## How can the diameter of a rainbow be measured?

How can the diameter of a rainbow be measured?

To quote from the great SNL philosopher, Mango, "Can you touch a rainbow? Can you put the wind in your pocket? No! Such is Mango." I think he has it right. I don't know much about the optics of rainbows, but I'm pretty sure they move relative to the observer, so they do not have an objective diameter.

At least that's what I thought until I found this answer on the magical internet here:

"It's probably not impossible, but it is difficult. A rainbow looks circular because it's basically the circle where a cloud of rain droplets intersects with your cone of vision, like the circle on the end of an ice-cream cone. Imagine said ice-cream cone with the point in your eye (don't actually try this experiment unless you're looking for a career in piracy). Now make the cone bigger and bigger until the round end hits the cloud of raindrops that are reflecting the sun's light. The big circle on the end of that cone is where the rainbow appears to be -- as someone else pointed out, you can only see the top half of it (because the other half is below the surface of the earth). The raindrops reflect light at about a 40 degree angle, so you can calculate the diameter of the circle if you also know the height of the cone (because the height of the cone, the radius of the circle, and the 40 degree angle are all part of a right angled triangle). The challenge is knowing the height of the cone, which is how far away the cloud of raindrops is from you. If you can work that out then, yes, you can measure the diameter of the rainbow (diameter = (2*distanceToCloud) / tan 40)."

Also, if the ends of a perfect rainbow appear to touch the earth at two known landmarks (say, two buildings), perhaps the diameter is the distance between those landmarks?

## I have come across a dilemma, I could not find the question on the site

I have come across a dilemma, I could not find the question on the site presently so I hope it has not been answered yet. If an atom is the smallest piece of matter that we are aware of, doesn't some form of matter have to make up an atom? And whatever the form of matter that makes up an atom, would have to be made up of some other form of matter and that matter would have to be made up of a kind of matter as well, and on and on forever. Where does that stop? How can a human being ever comprehend something like this? Thank you.

This is a wonderfully knotty question that has occupied philosophers at least since Zeno of Elea in the 5th century BCE. One way of interpreting Zeno on this is to say that the problem shows that space is illusory. David Hume later, like the atomists ('atom' meaning uncuttable) seems to have thought that there must be a point at which the cutting stops, at least so far as the world of experience goes. I might say that Zeno is right that certain ways of conceiving space are flawed, including the way the problem as you pose it conceives of space--that is, as continuous all the way down, becoming just a finer and finer Cartesian grid if you will, always subject to the same sorts of properties or ways of conceiving things (like length, height, depth, etc.). It seems, however, that once we reach the sub-atomic realm these ways of conceiving things just don't hold, so that it becomes impossible to apply mathematical divisions of space. Space seems dependent on, you might say, the ability of energy to propagate and knot in various ways and as we approach quarks (which have perhaps no spatial dimensions) it simply is not able to do so. So, in a sense the atomists are right but not because there is finally an uncuttable (or indivisible) particle, either conceptually or really. Rather, they're right because after a certain point the idea of cutting becomes in applicable.

## If quantum mechanics or other fundamental theories of physics have it that small

If quantum mechanics or other fundamental theories of physics have it that small physical entities which make up everything else do not behave deterministically, does that indeterminism inherited by all other larger entities, whether those are molecules, gases, instantiated computer programs, and people? In general, does indetermism on one "lower" physical level imply indetermism on a "higher" one?

The answer to the general question is that indeterminism at the "lower" level doesn't have to mean indeterminism at "higher" levels. Here's an abstract way to think about it. Suppose some theory has a set of possible states -- call it S -- and a strict deterministic law governing how the states change over time. Let's suppose that this theory is both true and know to be true. But suppose, unbeknownst to us, each of the states in S can be realized in many different ways, at some sub-microscopic level that we don't have access to. And suppose that even though the law that tells us how we get from one state in S to another is deterministic, there's no deterministic law governing exactlywhich way states in S will be realized as the system moves from one state to another.We might never have any reason to believe any such thing, but it could be true all the same.

That's one story about how indeterminism at the micro level might not infect the macro level. Another way is a "for all practical purposes" version. There might be all sorts of blooming and buzzing at the fine-grained level, but all that might average out so that things at the macro level are extremely unlikely to depart from some deterministic rule. Thermodynamic systems are typically like that; the thermodynamic laws hold "for all practical purposes" even though what's going on under the hood is (or may be) indeterministic, and even though there is a teeny tiny probability that your cup of coffee will spontaneously freeze.

As for quantum indeterminacy, it certainly can infect the macro-world. Indeed, it does so every time someone in a lab performs a quantum mechanical measurement. (What the instrumnet registers depends, crudely, on which way the quantum jumps.) But there are good quantum stories about why most of the time, in most circumstances, macroscopic things behave deterministically to a very good approximation

## If the universe is expanding, what is it expanding into ?

If the universe is expanding, what is it expanding into ?

## Astrophysicists maintain the idea that time and space came about with the Big

Astrophysicists maintain the idea that time and space came about with the Big Bang. Is there any way in which this notion can be related to Kant's concept which states that time and space are not objectively real, but that both are transcendental conditions of the perception of objects in terms of phenomena? Yours, Stephan R. (Aachen, Germany)

First, it should be noted that not all astrophysicists agree that time and space began with the Big Bang. There may be no meaningful way to measure or study space and time before the Big Bang, but that does not necessarily mean that there is no such thing. Scientists can agree on empirical findings and on the theories that best predict further findings without agreeing about the nature of the reality that underlies those findings. (This is especially clear in the case of Quantum Mechanics, where several competing interpretations have scientific adherents.)

Kant claimed that an entirely empty space, and an entirely empty time, are perfectly conceivable. So if the reason behind believing that space and time began with the Big Bang is the belief that the intelligibility of space and time depend on the presence of objects in space and time, he would disagree. Indeed, this view is the explicit target of several of his arguments in the Transcendental Aesthetic of the Critique of Pure Reason.

On the other hand, (in the Antinomies of the Critique of Pure Reason and elsewhere) Kant also warned against the attempt to decide whether space and time are finite or infinite. That is not a decidable question according to him. So I imagine that he would be sympathetic to those who simply refuse to speculate about space and time before the Big Bang.

Finally, the claim that space and time are transcendental conditions on the perception of objects does not by itself require a denial of their objectivity. We may need to experience objects in space and time in order to experience anything at all, but that leaves open the possibility that space and time also exist quite apart from our experience. Kant does sometimes assert that space and time are forms of experience that are imposed by us; but he also offers powerful arguments against the view that objects do not exist independently of us in space and time; how to reconcile these two claims is a disputed area of Kant scholarship.

## Is it true that "Things fall because of gravity?" "Gravity" is just a

Is it true that "Things fall because of gravity?" "Gravity" is just a placeholder word for the tendency of things to fall. So to say "Things fall because of gravity", is to say "Things fall because of their tendency to fall." Which is vacuous. A better explanation would be "Things fall because they have mass and are nearby another massive object (the earth)." Am I right here?

This sounds like an accusation that was regularly thrown at Medieval Aristotelian physicists. Aristotelian physics was built around the "teleological" principle that things have natural tendencies to strive to achieve certain goals or destinations. Why does a stone fall? Aristotle would say that the explanation for this rests on the fact that it is in the nature of an earthy body to move towards the natural place of such bodies, which (he believed) is in the centre of the cosmos. But you're quite right, this does sound rather vacuous, to say that it moves as it does because it has a natural tendency to do so. The Aristotelians sought to explain natural phenomena in terms of what came to be known as "occult qualities" and, although the term "occult" might not have carried quite the connotations it has now, it was used perjoratively by many non-Aristotelians to point to the fact that these supposed qualities really weren't explanatory at all. The Aristotelian approach was famously lampooned by Molière, the seventeenth century French playwright, who had one of his characters ask why it was that opium should put people to sleep. Because it has a dormitive virtue, came the reply. Which is just a fancy way of saying that it puts people to sleep because it has the power to put people to sleep. Gee, thanks for that: now everything's clear!

But let's skip ahead to Isaac Newton. The one thing that everyone knows about Newton is that he explained how gravity works, right? Well, not quite. Newton's achievement -- and it certainly was a colossal achievement -- was to discover a law that would enable one to calculate the effect the proximity of one massive body would have on the motion of another massive body. But, as to how it produced this effect, on this Newton remained deliberately silent. In the General Scholium to the 1713 edition of his Principia (wherein he had presented this law), Newton wrote: "I have not as yet been able to deduce from phenomena the reason for these properties of gravity, and I do not feign hypotheses. For whatever is not deduced from the phenomena must be called a hypothesis; and hypotheses, whether metaphysical or physical, or based on occult qualities, or mechanical, have no place in experimental philosophy."

Newton's attitude to the nature and scope of experimental science ("natural philosophy", in his day) would subsequently come to dominate the field. But does this mean that scientists never truly explain anything after all, but merely subsume phenomena under laws of nature? Well, perhaps we should take a narrower view of the nature of explanation itself: many modern philosophers of science would say that to subsume a phenomenon under a law of nature is to explain it. They have come up with something they call the "D-N" ("deductive-nomological") model of scientific explanation. Although there are some notorious problem-cases that this model can't neatly handle, which are extensively discussed in the literature, for present purposes we will just put those to one side. In this model, if you want to explain some phenomenon, what you need to do is (i) specify some initial conditions and (ii) find a law of nature ("nomological" means "pertaining to laws"), such that the phenomenon is (deductively) entailed by those initial conditions in conjunction with that law. It is not quite enough just to say that a stone falls because it has mass and is near another massive object: you additionally need to point to a law (such as Newton's) to connect these various concepts up together. Why is A the case? Because B was the case, and it's a law of nature that, whenever you get B, A will surely follow.

Such an explanation is not vacuous, because it really does add some genuinely new information-content that goes beyond the phenomenon to be explained. It is set out in different terms (at least in part), and it refers to a principle that encompasses a very much wider domain than just that one specific phenomenon. But is it enough? Can we regard this as a complete explanation of the phenomenon? No, unfortunately, we cannot. As Newton recognised, there are still further questions to ask. How is the gravitational influence of the Earth propagated across the intervening distance? Why should the laws of nature be like that at all? These were the issues that Newton declined to feign hypotheses about, on the grounds that he didn't (yet) have enough experimental data to settle them. But the important thing to appreciate is that there are always going to be further questions to ask. As small children tend to discover fairly early in life, they can quite effectively infuriate their parents by repeatedly asking "Why?" Whatever answer their parents might give them, explaining one thing in terms of something else, it will always invite a further question: "But why is that the case?" And this process can go on indefinitely, for as long as the child -- or the scientist -- has the patience to keep pressing. Equally, though, it can stop at any time. Explanation is always tied to specific, subjective interests, and, sooner or later, the questioner will be satisfied. "Why A?" "Because of B." "But why B?" "Because of C." "Fine, but why C?" "Because of D." "Ah, now I see: thank you." The questioner could perfectly well carry on: but they don't, because things have now been sufficiently explained to their own satisfaction.

Is there such a thing as an "ultimate" explanation, one that would close off all further "Why?" questions once and for all? I doubt it. Although it is not absolutely guaranteed, it does seem fair to assume that the natural world, as far as both its individual facts and its universal laws are concerned, is contingent through and through. It is one way, but it could have been another way -- which invites the question of why it is this way rather than that. But does the fact that the process of explanation might never reach an end-point mean that it cannot get anywhere at all? Certainly not. If we manage to subsume a phenomenon under some general law, as in the D-N model, we have provided some genuine information. If, let's say, we then manage to unify that law with others, and show that this phenomenon is actually an instance of a still-wider class of phenomena than we had previously suspected, this would be a major advance, pushing our explanation to a more fundamental level and increasing its information-content still further. Cutting-edge physicists right now are trying to come up with a set of equations that will enable them to connect gravitation up with the other three fundamental forces they countenance (electro-magnetism, and the strong and weak nuclear forces). And success in this project would be a major breakthrough, providing a tremendous new insight into the nature of the universe, even though it would then just invite a new question: why should those be the equations that govern how things work, rather than some others?

## Hi.

Hi. This is a question about Logic. I've read in a book by Michio Kaku, _The Physics of the Impossible_, that it may be possible to receive a signal before it was sent. This to my way of thinking would violate the logic behind causality. And yet on a social level we are effected by what happens in the future. An example would be Christmas shopping. My question is can an effect precede a cause, and if so what does that mean in relation to actuality and reality? Cheers, Pasquale

We normally assume that causes can't precede their effects, but this isn't a logical truth, and in fact it's possible to tell coherent stories where the principle fails. By "tell coherent stories," I don't just mean tell science fiction. As your author may point out (I haven't read the book), it's possible to say how causal loops and backward causation might fit into physical theory, even though there's no strong case for saying that such things actually happen. So there's no issue about the "logic" of causality being violated.

As for your Christmas shopping, this isn't really an example of the future affecting the present. Your present intentions do the causing. You want to make sure that come Christmas day, Granny gets that gorgeous pair of Manolo Blahniks, and it's that present desire and intention that gets you to head off to the shoe store. Granny's beaming grin isn't reaching back from the future to get you to the mall.

## How does the temperature ever change? If we assume that temperature is a

How does the temperature ever change? If we assume that temperature is a continuous measurement, then we know that it has an infinite number of potential values. In order for temperature to transition between two values, it must then pass over the infinite set of values that lies between whichever two values the temperature is transitioning between. It now seems that temperature should not be able to change at all because before it may change to a given value, it must first reach a value between the desired and the current. Since we can make this claim infinitely, it would seem that temperature becomes "trapped", in a sense, at its current value, unable to change at all. Of course this problem can be applied to other concepts as well, and we might easily draw comparisons to Zeno's ancient thought experiment of Achilles and the tortoise. But the logic here is slightly different; the desired temperature is not continuously fleeing from the present as the tortoise is from Achilles. I simply raise the...

I'll have to confess that I'm one of those people who was early on seduced by a particular sort of solution to this sort of problem, and since then I've never been able to feel the force of the puzzle. Here's a somewhat fanciful example that conveys the idea.

Suppose that we have a body whose temperature at 12:00 midnite is o° Celsius. And imagine that the body's temperature is increasing at a steady rate as follows. Let Tt be the body's temperature at time t. And let the temperature at any given time over the hour after midnite be given by

Tt = t

where t is the number of minutes after midnite. In other words, the temperature rises steadily at the rate of one degree Celsius per minute. It goes through all the intermediate values in finite time. If the arithmetic of real numbers makes sense, so does this.

And so I can't find the puzzle. Pick any time you like over that one hour period. There's an answer to the question "What is the body's temperature at that instant?" And in this example, the temperature is never the same at two different instants. That means that the body's temperature is changing.

As the question was stated, it assumed that the idea of a continuum makes sense. That's good. In particular, there's no good reason to think that the concept of the field of real numbers is paradoxical or contradictory. But if so, then we can use the real numbers to represent time, we can also use them to represent temperatures, and we can straightforwardly describe relations between the two: at time t, the temperature Tt is thus-and-such. Nothing is missing. At every moment, there's a temperature, and it's no more peculiar that those temperatures need not be the same than that the values of the real numbers differ at different points on the real line.

I'd add: this solution accepts the idea that temperature changes continuously, but even if it didn't, we could use the same kind of solution. So long as there's a unique temperature for each moment, there's no conceptual puzzle in the idea that the temperature changes -- i.e., that it isn't the same at each instant. It could be, for instance, that in the half our from midnite up to but not including 12:30, the temperature is a constant o°, and then at every later instant, it's a constant 1o°. Of course, there's no last instant during which it's still o°, but if there's a puzzle about that, it's a puzzle about the very idea of a continuum -- not a puzzle about change.

And in fact, we could go further. Even if time is granular, we can still make sense of the idea of change in exactly the same way: we need only imagine that the temperature is different at some moments than at others.

As noted, I was seduced by this approach early on. Since I'm thoroughly in the grip of this picture, I can't make myself feel the pull of the problem. Perhaps I'm missing something, but I can't for the life of me see what it is.

## Does a concept, such as the Law of Gravity, exist? If there was no such thing

Does a concept, such as the Law of Gravity, exist? If there was no such thing as mass or time, would the Law still exist - just in case?

Take a law-statement of the form "All As are Bs" (I'm not saying that every law-statement has to be of this form: but it will do no harm to concentrate on this type of case). Then we can ask a pair of related questions. First, what kind of fact(s) make this type of law-statement true? For if the statement isn't even true, it certainly isn't a law. And we can ask, second, what makes the statement a law-statement. For not all true generalizations are laws: some are just accidentally true.

Different philosophers offer different package answers to this pair of questions, and the issues here are very hotly contended. It would be difficult to say much about them here, and we'll have to shelve any extended discussion. But let's see if we can make just a few preliminary comments relevant to the question originally posed.

First: note that sometimes when people talk about laws they mean law-statements; sometimes when people talk about laws they mean the facts that make the law-statements true (facts that existed/will continue to exist independently of our ever stumbling across them and putting them into words). Now, talking of laws as "concepts" strongly suggests the first understanding of "law" -- though strictly speaking law-statements involve concepts as ingredients, but are not themselves properly described as concepts. On the other hand, talking of laws as "existing" strongly suggests the second understanding of "law", and raises issues about the existence of law-facts. So the way the original question is framed is perhaps in danger of confusing the two senses of "law".

Second, however, it certainly seems that it sometimes could be a law that all As are Bs even though there happen to be no As. Newton's First Law says that any object subject to zero net forces will move in a straight line with constant velocity. What if the world happens not to contain any object which is subject to zero net forces? Would we want to say on this ground that Newton got it wrong, and that there is no such law-fact? Arguably not. In this sort of case, the non-existence of As wouldn't seem to imply the non-existence of laws about As.

But that's not the end of the story. It was once believed, for example, that the statement "combusting materials give off phlogiston" was a law. It isn't, because it is false. There is no such stuff as phlogiston. And since there is no such stuff, we are rather inclined to say, there can't be any genuine laws about it either. In this sort of case, the non-existence of As does seem to imply the non-existence of laws about As.

What makes the difference between these cases? Now, that's a good question. But I suspect that taking it very much further would mean having to tackle head-on the tricky big issues that we shelved! (You can perhaps at least get a sense of the complexity of those issues by looking here.)

## I seen a question that went, "Can there be an event that is entirely random?" I

I seen a question that went, "Can there be an event that is entirely random?" I put a little bit of thought into this and concluded that the "Big Bang" theory, about the fact that the singularity became the universe (which is explained in the opening chapters of Bill Bryson's _A Short History Of Nearly Everything_) must be the only ever event that was random because no one can say why it happened and why it did not simply stay as a singularity forever. I still don't know if that is right because it wouldn't be classed as staying in that state "forever" as time did not exist. But it can be argued that it was not an event as it was the thing (if it can be referred to as a thing) that created time, on which events are obviously based. Also, if this is true wouldn't that be detrimental to the belief of free will? So this may be an answer, I'm not sure but I just wanted to know an expert's opinion on it as I am just a 17 year old student. Also I don't know if it was ok to post the title of a book on this so...

Thank you for your very good question. You have nothing to apologize for, and we're grateful to you for asking. I don't think I'll be able to respond to everything in what you ask, but here are a few thoughts:

Concerning the big bang, you write, "no one can say why it happened and why it did not simply stay as a singularity forever. " I should point out that even if it is true that no one can explain why it occurred, that doesn't mean there is no answer to be found. Perhaps no one can answer this question *now*, but someone (maybe you!) will someday find an answer. If that's right, then the only "randomness" here is due to our own ignorance.

Please let me mention also that contemporary physics holds that there is a very common form of indeterminacy, that is, of randomness. I mean what quantum theory has to say about the decay of an atom such as (some forms of) uranium. Whether such an atom decays at a given moment is, according to contemporary qm, entirely random. The most we can say is that there was a certain probability that it would decay within a certain period of time.

Next, I don't quite see how you are connecting randomness to the issue of free will. In fact, some philosophers concerned with free will hold that it creates its own form of free will: On this "libertarian" theory, whether I choose to perform an action or not is an event that cannot be determined by prior causes outside myself. That's a controversial view; many other philosophers will hold that freedom can occur even if the entire universe is deterministic.

So as you can see, I don't have any conclusive "answers" for you, but just some suggestions for connections. Also, if you wanto learn more, there are a lot of good things to read about this. For instance, Robert Kane has published an excellent book called _A Contemporary Introduction to Free Will_, which you may enjoy reading.

Don't stop wondering!