## Kurt Gödel's incompleteness theorems represent one of the foremost achievements

Kurt Gödel's incompleteness theorems represent one of the foremost achievements in mathematical logic's proud history. AskPhilosopher's very own panelist Peter Smith obviously is greatly intrigued by these theorems. Suppose out of jealously -- though I doubt he would succumb to such a vice! -- he decided to build a time machine. Imagine, moreover, that he went to a time before Gödel had proven the theorems and gave the great logician, say, the idea of Gödel numbering, the key to proving the incompleteness theorems. My question thus is as follows: Who would deserve credit for proving the incompleteness theorems? Gödel seems to have gotten the idea from Peter; Peter seems to have gotten the idea from Gödel. Is it possible that neither would deserve credit?

What a lovely question!

The first thing to ask is whether the story is internally consistent -- unlike a story in which John kills his grandfather before Grandfather fathers John's mother. That appears not to be a problem here; there's no obvious hint of an event having to have happened and not happened. Instead, we simply have surprising set of internally consistent occurrences.

On, then, to responsibility. Since we have a causal loop here, there's no clear way to say which of Peter or Kurt is causally responsible for the theorem's having come to be stated and proved. If pressed, we might say that each gets equal causal credit,m though your mileage might vary on the apportionment. Indeed, neither is the originator or initiator; the most that can be said is that both were crucial parts of the process through which the theorem came to be.

What of intellectual credit?

Well, Peter didn't think the ideas up on his own. He learned them by reading about them, having been taught them, working through the proof and what not. So he doesn't get any intellectual credit. But as this story goes, Gödel didn't think the thing up either. (To make the story as perplexing as possible, we should assume that Peter simply gave the whole shebang to Kurt, rather than just giving him a crucial bit.) And so Gödel gets no intellectual credit either. Somehow, the world just burped forth Gödel's theorem, praise be to the Lords of Time!

Wonder if it really happened that way?

Thank you for your question. The standard answer is that an instant lasts for no time at all. That is to say, the start of an instant and the end of an instance occur at exactly the same time. An instant is indivisible; it has no separate beginning, middle, or end. You might think of time as like a number line, with (for instance) zero as the time when you started reading this sentence and 1 as the time when you arrived at the end of it. Then each number between zero and 1 corresponds to an instant of time. None of those instants is any length of time at all.

Of course, that an instant of time lasts for no time at all might lead you to wonder how a span of time lasting, say, for an hour could possibly consist of a bunch of instants each lasting for no time at all. This is closely related to some of the paradoxes first proposed by the Greek philosopher Zeno thousands of years ago.

Bear in mind as well that between any two instants of time, there is another instant of time -- and, indeed, infinitely many instants of time, since you can repeat this thought infinitely many times. But that should be no more surprising than that between any two numbers, there is another number -- and, indeed, infinitely many numbers. If an instant of time lasted for any length of time (other than zero) and all instants lasted for the same length of time, then you couldn't fit infinitely many instants between any two instants.

So in thinking about how an hour of time could consist of many instants, each lasting for no time at all, you have to bear in mind that an hour consists not just of many instants, but of infinitely many instants. I hope that this answers your question (a little).

## Doesn't time travel involve space travel too? If I travel back in time one year,

Doesn't time travel involve space travel too? If I travel back in time one year, say, in order to be in the same 'place' as I started, I'd need to travel across countless millions of miles of space, since the planet has moved during the last year. Since such instant space travel contradicts Einstein, how come so many philosophers seem to think it's possible? Martin, Wales, UK

Nice conundrum. Here is a stab at it. If, in the example, time travel is traveling back one year of time in an instant of another time dimension--call it metatime--then Einstein has not been contradicted. He is silent about how much space can be covered in an instant of metatime. So time travel, conceived this way could be possible even given our actual laws of nature, if there is metatime. If, however, there is no metatime, then traveling back in time would be a case in which what would normally be a later stage of one's life occurs before what would normally be an earlier stage (see David Lewis, "The Paradoxes of Time Travel"). For this to be possible, the laws of nature would already have to be different than ours in such a way as to also allow that what would normally be the very next stage in ones' life occur far away from the current stage. If it is conceivable that the laws of nature be different than what they actually are then time travel would be conceptually possible. And this is the sense of 'possible' most philosophers appeal to in saying that time travel is possible. In other words, it is conceptually possible that something happen that contradicts Einstein.

## If time travel were ever mastered, might it be possible to change the past in

If time travel were ever mastered, might it be possible to change the past in manners which wouldn't create paradoxes? Or are all possible changes inherently paradoxical? Also, if the past were successfully changed, is it possible that all of history would change, and we would have no recollection of the original timeline? Or is this idea inherently flawed? Thanks.

Any "change" in the past is inherently paradoxical (to say the least). In fact, I think it is actually worse than that: Such changes would involve making it both true and false in the history of our world that the changed event did (or did not) take place. That's a contradiction, not a paradox.

On the other hand, one could go back in time and do what one actually did in some time long past (or do what one actually will do, some time long in the future). If it is actually possible to go back in time, for example, and be one's own father, then one would live in a universe in which that is (and always was) precisely what happened. What are called "looping" universes, in which time did not flow linearly, but in a closed loop, would make such apparently strange events possible. And though we have good reasons for supposing that we do not live in a looping universe, it does not seem that logic makes such an idea impossible.

To find out more about this topic, have a look at an article by David Lewis entitled "The Paradoxes of Time Travel" which was published in the American Philosophical Quarterly, vol. 13 (1976), 145-152. Lewis also cites two stories by Robert Heinlein in which the picture of time travel that he develops appear to be assumed.

## I ask this in regards to (what I perceive to be) the paradoxical nature of time

I ask this in regards to (what I perceive to be) the paradoxical nature of time and its origins. Two things seem particularly troubling here: A) How could time have had a beginning? Isn't the concept of a beginning only meaningful when examined from a frame in time? B) If time did not have a beginning, wouldn't we have traversed an infinite period of time in order to get to the present moment? Isn't that as inherently impossible as, say, eating an infinite amount of cottage cheese? One thing is apparent: time exists! From this I can gather there is some flaw in my reasoning. I suspect it resides in B, though I cannot seem to articulate the precise reason why, but I am open to the possibility that A is somehow fallacious as well. Or, perhaps, both A and B are false. Anyway, you guys run a great site! Thanks for answering (if you indeed choose to do so).

I already addressed your second concern in response to a previous question on this site. I'd invite you to take a look at my answer there.

As to the first concern, when we speculate about a possible beginning to time, we are doing so from a frame in time. We start at the present, and we conceptually project ourselves backwards through the period that intervened between the present and that supposed first moment. Was there a time one year ago? Yes. Was there a time two years ago? Yes. Was there a time thirteen billion years ago? Yes. Was there a time fourteen billion years ago? No! The supposition of a beginning to time means that there exists a number n, such that there was a moment of time n years back from the present but no moment n+1 years back. The supposition of an infinite past simply means that there is no such number.

## Not to be silly…but if I could build a time machine would it be possible for me

Not to be silly…but if I could build a time machine would it be possible for me to go back in time and stop myself from building the time machine?

Not a silly question at all, absolutely not! But the answer is no.

My own thinking on matters like this has probably been most influenced by the late, great American philosopher, David Lewis, particularly his article 'The Paradoxes of Time Travel'. It first appeared in the American Philosophical Quarterly, 13 (1976) 418-46, and is reprinted in his own Philosophical Papers, vol. II. I recommend it.

## We can only live in this "here&now moment"...in fact, there is no way we can

We can only live in this "here&now moment"...in fact, there is no way we can ever live out of "IT"...is it not?

'We can only live in this "here and now" moment . . . in fact , there is no way we can ever live out of it . . . is it not?'

I am not sure what is supposed to meant by living in the present instant ("moment" I think has more to do with action). Living at an instant seems as impossible as living at some other time, because there isn't even time to draw breath in an instant. In any case I do not believe that there is something called "the present instant", so I don't see how we could live in it (at it?)

It (the present instant) is an abstraction, and it is not, in reality! I do believe there are present times, though, such as the present day or hour. The trouble with the instant is that it is not a time.

## Everything that happens, why does is it happen at the moment that it does and

Everything that happens, why does is it happen at the moment that it does and not the moment before or the moment after?

Why does there have to be a reason? Maybe some events occur when they do just by chance. There seems to be nothing incoherent about that idea. Indeed, that's how we think that the world actually works. For example, the law governing the radioactive decay of an unstable atomic nucleus seems to be merely chancy. An atom of polonium-214 has a fifty/fifty chance of decaying in the next 3 minutes or thereabouts. But nothing determines when it actually decays. There is, according to our best scientific theory of the matter, no answer to the question of why the decay happens at the moment that it does rather than a little while before or a little while after. Why should there be?

## How does our approach to knowledge about the past differ from our approach to

How does our approach to knowledge about the past differ from our approach to knowledge about the future, keeping in mind that there is an element of uncertainty in both?

Our knowledge of the past derives from perception, memory and inference, in the sense that these are answers to the question, 'How or by what means do you know?' (There are other ways, for example report or testimony). But our knowledge of the future has in it no elements of memory or perception. So as one might therefore expect it is harder to come by knowledge of the future, and we have less of it per hour, if you want. We typically can know more about a past hour than about a future hour, though by no means all of the past hours, for example those in past centuries. If I know p, and p is a proposition about the future, I cannot know it by memory, special cases apart. (A special case would be that I come to know that I am going to Africa next summer - a piece of knowledge about the future - by remembering that I am going to Africa next summer. 'How do you know?' 'I just remembered it . . .' makes sense as a conversation.)

It seems to me, in spite of the assumption you make, however, that in some cases there may not be an element of uncertainty in either knowledge of the past or the future. There is no uncertainty that the cat will be roughly where it is on the sofa in one attosecond - cats don't move that fast - and there is no uncertainty that the cat has been sitting there for the last five minutes, as I have been watching it for the whole time. There is an interesting mistake (I myself think it's a mistake, anyway) to be avoided in this area. Why are there asymmetries in time with respect to knowledge? I am not sure the question put just like that makes sense. Why can we remember the past but not the future, for example? The simple answer is that if I remember something, then it must already have happened, so memory of the future is a contradiction. My own view is that even the alleged logical asymmetries between past and future are much more slippery than they seem at first glance, and we must be careful to get our tenses right. It is certainly true, for example, that the past exists, in the sense that past events have occurred - and what other sense are we considering? But then so does the future exist, in just the same sense: future events will occur.

## Time stretches back to infinity, therefore it cannot have reached NOW {let 2009

Time stretches back to infinity, therefore it cannot have reached NOW {let 2009 = NOW}. Manifestly, however, it has reached NOW. How can this be?

As a warm-up exercise, consider the following two infinite ordered sets of numbers. Firstly, take the negative and positive integers in their 'natural' ordering

... -4, -3, -2, -1, 0, 1, 2, 3, 4, ...

trailing off unendingly to the left and to the right. Second, take all the negative numbers, in increasing size, followed by zero and all the positive numbers:

-1, -2, -3, -4, ... o, 1, 2, 3, ...

Now, in both orderings, any positive number is preceded by an infinity of numbers (including all the negative numbers). But there is a very important difference between the two cases -- they have, as mathematicians say, different 'order types'. One big difference is this: there is no first member of the first ordering (i.e. for any given element of the ordered series, there's an earlier one); but there is a first member of the second ordering (namely, -1). To bring out another difference, suppose in each ordering we take one of the negative numbers, and we ask: can we start with that predecessor of 0 and by taking finite number of steps to the right, to successors in the ordering, eventually get to 0 and on to the positive numbrs? In the first case the answer is 'yes'. Pick any predecessor, e.g. -43. Then after a finite number of steps (43 of them!), we'll reach o, and we can keep on marching throuugh the positive numbers. In the second case the answer is 'no'. Pick any predecessor, e.g. -43 again. Then after a finite number of steps, we'll just reach another, bigger, negative number.

OK, now let's turn to the case of time. Suppose (to keep the argument simple, but without losing anything essential to the present issue) we take time to be discrete, with moments ordered by the 'before/after' relation. We could tag the moments with numbers, and suppose we use some positive number like 2009 to mark the present moment, NOW. And let's ask: is the 'order type' of the temporal sequence like our first ordered sequence of numbers or like the second?

If time were whackily ordered like the second number series, there would be a first moment in the ordering (the one tagged -1), though the presentmoment 2009 would still have an infinite number of predecessors. But in that case,we couldn't have got to NOW by a finite number of steps from any given moment, wherever in the past. But of course, if we do suppose that 'time stretches back to infinity', then the natural view, still assuming discreteness, is that time is ordered like the first number series. So NOW is again preceded by an infinity of earlier moments, but there is no first moment, and from any past moment, a finite number of steps from one moment to the next reaches NOW.

Of course, it is another question whether we do have to accept that time does stretch back to infinity: but the point I'm making is that there is no paradox in supposing that it does, if you give time a sensible order-type.