.
If we’re honest with ourselves, most of us would have to admit that the reason we “believe in” something as counterintuitive as quantum mechanics is more because we were taught it and less because we worked it out for ourselves by careful experimentation and reasoning.
I think that the reason is that we have evolved to have an intuitive sense of reality including rocks and especially other minds. It is so ingrained that we can’t help but believe they exist. Evolution has not found it necessary to endow us with any particularly deep intuition about numbers other than the first few and especially with anything that is infinite such as teh natural numbers. Its the same reason why we have such difficulty when science proves that the real world deviates from our naive intuitions – although I believe that rocks really are mostly empty space (in some sense) it almost takes an effort of will to believe it – its a different kind of belief than the belief that they exist which just IS. Similarly whilst I have no problems with ‘small’ natural numbers it takes some effort to believe in a number such as say 10 to the power 100 since I have no real concept of it.
Might I recommend Charles Chihara’s book on this topic,
http://www.amazon.com/Constructibility-Mathematical-Existence-Clarendon-Paperbacks/dp/0198239750/ref=pd_vtp_b_1
Doc Merlin:
Re: Empty space
The very idea of empty space is meaningless. As fundamental particles are either points or smears all space is empty, in one sense. In another sense all space is full of fields and virtual particles.
Yes, my point exactly. First rocks appear solid. Then on closer examination they appear to be mostly empty space. Then on still closer examination they appear to be fluctuations in a quantum field. Given that our intuitions about the physical universe are so dramatically wrong, and then when we fix them turn out to be dramatically wrong in another way, and so on through several levels, why would anyone have faith in the reality of rocks and not of the natural number system?
I believe in rocks and minds. But I do not *know* they exist in an absolute sense. I may be a computer simulation. I must operate in my world as though they exist because to do otherwise is to be unable to function. I also believe in the natural numbers – but I do not necessarily *know* they exist in an absolute sense. I can conceive of my existence without rocks (for example as a simulation or a dream), but I find it harder to conceive of the absence of the natural numbers.
Andy Wood:
What’s your take on moral realism, where statements about right and wrong are true or false in the same way that mathematical statements are true or false?
I have a great deal of difficulty seeing how this can be the case. Our sense of morality is a product of evolution, and would surely have been very different if the history of evolution had been different. In that sense, morality — unlike, I think, mathematics — is very much a human invention.
Maurizio Colucci: But when you argue something, you are committing yourself to the logical rules that hold in this universe.
You can in principle restrict yourself to arguing from rules that you believe would hold in any universe. Or more precisely, rules that already exist outside the universe.
Take the rules of propositional logic, for example. I can use these rules to reason about euclidean geometry without asserting that they are “in” euclidean geometry. I can use them to reason about the real numbers without asserting that they are “in” the real numbers. Why can’t I use them to reason about the universe without having to assert that they are “in” the universe?
why would anyone have faith in the reality of rocks and not of the natural number system?
This was debated between Plato and Aristotle 2350 years ago, and never completely resolved. Steve, you are in Plato’s camp. Most people found Aristotle more persuasive.
I wonder, with Aristotle and Roger, if our notions about natural numbers really are prior. My guess is we learn numbers from counting, not counting from the peano axioms. We formulate abstractions, like the number 4. I am more confident in the deductions I make about those abstractions than I am about any assertion about any physical thing. 2 = 2 = 4, absolutely no doubt; Bob Murphy writes a blog, very very small doubt. But Steve seems to want me to be sure that the abstractions — the number 4, addition, lower bound, etc — explain the physical things. That involves a bending back I don’t find convincing. It’s like Anselm’s proof.
I guess it boils down to this. Suppose that intelligent life in the universe is wiped out. I am confident that rocks would still exist, but I’m not so confident that numbers would still exist. Maybe they would, but I am not as confident as with rocks.
Part of the problem is that we are so provincial. Baryonic matter is only 4% of the mass-energy of the universe, but it is the only part we really know.
The trouble (in my mind at least)is that when you say “The natural numbers exist” it’s not entirely clear what you mean by “exist”. This is more a problem with natural language than anything else. The natural numbers certainly exist as mathematical objects/ideas but I’d be very surprized if we found physical objects (40 foot pillars of fire in the shape of each number, for dramatic effect) corresponding to the natural numbers. If you’re willing to make precise what you mean when you say the natural numbers exist then we can get on with the discussing weather rocks and natural numbers “exist” in the same sense.
Wow, there’s so much great stuff in this thread, I don’t know where to begin. Re. the Mathematical Universe, I think there are four possible views:
1. The natural numbers, and mathematics in general, are a pan-universal construct, and the physics of our universe are built upon that foundation. There could exist other universes with the same mathematics, but different physics. I think Max Tegmark is in this camp.
2. The natural numbers are a product of the physics of our universe. The physics are the foundation, and mathematics is a byproduct of the universe’s physics.
3. Mathematics and physics are the same thing. One is not built on the other, they are simply different external perspectives of a single model.
4. Physics exists, and mathematics is a human invention created to explain physical laws.
I subscribe to #3. Personally, I think that anyone who subscribes to #4 doesn’t have their head screwed on straight.
Andy Wood: I don’t see how one could argue that ethics are objective, unless they posit a God that establishes those ethics. Take slavery, as an example. 2,000 years ago I suspect you would have been hard pressed to find someone who believed that slavery was morally wrong. Even slaves, when freed, kept slaves. The immorality of slavery is a perspective that arose during the enlightenment, and was a result of religious and economic changes. Today, most people in the world, and all westerners, believe that slavery is wrong.
So, if ethics are objective, slavery must always have been wrong, and we only recently came to recognize this. And if you accept that, then you also have to accept that some of our views broadly held today may also be wrong. For example, 200 years from now people might universally abhor the exploitation of animals as food and other products.
My personal view aligns with Steve’s. Ethics are purely a cultural product that change as culture changes.
” Ethics are purely a cultural product that change as culture changes.” i wonder if that is really Steve’s view. It’s not mine, but substitute “biological” for culture, as biology embraces culture as well as culture-independent features …
Sam Harris’s latest book is on just this point
Physical existence is different than the existence of qualities, so aren’t simply dealing with a linguistic confusion here- trying to use the words “exist” and “existence” to mean different things?
While we’re talking about ethics and the more philosophical side of mathematics and physics, I have a question about the anthropic principle. The anthropic principle seeks to answer the question, “why is this universe so perfectly tuned to support life?” For example, that the mass and rate of expansion of the universe are so perfectly tuned that the universe didn’t immediately collapse back into a point after the big bang, and at the same time didn’t expand so rapidly that matter couldn’t coalesce into stars and planets. The principle being that, if the universe wasn’t perfectly tuned to our existence, we wouldn’t be here to observe it.
Makes sense to me. And yet, I often read physicists appear to have the view that having to resort to the anthropic principle is somehow a failure; that we need to have a better explanation. I don’t understand why we need a better explanation.
Steve:
“Our sense of morality is a product of evolution, and would surely have been very different if the history of evolution had been different. In that sense, morality — unlike, I think, mathematics — is very much a human invention.”
Thanks for this. That’s my view. But I now have a couple of supplementary questions:
Have you ever argued about this with David Friedman?
If so, was the outcome consistent with Aumann’s Theorem?
Al V.: I think there are four possible views
All of your possibilities are variations on your favorite, that “Mathematics and physics are the same thing”. I don’t agree with any. Math and physics are very different things.
a. Some find the physical universe being subject to verification and hence the constant revision of theory to be the very fact that makes its existence much more plausible than the beings of a purely abstract universe.
b. Some assert the possibility of doing physics and all science without math opening arguments for divorcing mathematical existence from any talk of its usefulness or marriage to science
Totally agree w/Jonathan Kariv here. The term “existence” has a common sense definition for physical objects but not for abstract ones. Until the term is defined more precisely, commenters will be arguing past each other.
So your argument is essentially “My God (natural numbers) is superior to Bod’s God, and so displaces Him. Natural numbers are a superior God because having always existed, they provide for my existence without requiring anything of me morally speaking, although how they have wrought a universal set of morality is as yet unclear.”
This is a terrible argument.
@Marcelo, J. Kariv:
While ‘existence’ is not well defined here, most, rightly so, have been able to discuss this question on a working definition that roughly comes down to:
numbers are a human construction vs. numbers numbers are the way they are without our being their describing their properties
There are metametaphysical debates about what existence means and even about whether the question of existence has an answer – but I think commenters have been trying to steer away from this and that we can more or less talk at, not past, each other without a perfect definition
I do have to laugh at the idea that belief in god is anything like belief in numbers. Only a christian believes the equation 1+1+1=1 (holy trinity).
Al V.: I don’t see how one could argue that ethics are objective, unless they posit a God that establishes those ethics
Positing a God does not solve the problem of establishing objective morality, check out the Euthyphro Dilemma:
http://en.wikipedia.org/wiki/Euthyphro_dilemma
@Todd, do you say that Positing a God does not solve the problem of establishing objective morality because “our duties to others … were arbitrarily willed by God and are within his power to revoke and replace”? That would seem to argue that ethics are subjective, but subjective to God alone.
@Roger, Could you explain your views? Are you saying that math and physics are unrelated to each other? I would argue that either math is based on physics, or physics is based on math, but I’m unclear how you could argue that they are unrelated to each other.
No, I do not think that either math is based on physics, or physics is based on math. They are separate fields.
@Steve “my belief in the natural numbers is a starting point”
@Doc Merlin Nonsense! Its deduced from the way your brain works
I think Steve would admit that while he originally came to an intuition about the natural numbers via his brain, he now believes that in fact his brain, and everything else, came about via the natural numbers (and all other mathematics). He would note that “the natural numbers” and “intuition about the natural numbers” are quite distinct things, so there’s no contradiction there.
He would say that the “Primum Movens” (First Cause) is not the Abrahamic God, nor the physical universe, nor the Big Bang, nor the laws of physics, nor mathematics-as-developed-by-humans, but rather it is the platonic ideal all-possible-mathematics.
This is fundamentally, although Steve doesn’t like the word, an article of faith on his part – although he does present arguments to support it.
Steve, do correct me if I’m wrong.
I think that the idea of natural numbers and the idea of God are basically identical.
Natural Numbers have always existed. They existed 20 billion years ago, long before our universe started.
God existed 20 billion years ago. Of course, our concept of time and God’s concept of time are not at all the same.
Mike H: For someone who is extremely skeptical of my position, you did an excellent job of summarizing it. Thanks.
You’re welcome :-)
Al V.,
This really isn’t on topic, but yes–if God arbitrarily established morality as we know it, and could thus change it, then it isn’t objective.
Given the morality found in the sacred books of various religions, at least those I’m familiar with, it is astounding to me that any mainstream theist can claim that morality comes from God with a straight face.
@Steve – Encouraged by your kind words, I’ve written a critique of your position here : http://www.dr-mikes-math-games-for-kids.com/blog/2012/02/is-math-the-primum-movens/
Interesting post, Steve. I loved almost everything in the post itself, though in the comments here you seem to be going a little awry. E.g. you casually assert that morality is a product of evolution, and would be different if evolution were different. How do you know the same isn’t true of our thoughts about the natural numbers? Sure, I can’t conceive of a “normal, functioning human” not grasping a Euclidean proof, but it’s also hard for me to conceive of a “normal, functioning human” thinking it’s cool to drown babies for sport.
Anyway, one clarification: I am NOT saying, “If you are an atheist right now, I have a good argument for why God exists: It explains why mathematics is so elegant.”
Not at all. I have a bunch of other reasons for believing in God, and now that I do, I notice that as a bonus it resolves a question that is problematic for the atheist. But since I can’t explain where God comes from or why He should be rational, I can’t ultimately explain the existence of elegant mathematics either.
Steve, I think the resistance to mathematical Platonism or realism on the part of many of us has nothing to do with wanting to seem sophisticated (though I am curiously attracted to neo-Meinongianism…) but rather because we don’t want to give up on physicalism (which has great beauty and simplicity and a good track record). I know your move kills two birds with one stone (solves the creation problem as a bonus) but it just seems – I don’t know – a bit too simple or arbitrary. Too neat. I mean, why should we think we currently have the knowledge required to answer such very big questions. I think we should be able to make progress on the question of the nature of mathematics however. Sure, mathematics surprises us. It’s out there in a sense. But I shy away from a full-blooded Platonism.
I’d like Steve to comment on the continuum hypothesis. after all, if math made tghe world then it’s because mathis true. Is the CH true? I ask because it is independent of the axioms of set theory. And there are an endless number of other hypotheses also independent. So which is the true vine, er I mean the true math?
One other thing. Steve you wrote:
Whereas it’s easy (trust me!) not to believe in God, because God is quite unnecessary for understanding the Universe.
Right, but that’s exactly what the skeptics of the natural numbers would say, too. They are able to go through life, balancing their checkbook, holding a job, watching the Super Bowl, etc., even with their skepticism. You might say, “But they’re living a lie! They can’t actually reconcile their behavior and attitudes with their stated view on the existence of natural numbers!!”
Right, just like I can do with your professed atheism.
Steve, I don’t have any major problems with your point of view; indeed I’m also a Platonist (a much weaker one than you), in that I believe in the objectivity of mathematical truth (and all truth for that matter). But there are a few issues I have with your arguments for your position:
1. You often relate, and sometimes even identify, the consistency of Peano arithmetic and the existence of the natural numbers. You do this on the basis of Godel’s completeness theorem, which states that consistency is equivalent to having a model. But this assumes set theory: it just asserts that we can represent natural numbers using sets. But if someone didn’t believe that natural numbers exist (but did accept the consistency of PA), they would probably not accept the existence of a universe of abstract sets either.
2. You often say that almost everyone, even average everyday people, believes in the consistency of arithmetic. You give the example of adding up a list of numbers, and getting two different answers if you add them up in two different ways: people will assume that they made a mistake, not that arithmetic is inconsitent. That is certainly true, but this doesn’t mean they think it’s impossible for Peano Arithmetic to be inconsistent. All they are committed to is the consistency of a much weaker theory, what logician Peter Smith calls “baby arithmetic”: the theory of sentences of arithmetic with absolutely no quantification or variables. I don’t think that the person on the street so is committed to the consistency of all of PA.
3. You claim that humans have the ability to directly perceive natural numbers just as clearly as they can perceive physical objects. In your economics posts you often criticize taking people’s self-reported perceptions at face value, and instead encourage empirically determining their motivations and behaviors. Applying this standard, do you think there is any way to empirically test this “extra-sensory perception”?
I have wanted to ask you about these things for a very long time, ever since I read The Big Questions (and some old blog posts of yours).
@Bob Murphy: You say “But since I can’t explain where God comes from or why He should be rational, I can’t ultimately explain the existence of elegant mathematics either.” That leads me to the question, do you see the existence of God as a contingent truth or a necessary truth? In other words, are you a rationalist in the tradition of Dostoevsky, who thought we just happen to live in a universe in which it’s an empirical fact that there’s a being called God? Or do you view God the way Steve views numbers, namely that they must exist in any possible universe? I think people like Thomas Aquinas believed in the latter: they believed that the existence and characteristics of God could be deduced by unaided reason alone.
Bob:
Right, just like I can do with your professed atheism.
Yes, I absolutely concede this analogy.
Ken B:
Is the CH true?
It is true in some models of ZFC and false in others. It is true in an absolute sense that both kinds of models exist.
Keshav Srinivasan:
1. You are absolutely right that the equivalence between the existence of natural numbers and the consistency of Peano arithmetic assumes some small fragment of set theory. Of course, any argument one makes about anything must start somewhere, and I think this small fragment of set theory is a pretty minimal starting position, but of course I have no full-on response to the extreme skeptic, just as I have no full-on response to the extreme skeptic who doubts my consciousness.
2. You are certainly right that a belief in the consistency of baby arithmetic is not as strong as a belief in the consistency of PA. On the other hand, I would be astonished if any random layman off the street, having understood the content of the Peano axioms, would doubt that they are true statements about the natural numbers, as he understands them.
3. One reason I tend to doubt self-reported perceptions is that most people, most of the time, have little incentive to examine their perceptions closely. I am much more sanguine about accepting the self-reported perceptions of professional mathematicians regarding mathematics.
4. Great questions, every one of them. They deserve fuller answers. I hope to find time.
Bob Murphy: “Sure, I can’t conceive of a “normal, functioning human” not grasping a Euclidean proof, but it’s also hard for me to conceive of a “normal, functioning human” thinking it’s cool to drown babies for sport.”
Do you really mean to say that these are equally inconceivable?
That you can even make the latter statement is strong evidence that our morality has evolved drastically; there was a time that such things were not at all unexpected (see Psalms 137:9, for just one example). Indeed, I think one would be hard pressed to conceive of any morally reprehensible act which has not been committed by humanity on a reasonably large scale at some point in our history.
The natural numbers, however, have been completely consistent throughout. Our understanding grows, and sometimes needs to be corrected, but the underlying truth has never changed. Or would you contend that it has?
I’m going to have to agree with the people who say this is a definition problem (which was going to be my reply even before reading those other posts). Numbers don’t exist in the same sense that physical objects exist, and the fact that people don’t agree that numbers exist has a lot more to do with them having different definitions of “existence” than you do than it has to do with disagreement about the nature of numbers.
In fact, I can’t myself figure out exactly what Steve means when he says that an abstract concept exists. The best I can figure out is that he considers a concept to “exist” if it is hard to make useful logical statements that do not presuppose that concept.
(Then consider the question: does Euclid’s parallel postulate “exist”?)
A follow up to my book recommendation, I think you err when you say that “They explain too many things that can’t be explained without them”.
We actually don’t need them. We can “translate” the object-laden language of mathematics into a new language that makes no reference to objects but still has the necessary logical structure to aid us in physics and the other natural sciences.
Ken Arromdee:
Numbers don’t exist in the same sense that physical objects exist
This is precisely what I deny.
A number is a component of the system of natural numbers. Physical objects are components of a particular physical universe. The system of natural numbers is a mathematical object. A physical universe is a mathematical object. So numbers and physical objects have exactly the same ontological status.
Assuming you disagree, I realize that I won’t change your mind by repeating myself. Do you realize that you won’t change my mind by repeating the contrary?
I don’t understand what it means to say that a physical universe is a mathematical object, and I’m pretty sure that most of the people on the street wouldn’t either.
E.P. Wigner discussed the “unreasonable effectiveness of mathematics
in the natural sciences” in 1967. It’s not just a question of whether the natural numbers exist. What makes me believe that math and physics are the same thing is calculus. Newton deduced that F = m*dv/dt = dp/dt, and p (momentum) = m * dx/dt. E = (m*v^2)/2, and dE/dv = p. If physics is not somehow inextricably tied to math, or the same thing as math, why would these laws be so clean? Why isn’t E = m*v^1.6513? It’s too clean to be just a coincidence.
Roger wrote: “This was debated between Plato and Aristotle 2350 years ago, and never completely resolved. Steve, you are in Plato’s camp. Most people found Aristotle more persuasive.”
I think Steve stops well short of being in Plato’s camp. In The Republic, mathematics was the metaphor for the theory of Forms, but mathematics was not the highest rung on the Divided Line. Atop the divided line, above both math and the forms, was “The Good.” In other words for Plato, it was ultimately a moral universe, not a mathematical one.
Plato thought a mathematical understanding of the universe was essential (all who enter the Academy had to have a grounding in math – something to that effect was carved over the doorway) but he also offered some insight into what people who stop at a mathematical conception of the universe are susceptible to.
@Todd, thanks for your response – I like your point. You are correct that the Biblical God could not have defined an objective morality, but in theory there could be a religion that has one, although it would have to deny the validity of all other religious perpectives. I don’t know enough about comparative religion to know if one exists.
Do the natural numbers explain my experience of this universe?
1. Do the natural numbers make a universe NECESSARY? That is, are the consequences of natural numbers so pervasive as the preclude the possibility of no universe?
2. Do the natural numbers make THIS SPECIFIC universe necessary?
3. Do the natural numbers preclude the possibility of alternative universes? If so, do the natural numbers explain why I perceive this universe and not some alternative one?
I sometimes think that Lansburg is arguing that our universe is merely one manifestation of the natural numbers – suggesting that the natural numbers may produce other manifestations. If so, what would account for the fact that I perceive one universe but not another?
Presumably this is not a matter of chance because, in a deterministic universe, there is no such thing as chance. Then again, maybe the question has no meaning because, in a deterministic universe, the idea that “I perceive” has little relevance; *I* has no agency….
Neil wrote: I do have to laugh at the idea that belief in god is anything like belief in numbers. Only a christian believes the equation 1+1+1=1 (holy trinity).
First, I don’t consider myself a Christian (although growing up in the US an awful lot of that gets imprinted on a person no matter what they do). But, just as you can look at a rock more closely in the way that a physicist would and realize that it is mostly space (although when you drop one on your foot, it seems solid enough — same principle applies when trying to hit a golf ball through trees which are mostly space too, btw, but I digress…), the mystics tell us that when they look at a rock, they see that it, in fact, exists only relation to everything else, and it is the perception of separateness that is the illusion. To the mystic the only real number is one, as everything is hitched to everything else and cannot be separated.
From that perspective 1+1+1 could equal 1 and the only real number is one.
To tie this thought to the other posts about Plato, there is a dialog that addresses this exact question. A very young Socrates meets his dialectical match in Parmenides…The Unitary vs. The Forms…Good stuff!
nobody checks in – yay!
Tegmark argues that there could be four levels of universes:
1. Our universe, including regions beyond our ability to detect them.
2. Other universes that have the same physical laws as ours.
3. Other universes that have the same mathematics, but different physics than ours.
4. Other universes that have different mathematics.
Thus, “our universe is merely one manifestation of the natural numbers”, in Tegmark’s view. If, however, math and physics are really the same thing, then obviously #3 is eliminated. But I don’t know if #4 would be.
However, if math and physics are the same thing, then the natural numbers would require the existance of some universe, wouldn’t they? Perhaps not this specific one.
If mathematics are fundamentally “real”, then it’s hard to avoid believing in a Tegmark type 4 multiverse, where every mathematically self consistent physics has the same level of “reality”. And yet, my own memories are suspiciously incompatible with such a multiverse. If I roll a blue marble across my desk, there’s only one way for it to roll in a straight conservation-of-momentum-mass-and-energy preserving line, but there are countless ways for it to veer around in a zig-zagging conservation-except-with-an-arbitrary-exception-or-two path. Out of all those googols of universes, how did my consciousness happen to improbably end up in the one without any of those arbitrary exceptions tacked on? If there’s no reason to prefer the simple underlying rules we see to the complicated underlying rules we can imagine, why do we keep seeing the simple rules made manifest? Why should I be confident that my marbles are blue and green rather than bleen and grue? Who made Occam God?
What does “accept the existence of” mean? If I did not “accept the existence of” rocks, how would that fact manifest itself? And if I did not “accept the existence of” natural numbers, how would that fact manifest itself?
The model of the world that I carry in my head includes hypotheses about rocks and natural numbers. In this sense, Landsburg may have a point: why would I disparage the hypotheses I have about one thing more than the hypotheses I have about the other?
Presumably the model in my head also includes a long list of cognitive biases and false beliefs. What conclusions should I draw from that?
I sense Landsburg is arguing, “If you are willing to rely on the idea of natural numbers for some purposes, then it’s only logical that you would rely on this idea for ALL purposes. And, if you think about it long enough, you’ll come to the conclusion that natural numbers explain everything.”
And HERE is where my skepticism about natural numbers may arise: Landsburg is asking me to extend my hypotheses about natural numbers and apply those hypotheses to unfamiliar contexts. But does the fact that I rely on an idea for one purpose mean that I should rely on it for ALL purposes? After all, I don’t have a world view grounded in TRUTH; I have a view grounded in UTILITY. And heuristics that work in one context may not work in another.
Moreover, I’m not yet persuaded that natural numbers explain everything. Or, in Landsburg/Arromdee terms, I don’t understand what it means to say that a physical universe is a mathematical object.
I sense Landsburg argues that rules of mathematics render all applications of the idea of natural numbers equally familiar; math knows no context. And we merely need to reflect upon the idea of natural numbers (and the ideas that can be derived from them) long enough to appreciate that they explain everything.
Maybe so; I haven’t put the effort in, so I can’t preclude the possibility. And I haven’t dedicated my life to becoming a monk, so maybe I’m not qualified to evaluate claims about faith. Suffice it to say, I’m not there yet.
Finally, a word on language: The linguistic school known as General Semantics discourages use of terms such as “reality” and forms of the verb “to be.” These terms tend to generate more heat than light. People often use these terms to accord a privileged status to some ideas relative to others. Where meaning matters and language may tend to obscure ideas, I challenge people to at least attempt to phrase their arguments without reference to such status terms.
Admittedly, this can be a pain in the ass – er, that is, doing this can cause me discomfort – but I find it a useful exercise anyway. The question “Who is a Jew?” may provoke needless arguments, whereas the question, “What document should Israel require from someone seeking citizenship?” may provoke needful arguments. No, we don’t avoid arguments entirely, but we at least get to have useful ones.
Ken Arromdee:
I don’t understand what it means to say that a physical universe is a mathematical object
Is this because you didn’t understand the explanation in The Big Questions, or because you haven’t read it?
@Steve: I agree completely about the CH. Now which model created the world as we know it? Endless true but incompatible models, but you claim they create this particular universe.
Surely we can phrase the CH in this world. Or it might be simpler to ask a question about large cardinals. Do inaccessible cardinals exist in this world? And so it will be true or not. Same for any independent-of-your-axioms assertion. What I am getting at is you have an embarassment of riches: too much mathematics for one world. How do you choose which math makes the world when your argument is, math is true regardless of the world. If the world is one mathematical object, why that one instead of the uncountably countless others?
Ken B:
If the world is one mathematical object, why that one instead of the uncountably countless others?
One thing I like about my view of things is that it eliminates this embarrassing question. The uncountably countless others all exist; we just happen to live in this one.
Roystgnr: Your point is an extremely good one. (Though I think you meant Tegmark 3, not Tegmark 4. My own view is most like Tegmark 3.) I suspect one could formulate a good answer by arguing that the universe with the simple rules is the universe most hospitable to conscious life. But of course that would require a whole lot of arguing. So I’m going to say that you’ve raised a highly cogent objection.
I wonder if and how this is related to a view, emerging in physics, that the fundamental constituent of the universe is a unit of information (qubit). We tend to think of information as non-physical, involving arrangements and relations between physical objects, but not as a physical object itself. However, Landauer showed that information is physical.
If information can be physical, I suppose numbers can be physical.
I will be much happier when the mystical or murky aspects of the identity between mathematics and the physical universe are removed. I can kick a rock. I can erase information. I don’t know how I can physically interact with numbers except by thinking about them. Thinking, of course, is a physical act, but one that dies when brains die.
It is easier for me to believe in the existence of things that I believe have been revealed through the scientific method. E.g. I believe in the existence of viruses.
For things not revealed through the scientific method, it is hard to know what is real vs. what is made up but seems to work.
The rule that says 1 + 1 = 2 and it’s implications seem to work for our world as does the rule that says it wouldn’t be good to destroy as much life as possible on earth with nuclear weapons.
If someone could point me to scientific literature showing that the existence of natural numbers was revealed through the scientific method, then I would be more inclined to believe in the existence of natural numbers.
Of course, it would also be good if someone could point me to scientific literature that uses the scientific method to prove that the scientific method is a good way to answer questions.
More seriously, why would a person use the rules of logic to try and prove a point if they didn’t believe in the existence of logic?
And why would someone who believes in the existence of logic not believe in the existence of natural numbers?
Steve: I suggest my objection and Roystngr’s are basically the same. There is nothing in mathematics that seems to require this one world rather than the others. And when you riposte, “well the others exist just without us” I’d say you run smack into Jonathan Kariv’s objection. What you mean by ‘exist’ is has a description. Which sounds like St Anselm of Rochester.
Ken B:
What you mean by ‘exist’ is has a description.
I confess to not having a clear definition of the very “to exist”. But what I mean to assert unambiguously is this: There is no reasonable definition of the word ‘exist’ according to which rocks exist and natural numbers don’t.
So if you want to hone in on the meaning of existence, let me start with a couple of questions:
a) Do you believe that rocks exist?
b) If so, what do you mean by the word “exist”?
I am almost certain that if your answer to a) is yes, and that if you have a coherent answer to b), then I will be able to respond that I believe natural numbers exist in exactly the same sense.
Neil:
I can kick a rock. I can erase information. I don’t know how I can physically interact with numbers except by thinking about them.
I write down economic models all the time. The people in those models have no way of physically interacting with me. It would be rash of them to conclude that I don’t exist.
On the concept of existence:
…said God to the atheist….
@ nobody.really:
If we assume that logic exists, the ability to create somethings doesn’t imply the ability to create all things.
@Steve : “@Roystgnr: Your point is an extremely good one.”
Yes, it’s similar to the point I made in my critique. However, you can’t stop at Tegmark 3, you need to accept Tegmark 4.
After all, if you say “the universe is real because it is a mathematical object, and all mathematical objects exist”, then you must either be talking about “All Possible Mathematics”. Since other systems of mathematics can also be described within ours, they are also real, as are the objects they describe. There must then be universes where the natural numbers are finite in number, where modus ponens doesn’t hold, etc. Terry Pratchett’s Discworld is real. The God of the Bible is real, as are all other gods, since each of these can be described by All Possible Math.
Everything must exist – not just everything possible, since mathematics is perfectly capable of describing inconsistent and impossible things.
You haven’t eliminated God by your explanation, you’ve necessitated His existance.
…. unless there’s some Cosmic Mathematician who/that gets to decide which bits of All Possible Mathematics get to create reality and which bits don’t….
Steve said.
I confess to not having a clear definition of the very “to exist”. But what I mean to assert unambiguously is this: There is no reasonable definition of the word ‘exist’ according to which rocks exist and natural numbers don’t.
OK what if I define exist as “manifests in the physical universe”. Rocks exist in that sense but the natural numbers don’t.
Jonathan Kariv:
OK what if I define exist as “manifests in the physical universe”. Rocks exist in that sense but the natural numbers don’t.
Well, okay. You could also define exist as “manifests in the ring of even numbers” and conclude that the number 2 exists in that sense but the number 3 doesn’t. But this feels like a cheat, and your definition feels to me like a cheat in exactly the same way. The physical universe is, according to every physical theory with any currency, some sort or another of a mathematical object (e.g. a manifold equipped with various fiber bundles, connections on those bundles, etc.). So you have defined “existence” as “manifesting” in one arbitrarily chosen mathematical object as opposed to any other. The physical universe seems to me to be no more special than the ring of even integers.
I’d say that all our theories describe the universe as mathematical objects because mathematics is the only real tool we have for making models of the universe and not because the universe is a mathematical object. Suppose for a second that there was another “universe” that was not a mathematical object existed, suppose further that we (all of human kind) where transported there and charged with developing physical models of it. What would you expect these models to look like?
Secondly even if I grant that the universe is a mathematical object, I’d still like people discussing existence to keep track of in what mathematical framework things exist. You might not think the physical universe (the object we inhabit) is special but I think it’s clear that most people do, and exist *does* have this colloquial meaning of existing in the physical universe.
Steve, is your belief that humans can perceive the natural numbers compatible with your belief that the natural numbers must exist in any universe? There are different ways humans acquire beliefs:
1. There are some beliefs (what philosophers call “a posteriori”) which we acquire through ordinary sensory perception of the physical world, for instance the notion that rocks are hard solid objects. Since this information came to us through the senses, it is a contingent fact: we can easily imagine a universe in which our senses revealed rocks to be soft and fluffy.
2. There are other facts (called “analytic a priori”) which require no perceptions of any kind, namely tautologies whose truth is an immediate consequence of the meaning of the words involved, like “Bachelors are unmarried.” We obviously do not need to make much in the way of philosophical or ontological commitments in order to accept that these kinds of statements, because they are true by definition, must necessarily be true in any conceivable universe.
Presumbably you believe that mathematical statements belong to neither of the above categories, but to a third category of statements (called “synthetic a priori”), which are not discovered by interacting with the physical world or by simply examining the meaning of the statements, but rather through some special mode of intuition or perception. In simplistic terms, you think that human knowledge of mathematics comes from our ability to view Platonic heaven. But if that is the case, then don’t the same kinds of limitations apply as in category 1? Just as our senses could have revealed rocks to either be hard and rigid or soft and fluffy, couldn’t our special perception of mathematical objects have revealed the prime numnbers to have either finite cardinality or infinite cardinality? Just as we have little reason to believe that the observed properties of rocks are a necessary feature of any possible universe, what justification do we have to believe that the properties of numbers are necessary?
It’s worth noting that not all extreme Platonists believe that mathematical truth is necessary truth. To take a famous example, Kurt Godel saw the laws of physics and the laws of mathematics on even footing: he saw both as true, both discoverable through human perception, but both accidental features of the world which could have turned out differently.
Steve’s comment that for him the physical universe is no more special than the ring of even integers is very revealing. It shows (if he really means it and is not just being provocative) a view of the world which I find quite alien. Clearly he loves (as many mathematicians do) the abstract world of pure math and it may well have the significance he attributes to it. It may well be that the intuitions which I have about existence and non-existence are just biases. But I will hold to my sense that the physical world is unfathomably rich and real (more real than – or somehow prior to – math) until persuaded otherwise. Because we all have our biases, mathematical Platonists included. I concede, however, that the existence of mathematical truths does pose serious challenges for physicalism but each of us has to go with his/her own intuitions until they prove sterile or incompatible with things we know to be the case. And I am not yet at this stage.
Steve: You remarked to JK, “The physical universe seems to me to be no more special than the ring of even integers.” I understood you to be asserting more, that the natural numbers/mathematics explain or entail the existence of the physical world. That is why roystngr’s objection (and mine) have force.
As for rocks I think there is a trick here. Rocks are an abstraction. I believe in some real substrate to the world that sometimes takes the form of rocks. Whether it’s really some 26 dimensional string I have no idea (I am confident it isn’t a thought in the mind of Bob Murphy’s imaginary friend though).
I have no issue with saying, fretting about the existence of the integers is odd if you don’t also fret about the existence of rocks. Rocks as I said are abstractions too. I don’t see that implies the integers make the rocks as it were, or that the integers are more existent somehow.
The medieval scholastics thought anything that had a name was real. This allows contradcitory names, like the yellow red ball or the square circle. You sidestep that problem, but it seems your notion, “I can write math for it so it’s real” is pretty similar. So do you find Anselm’s proof convincing? I’m betting not, and I also bet the reason is related to JK’s objection [which is the strongest one].
Going back to the rock vs. natural numbers, I agree with Steve that the natural numbers are more real than the rock. In M-theory (very much just a theory, but useful for this discussion), there are 11 dimensions, of which only perceive 4. Thus, that rock is an 11 dimensional object, but we are limited to only seeing 4 of those dimensions. If I stand on the street and look at a house, seeing one face of the house tells me something about the house, but only a fraction of all information about the house. When I kick the 11 dimensional rock, I am only interacting with a fraction of the rock – or perhaps I should say, I only perceive an interaction with a fraction of the rock.
@Steve, the one thing I struggle with in regards to the Mathematical Universe is incompleteness. If math is incomplete, and this is a Mathematical Universe, or one of many Mathematical Universes, then is physics incomplete? It seems that either (a) there are going to be some aspects of physics that are unprovable, or (b) physics can only employ a subset of mathematics. I have a hard time imagining either one, so is there a third possibility?
Is this because you didn’t understand the explanation in The Big Questions, or because you haven’t read it?
I don’t purchase books in order to comment on blogs. I did my best to understand you from the blog post on which I am commenting. As far as I can tell, you believe the physical universe is a mathematical object because you can make mathematical statements about it.
But the mathematical statements we can make about physical statements are a shorthand. I may say “I add 2 objects and 3 objects and get 5 objects”. But what I’m really doing is abstracting numbers from the stacks of objects, then operating on the abstracted numbers, then instantiating the abstraction with respect to objects again. “Adding” is what I call the operation that, after this procedure, produces a description of the physical objects I get when I combine the stacks.
I’m not making any mathematical statements about physical objects. I’m abstracting the objects to get numbers, then making mathematical statements about numbers, then instantiating the numbers.
Do the operations on natural have to exist?
That should be *…natural numbers…*.
Prof. Landsburg,
I think you have drawn the wrong conclusion from the fact that we often revise our beliefs about of rocks and rarely revise any opinion about numbers. But, before I get to that, I should say that I cannot follow the structure of your argument. It seems to go like this:
1. We have been wrong about what rocks are made of but this does not convince anyone that rocks do not exist.
2. Therefore, numbers exist.
That cannot be it, I know. But I cannot tell what else it is. If people argued that numbers do not exist because we have been wrong about what they are made of, I might get your point. But no sceptic about mathmatical objects has ever made this argument.
Actually, an argument for scepticism about mathematical objects starts from the peculiar fact that we are so rarely wrong about them.
Compare atoms and centres of gravity. Both are posits of theory — initially, at least. But in the case of centres of gravity, we are disinclined to take them ontologically seriously. This is because, once you’ve got the idea of centres of gravity from the theory, there will be no surprises about them (except insofar as logical implications can sometimes surprise us). Centres of gravity are purely “theoretical entities”. Atoms, by contrast, are the kind of thing we might learn more about. For example, as with rocks, we might revise our opinions about their parts. That is precisely why we are inclined to take them ontologically seriously.
If you look at the matter this way, the fact that our perceptions of mathematical facts are “largely eternal” is one of the things that makes mathematical objects suspect. This is not to deny that mathematical statements are true and, hence, that there are mathematical facts. But there are ways of understanding necessary truths that do not require the existence of consituent objects, such as numbers.
Sokona: Your last two paragraphs are exceptionally thought-provoking. I will therefore think about them. Thank you.
I’ll ask something I buried inside another post: Does the parallel postulate “exist”?
The comments about what “exist” means reminds me of a quote from Metrodorus of Chios (borrowed from the Stanford Encyclopedia of Philosophy): “None of us knows anything, not even this, whether we know or we do not know; nor do we know what ‘to not know’ or ‘to know’ are, nor on the whole, whether anything is or is not”