## Mathematics seems to accept the concept of zero but not the concept of infinity

Mathematics seems to accept the concept of zero but not the concept of infinity (only towards infinity); whereas Physics seems to accept the concept of infinity but not of nothing (only towards zero). Yet there is a discipline of 'mathematical physics' . Is there an inherent fault in mathematical physics?

### I'm pretty sure that

I'm pretty sure that mathematicians and physicists would both reject the way you've described them.

Mathematics not only accepts the concept of infinity but has a great deal to say about it. To take just one example: Cantor proved in the 19th century that not all infinite sets are of the same size. In particular, he showed that whereas the counting numbers and the rational numbers can be paired up one-for-one, there's no such pairing between the counting numbers and the full set of real numbers. Thus, he proved that in a well-defined sense, there are more real numbers than integers, even though in that same sense there are not more rational numbers than integers.

Now of course, we sometimes talk about certain functions going to infinity in a certain limit. For example: as x goes to 0, 1/x goes to infinity, even though there is no value of x for which the value of 1/x is infinity. Rather, we say that at 0, the function is not defined. There are good reasons why we say that, though this isn't the place to spell them out. But examples like this don't show that mathematics rejects the concept of infinity.

As for physics, consider a very basic physical concept: an inertial system. The net force on an inertial system is 0. Physics assumes that the concept of an inertial system is a perfectly good one and indeed, a deeply important one. Whether any actual physical systems literally experience a net force of zero is another question; it depends on how things are actually arranged in the world. But physics doesn't rule it out a priori.

### I'm pretty sure that

I'm pretty sure that mathematicians and physicists would both reject the way you've described them. Mathematics not only accepts the concept of infinity but has a great deal to say about it. To take just one example: Cantor proved in the 19th century that not all infinite sets are of the same size. In particular, he showed that whereas the counting numbers and the rational numbers can be paired up one-for-one, there's no such pairing between the counting numbers and the full set of real numbers. Thus, he proved that in a well-defined sense, there are more real numbers than integers, even though in that same sense there are not more rational numbers than integers. Now of course, we sometimes talk about certain functions going to infinity in a certain limit. For example: as x goes to 0, 1/x goes to infinity, even though there is no value of x for which the value of 1/x is infinity. Rather, we say that at 0, the function is not defined. There are good reasons why we say that, though this isn't the...

## Quantum mechanics seems to suggest that there really is such a thing as a random

Quantum mechanics seems to suggest that there really is such a thing as a random number, yet all of philosophy and logic point to a reason or cause for everything, perhaps beyond our understanding. Is this notion of a random number just another demonstration of limited human understanding?

### I guess I'd have to disagree

I guess I'd have to disagree with the idea that "all of philosophy and logic point to a reason or cause for everything." There's certainly no argument from logic as such; it's perfectly consistent to say that some events are genuinely random. Some philosophers have held that there's a reason (not necessarily a cause in the physical sense, BTW) for everything, but the arguments are not very good.

On the other hand... quantum mechanics is a remarkably well-confirmed physical theory that, at least as standardly interpreted, gives us excellent reason to think that some things happen one way rather than another with no reason or cause for which way they turned out.

An example: suppose we send a photon (a quantum of light) through a polarizing filter pointed in the vertical direction. We let the photon travel to a second polarizing filter, oriented at 45 degrees to the vertical. Quantum theory as usually understood says that there's a 50% chance that the photon will pass this filter and a 50% chance that it won't. But quantum theory itself provides no account whatsoever of which will actually happen. And on the usual interpretation, there is no reason or cause; it's really random.

Now the usual interpretation of quantum mechanics could be wrong. There are deterministic interpretations, most notably "many worlds" or Everettian quantum mechanics, and Bohmian mechanics. No one is in a position to rule either of those out; all I can say is that neither of those approaches is to my taste. But even though both of them restore determinism, that's not really their motivation. Most people who work in foundations of physics are not bothered by the very idea of indeterminism and in fact, indeterminism wasn't by any means Einstein's biggest issue with quantum mechanics.

So to sum up: I don't think there are any good general arguments against randomness. I think the concept is coherent, and that it's a plausible fit for our best physical theory. It also happens to suit my own prejudices about quantum mechanics, but that's just icing on the cake. ;-)

### I guess I'd have to disagree

I guess I'd have to disagree with the idea that "all of philosophy and logic point to a reason or cause for everything." There's certainly no argument from logic as such; it's perfectly consistent to say that some events are genuinely random. Some philosophers have held that there's a reason (not necessarily a cause in the physical sense, BTW) for everything, but the arguments are not very good. On the other hand... quantum mechanics is a remarkably well-confirmed physical theory that, at least as standardly interpreted, gives us excellent reason to think that some things happen one way rather than another with no reason or cause for which way they turned out. An example: suppose we send a photon (a quantum of light) through a polarizing filter pointed in the vertical direction. We let the photon travel to a second polarizing filter, oriented at 45 degrees to the vertical. Quantum theory as usually understood says that there's a 50% chance that the photon will pass this filter and a 50% chance that it...

## Hi Philosophers,

Hi Philosophers, I have a burning question that is troubling me relating the religion versus science debate. I hope I articulate it well enough. Here goes. Mathematically, physicists are close to proving that a multiverse exists. Assuming they do prove this, and that as part of this proof it is deemed that infinity universes exist with both every conceivable and inconceivable possibility and outcome occurring throughout, then is it not fair to say that God certainly exists in at least one of these infinite possibility universes? Adversely, it is also fair to assume that God certainly does not exist in at least one of these universes? Then consider that if God certainly exists in at least one universe, and he is the all-seeing, all-knowing God that religion states he is, then how can he certainly not exist in at least one of the infinite universes? To say that God definitely exists is to, by definition of God, say that he exists everywhere and created everything, yet this notion within the multiverse...

Great question (and great response by Allen). Let me just add a tiny bit, by encouraging you to check out both Norman Malcolm's and Alvin Plantinga's work on the ontological argument. (The latter is a lot more technical and difficult, so start with the former.) From them you get something like the idea that if God exists at all, He exists necessarily (for God surely isn't a contingent being); to say that God exists necessarily is to say that He exists in every possible world. But now, if it's even possible that God exists -- i.e. the idea of God contains no contradictions -- then God would exist in at least one possible world. But if He exists at all He exists in every possible world, so if He exists in one PW He exists in every PW. Now is it possible that God exists? Does the idea of God involve any contradictions? Lots of discussion in the history of philosophical theology on that topic (lots of purported contradictions posed, then response to), but lots of people, even many ordinary atheists, think there's no contradiction in the idea of God, just merely that God contingently doesn't exist. So if you construe the multiverse theory to mean that every possible world exists (not sure it should be construed this way, but let's suppose), and if you think the idea of God involves no contradictions, then it sounds like the multiverse theory could support this line of argument toward God's existence.

hope that's useful!

ap

I think it's a bit optimistic to say that physicists are close to proving the existence of a multiverse, but we can set that aside. There are different ideas of a multiverse in physical theory, but none of the ones that cosmologists take seriously call for showing that literally every possible "universe" exists. Rather, what's at stake is the idea that the totality of the Universe writ large contains relatively isolated sub-parts that have many of the characteristics of the physical universe as we usually think of it. In particular, the values of various physical "constants" would vary across the different sub-universes. But the important point for your question is that this is entirely about physics and has nothing to do with God. God, as usually understood, is not a physical being at all, but a being who (among other things) underwrites the existence of physical things. God doesn't exist within this or any other physical universe on the usual theological view. Put it another way: if the God...

## The big bang theory says that time began with the big bang. Is that correct?

The big bang theory says that time began with the big bang. Is that correct? Then does that mean that those who describe the big bang theory as an idea that something comes from nothing are incorrect? If time began with the big bang doesn't that mean there never was a time when there was nothing?

I can't resist responding to one thing that Prof. Stairs says in his excellent reply: "If there's no such [necessary] being, then it might be that there's no explanation for why contingent things exist." I used to think that myself. But as I thought more about the question "Why do any contingent things exist?" I concluded that the question has a very simple answer -- indeed, many simple answers -- if it's a well-posed question in the first place, and those answers have nothing to do with any necessary being. I try to explain why in this paper.

Not quite correct. Cyclic theories still posit a Big Bang, but they also posit a cycle of expansions and collapses. This is not something I know much about but you can read a bit more here If we suppose that the non-cyclic Big Bang model is correct, then in at least one sense, the universe isn't a case of getting something from nothing: it's not an example of matter appearing uncaused in a universe where there are earlier times with no matter. Of course, that's consistent with there being no explanation of why there's matter/energy at all. That may not quite amount to getting something from nothing, but it's an idea that doesn't sit well with everyone. Some versions of the Cosmological Argument are meant to explain why contingent things (like the physical things we're familiar with) exist. The explanations typically appeal to the existence of a Necessary Being—one who's very nature requires that it exist. If there's no such being, then it might be that there's no explanation for why contingent...

## In a book by John Honner dealing with Niels Bohr's philosophy of physics, he

In a book by John Honner dealing with Niels Bohr's philosophy of physics, he finishes a sentence with "once the framework of complementarity is substituted for that of continuity and univocity." I can't find a definition of 'univocity' in the dictionary, and all google search results seem to apply to religion. Can someone help me with a definition that might apply in this context?

The best word to look up is univocal, which is roughly the opposite of equivocal. It means, more or less, unambiguous, or having one meaning. Without the whole sentence, it's a bit hard to be sure what the author meant, but the idea of complementarity, in part, is that we can't bring pairs of concepts such as position and momentum both to bear in a single experiment; we must choose. In classical physics, we can. There is, as it were, a univocal point of view that we can take on a physical system classically, whereas in applying quantum theory we must choose between incompatible experimental arrangements and "complementary" physical concepts.

The best word to look up is univocal , which is roughly the opposite of equivocal . It means, more or less, unambiguous, or having one meaning. Without the whole sentence, it's a bit hard to be sure what the author meant, but the idea of complementarity, in part, is that we can't bring pairs of concepts such as position and momentum both to bear in a single experiment; we must choose. In classical physics, we can. There is, as it were, a univocal point of view that we can take on a physical system classically, whereas in applying quantum theory we must choose between incompatible experimental arrangements and "complementary" physical concepts.

## Are dimensions exceeding 3 actually comceivable or are they purely intellectual

Are dimensions exceeding 3 actually comceivable or are they purely intellectual constructs? Is this even debated in philosophy?

If I understand your question correctly, it's whether there really could be more than three dimensions in physical space. The best answer, I should think, is yes. One reason is that there are serious physical theories that assume the existence of more than three spatial dimensions: string theory is the example I have in mind.

More generally, though, it's not clear why we should doubt that this is possible. The fact that we can't represent it to ourselves imaginatively doesn't seem like a very good reason. We can't represent curved space-time to ourselves imaginatively, but if general relativity is right, space-time does curve. We have a notoriously hard time representing quantum mechanical objects to ourselves imaginatively, and yet quantum mechanics is the cornerstone of much of our physics.

We can even say things about what it would be like to live in a world with more than three spatial dimensions. Consider: think of a plane in 3-space, and imagine a walled square in that plane. An object can enter the square without passing thought its walls; it enters from above or below. In a 4-dimensional space, if we picked a walled cube in a 3-dimensional "hyperplane" of the 4-space, an object could enter that cube without passing though its walls: from a region outside the plane. Those with better geometric skillz than I could offer more elaborate examples. (See also Edwin Abbot's Flatland for a delightful 19th-century exploration of this theme.)

One other thought: though we can in some sense "imagine" three dimensions, when you start trying to think about the details even of three-dimensional geometry, it's interesting just how fast imagination runs out and you're forced to adopt a more purely geometrical or mathematical approach. The limits of what we can picture or grasp intuitively aren't a very good guide to what's really possible, let alone true!

If I understand your question correctly, it's whether there really could be more than three dimensions in physical space. The best answer, I should think, is yes. One reason is that there are serious physical theories that assume the existence of more than three spatial dimensions: string theory is the example I have in mind. More generally, though, it's not clear why we should doubt that this is possible. The fact that we can't represent it to ourselves imaginatively doesn't seem like a very good reason. We can't represent curved space-time to ourselves imaginatively, but if general relativity is right, space-time does curve. We have a notoriously hard time representing quantum mechanical objects to ourselves imaginatively, and yet quantum mechanics is the cornerstone of much of our physics. We can even say things about what it would be like to live in a world with more than three spatial dimensions. Consider: think of a plane in 3-space, and imagine a walled square in that plane. An object can...

## I'm not sure if this is a question for philosophers or for physicists, but I'll

I'm not sure if this is a question for philosophers or for physicists, but I'll ask it here anyway. Do you think it is possible that there are other universes? I mean "other universe" in a very physical sense: any group of objects that have no past, present or future physical relations to the objects in our universe. For example, they don't originate in the Big Bang. And it is physically impossible that a photon leaves one of such objects and hits one of the objects in our universe. And those objects aren't at any distance from the objects in our universe (it cannot be said truthfully that those objects are or were n light-years away from any star in our universe). But I mean real, actual objects, and not merely "possible objects" (there is a previous answer on this subject in AskPhilosophers, but that's not what interests me)! Do you think that there can exist other universes in this sense?

Let's use a phrase from the philosopher David Lewis: concrete worlds. Let it mean complete, concrete universes. Lewis thought that there are concrete worlds other than our own, and that there is at least one for each way our world could be. Lewis also characterizes these words in the way you do: they aren't in our space-time, so they're not at any spatial or temporal distance from our concrete world, and they don't interact causally with our world. As Lewis understand things, they wouldn't be other worlds if these conditions didn't hold.

Lewis thought there were such things. He thought that making sense of ordinary truths about what's possible calls for them. My admittedly unpopular view is that this is the wrong way to think about possibility. Even if such worlds do exist, there may be (would be, I'd say) non-trivial modal truths about them. Any particular such world might not have existed, for example. Far as I can see,here's no home in Lewis's account of modality for these sorts of possibilities. In fact, I'm tempted to resurrect an old term and say that the Lewis-style way of thinking about possibility amounts to a category mistake.

But that's just me, and at least some of my colleagues would hoot in derision.

Be that as it may, it's one thing to say that these worlds would be the wrong sorts of things for the job Lewis assigns them (and I realize that's not your issue); it's another to say they couldn't exist. I can't think of any good reason to believe that such worlds are flat-out impossible. And though I'm open to persuasion, my instinct is to suspect that any "proof" to the contrary would be sophistical.

This might sound like a simple "yes" to your question, but it's not so simple as that. What's possible and what I seem to be able to describe or imagine may not be the same thing. It may be that, for reasons I'm not grasping, utterly other concrete worlds are not possible. All I can say is that if I were making philosophical bets, I'd go with saying they are. I'm just not sure what my odds would be...

Let's use a phrase from the philosopher David Lewis: concrete worlds . Let it mean complete, concrete universes. Lewis thought that there are concrete worlds other than our own, and that there is at least one for each way our world could be. Lewis also characterizes these words in the way you do: they aren't in our space-time, so they're not at any spatial or temporal distance from our concrete world, and they don't interact causally with our world. As Lewis understand things, they wouldn't be other worlds if these conditions didn't hold. Lewis thought there were such things. He thought that making sense of ordinary truths about what's possible calls for them. My admittedly unpopular view is that this is the wrong way to think about possibility. Even if such worlds do exist, there may be (would be, I'd say) non-trivial modal truths about them. Any particular such world might not have existed, for example. Far as I can see,here's no home in Lewis's account of modality for these sorts of...

## In theory of relativity all relations are derived based on one observer in a

In theory of relativity all relations are derived based on one observer in a moving frame relative to another frame. How statistically relevant it is to make conclusions based on just one observer? Who told it is valid?

I don't quite recognize relativity in what you're saying. Relativity tells us that an experiment in one inertial (non-accelerating) frame will look the same in any other inertial frame (that part also applies to Newtonian physics) and that the speed of light (in a vacuum) is the same in all inertial frames. (That part is a departure form Newtonian physics.) Relativity also tells us how to translate velocities, times, etc. between different inertial frames, and it gives an answer that's different from the Newtonian one. But the evidence for relativity has nothing to do with picking some one observer and giving that observer special status. On the contrary: that would go completely against the point of relativity. Further, there's no question of drawing experimental conclusions of whatever sort based on just one observer. Rather, what relativity says is that whichever observer performs an experiment, his/her state of inertial motion won't affect the physics. Whether the evidence from the observer's experiment is strong enough to let us draw general conclusions is a separate question, and one that has nothing to do with the theory of relativity.

I don't quite recognize relativity in what you're saying. Relativity tells us that an experiment in one inertial (non-accelerating) frame will look the same in any other inertial frame (that part also applies to Newtonian physics) and that the speed of light (in a vacuum) is the same in all inertial frames. (That part is a departure form Newtonian physics.) Relativity also tells us how to translate velocities, times, etc. between different inertial frames, and it gives an answer that's different from the Newtonian one. But the evidence for relativity has nothing to do with picking some one observer and giving that observer special status. On the contrary: that would go completely against the point of relativity. Further, there's no question of drawing experimental conclusions of whatever sort based on just one observer. Rather, what relativity says is that whichever observer performs an experiment, his/her state of inertial motion won't affect the physics. Whether the evidence from the observer's...

## Is length an intrinsic property or is it something which is only relative to

Is length an intrinsic property or is it something which is only relative to other lengths? Is an inch an inch? Or is it simply a relation between other (length) phenomena?

It is indeed an interesting question, and in fact it's more than one question.

To begin with, my colleague is correct: in special relativity, length is like velocity in classical mechanics: it's a "frame-dependent" quantity. However, the theory of relativity is also a theory of absolutes; between any two points in space-time there is a quantity called the interval, and it is not frame-dependent. To put it in the jargon of relativity, the space-time of special relativity has a metric -- a generalized "distance function" -- and that metric delivers an unequivocal answer to the question of whether the interval between w and x is equal to the interval between y and z.

But now we have a new question: suppose that relativity says that the interval between w and x is, in fact, the same as the interval between y and z. What kind of fact is that? Suppose that the two intervals have no overlap. Doing business as usual, so to speak, we come up with the answer that the intervals are equal, but we could use a different metric function that gave a different answer, and by making adjustments elsewhere, we could make the physics work out. The physics that made the adjustments might be more unwieldy; it might involve some peculiar "universal forces," for example. However, it's not immediately clear that this shows the usual way of doing things to be ontologically privileged.

The debate we're now describing has to do with the "conventionality of the metric," and some heavyweight thinkers, not least the philosopher of science Hans Reichenbach, have argued that the metric is indeed conventional. That is, it depends on choices we make that could, in principle, have been made in a different way.

This issue has a connection to questions about meters and such. We pick out a meter by reference to the standard meter stick (or at least, that's how we used to do it.) But there's some complexity here. Suppose I take my own meter stick, lay it against the standard meter, and find that they match. While they're in contact, there's no doubt that they have the same length. But what about when they're not? Does my meter stick retain its "real" length when I move it around? (Leave issues of relativity aside for the moment. We could say things in a more complicated way that took them into account, but the issue would not really change.) Or does it contract or expand? Or is there really no absolute metaphysical fact of the matter? The conventionalists would pick this last option. When we consider the whole package of our physics and our measuring devices and our assumptions about forces, we may say that (absent unusual circumstances) the measuring rod has the same length when it's "here" as it does when it's "there." But the conventionalist would insist that saying this ultimately rests on certain stipulations or conventions.

The literature on this topic is complex, as you might imagine. Even though he argues for one side rather than the other, I'd suggest that Reichenbach's Philosophy of Space and Time is a good place to start. It's an engaging book that's more accessible than it might appear at first sight.

It is indeed an interesting question, and in fact it's more than one question. To begin with, my colleague is correct: in special relativity, length is like velocity in classical mechanics: it's a "frame-dependent" quantity. However, the theory of relativity is also a theory of absolutes; between any two points in space-time there is a quantity called the interval , and it is not frame-dependent. To put it in the jargon of relativity, the space-time of special relativity has a metric -- a generalized "distance function" -- and that metric delivers an unequivocal answer to the question of whether the interval between w and x is equal to the interval between y and z. But now we have a new question: suppose that relativity says that the interval between w and x is, in fact, the same as the interval between y and z. What kind of fact is that? Suppose that the two intervals have no overlap. Doing business as usual, so to speak, we come up with the answer that the intervals are equal, but we ...

## Help me know if I have the Big bang theory down correctly. It consists of the

Help me know if I have the Big bang theory down correctly. It consists of the following ideas. 1. The big bang theory is usually or often seen as a naturalistic hypothesis where only physical reality is truly real. 2. The universe is a physical reality. 3. There was no physical reality prior to the universe. 4. The universe began with the Big Bang. 5. There was no universe (or physical reality) prior to the Big Bang. 6. It follows from 1-5 that nothing whatsoever existed prior to the Big Bang. 7. The objection of so how did the universe come about if there was nothing prior to the big bang is that time only began with the big bang. To speak of a beginning implies an occurrence within time. It is therefor circular to say that time began with the beginning(of time). I think that what is happening here is that a rejection of (traditional metaphysical)philosophy and even common sense means that science has become a new form of irrational religion in our day. What do you philosophers have to say about this?...

I'd say that you don't have the Big Bang theory down correctly if by the Big Bang Theory you mean what physicists mean. Whether a physicist accepts some version of the Big Bang account as a piece of physical cosmology and whether the physicist believes that nothing is real except for the physical are two different questions. Some physicist (perhaps even most for all I can say) believe that there's nothing outside the physical, but that's not a claim within physics. It's a metaphysical claim that the Big Bang hypothesis doesn't settle.

But suppose we're physicalists (i.e, suppose we think only the physical is real.) Is the combination of physicalism and the Big Bang inconsistent?

I don't see why.

You say that to speak of a beginning implies an occurrence within time. Let's concede that; it's logically harmless. The first moment, if there was one, was, of course, a moment in time. If time began, it began with the beginning; we can agree that this is circular (or better: a tautology.) But "Time had a beginning" is not a tautology at all, and the claim that there were no moments before the big bang is anything but circular. (It may be false; that's a scientific question. But the very fact that it could be false is proof that it's not a tautology.)

The idea that there was a first moment is peculiar the first few times you hear it. Once you get used to the fact that modern cosmology routinely appeals to space-time structures that are mathematically well-understood but not what Newton - let alone Euclid - had in mind, the oddness tends to fade away. No contradiction follows from saying that there was a first moment, and even on its face, the claim doesn't seem circular. The fact that "common sense" finds this odd doesn't tell us much; there's a great deal of respectable science that common sense finds odd, but then common sense didn't cut its teeth on the sorts of far-from-common situations that science has taught us to deal with.

I'd say that you don't have the Big Bang theory down correctly if by the Big Bang Theory you mean what physicists mean. Whether a physicist accepts some version of the Big Bang account as a piece of physical cosmology and whether the physicist believes that nothing is real except for the physical are two different questions. Some physicist (perhaps even most for all I can say) believe that there's nothing outside the physical, but that's not a claim within physics. It's a metaphysical claim that the Big Bang hypothesis doesn't settle. But suppose we're physicalists (i.e, suppose we think only the physical is real.) Is the combination of physicalism and the Big Bang inconsistent? I don't see why. You say that to speak of a beginning implies an occurrence within time. Let's concede that; it's logically harmless. The first moment, if there was one, was, of course, a moment in time. If time began, it began with the beginning; we can agree that this is circular (or better: a tautology.) But ...