Why not? We can conceive a nice large space filled with moving matter, all as in our universe, except that the laws of nature vary randomly in space and time -- which is really to say that there are no laws of nature. You could still use geometry to describe the trajectories of objects, but you could not simplify these descriptions with general formulas that cover, say, the force that objects exert on one another. Nor of course could you project any descriptions into the future (predict what will happen) nor even describe with any accuracy what is happening elsewhere or what was happening in the past (because you would have no firm ground for reasoning backward from the data you have to their origins). So it seems that we can conceive such a world. But whether a cognitive subject could have experience of such a world, could hold it together in one mind, that's another question, one that is very interestingly examined in Kant's Critique of Pure Reason .

Acceptable to whom? I don't think the evidence you provide would or should convince mathematicians. They justify their beliefs about conjectures like this by appeal to proofs or counter-examples. So long as neither is forthcoming, they will rightly suspend belief. But for the rest of us, perhaps the kind of "observational" evidence you suggest might be convincing. Here it would not help much to argue that, because we have found Goldbach's conjecture to be correct up to 10^n, it is probably correct all the way up. One reason this would be unhelpful is that the as yet unexamined even numbers are infinitely more numerous than the examined ones, so we will always have examined only an infinitesimally small sample. Another reason this would be unhelpful is that the examined even numbers are not a representative sample -- rather, they are all very much on the small side, as far as numbers go. So the probabilistic argument would have to go differently. Let's first establish that the two prime...

Suppose someone had made the analogous argument about dividing a line of 1 unit into three equal parts. She tells us that "the length of each of these parts is 1/3 which is 0.333333333333 .... It is never ending. So theoretically we cannot determine the exact length of these parts." I think this would be a bit overblown. We know that the length of each of these three parts is exactly 1/3, and we also know that, while this leads to an infinitely long expression in the decimal system, it would not do so in the duodecimal system (which is based on the number 12 rather than the number 10). I want to suggest that you consider a similar response to your question. Yes, there is a notation in which we cannot express the length of the hypotenuse you have in mind with a finite number of signs. But there are other notations in which this is possible -- we can just call it "sqroot(2)". So, contrary to what you are saying, we can determine the exact length of that hypotenuse. You can refresh...

Infinite numbers are not found in nature but rather constructed through mathematical axioms and reasoning. This is somewhat analogous to how we can also construct the natural numbers by starting from 1 and then adding 1 again and again. We start from a set of cardinality aleph-naught, for example the set of all natural numbers. (Cardinality is a measure of how many elements the set contains; and aleph-naught is countable infinity: the cardinality of any set whose members can be mapped one-to-one into the natural numbers.) We then construct a set of higher cardinality, for example the power set of the set with which we started. The power set of any set S is the set of all the subsets of S -- and Cantor showed that the powerset of the set of all natural numbers has higher cardinality than the set of all natural numbers (i.e., that the powerset of any countably infinite set is uncountably infinite). Sets of even higher cardinality can be constructed through replication of a simple principle, much like ever...

Yes, in my view. Suppose there were no difference between the number 3 as an abstract object and the number 3 as used to express a certain length or volume. This would mean that there is no difference between 3 meters and 3, and no difference between 3 and 3 liters. Would it then not follow (by transitivity of no difference ) that there is no difference between 3 meters and 3 liters?

I understand well what you're saying. Points have zero extension, and lining up a bunch of them won't get you beyond zero extension. It's like adding up zeros: 0+0=0 0+0+0=0 and so on. There's no reason to think that adding infinitely many zeros together would get you anything other than zero. And likewise with the lining up of points. But when we are dealing with infinities, things are often tricky and counter-intuitive. So let's see whether we can construct an argument for your teacher's conclusion. Consider this. We begin with a line -- let's say it is 32 inches long -- and we divide it into two equal segments, these again into two equal segments, and so on. Dropping the inches, we can write this as follows: 1*32 = 2*16 = 4*8 = 8*4 = 16*2 = 32*1 = 64*1/2 = .... Here the number before the "*" signifies the number of segments and the number after the "*" signifies the length of each segment. Now the question is this. If we keep dividing an infinite number of times, then what...

In a finite lifetime, you won't be able fully to inspect an object with parts that are infinitely far from you, at least if we assume that you are limited by the speed of light. But there's other evidence. For example, you may be able to measure the gravitational pull of the ladder. If this pull turns out to be exactly what our theory would predict for a ladder that's like the piece of it we have before us (same material, thickness, density, etc.) and infinitely long, then this would be evidence for infinite length. (Note here that the gravitational pull exerted by any one inch of ladder declines with the square of its distance from you. So no matter how long the ladder its, its gravitational pull will not be infinite.) It's also possible that the ladder is expanding (as our universe is), or perhaps contracting. In that case you get a nice Doppler effect: a transformation of light reaching you from distant parts of the ladder -- the farther the light has traveled, the more strongly transformed it...

In relation to my earlier answer, the following article from the Economist may be of interest. It's advertised as follows: "Can the laws of physics change? Curious results from the outer reaches of the universe." The link is www.economist.com/node/16941123?story_id=16941123&fsrc=nlw|hig|09-02-2010|editors_highlights This is not exactly what I had in mind, but relevant nonetheless. BTW, this question is probably best classified under "physics" rather than "mathematics."

I think your #1 should go. If you drive a sled through the snow, the two lines you draw in the snow will never meet, never get closer to each other, even if you drive on forever. If you have two intersecting lines and close the angle toward zero, then at the limit the lines will have the same direction ... but at that limit they will also coincide (be identical) and hence not be parallel.

You are asking what we call a counterfactual question. Some such questions present little difficulty. For example, if your parents had never met, you would not be here asking hard questions. Your counterfactual question is much harder, because you are asking us to imagine something that is quite remote from the world we know. You are asking what human beings and human life would be like if we lacked the idea of counting. Given the kind of intelligence we have, it's a foregone conclusion that we would count, I think. So you are really not asking about human beings, but about some less-endowed or differently-endowed beings (perhaps some distant pre-human ancestors) who are otherwise similar to us. There isn't just one such species we might imagine (or discover). And the answer to your question could then be different for different non-counting but otherwise human-like species. I would doubt, though, that beings whose mental faculties do not lead them to count would do much else that we would call...

Axioms are the foundation of a theory, that from which all its claims are derived. What then grounds or justifies the axioms themselves? In practice, axioms are justified in large part by their implications. This may sound circular, but isn't on reflection. As we start theorizing in some particular domain of inquiry, we already have firm ideas about some truths and falsities, and we want to formulate our axioms so that they confirm these antecedent commitments. This approach is captured by the term axiomatization . We are to axiomatize our antecedent commitments, that is, we are to formulate a small set of axioms from which we can elegantly derive the much larger and messier set of propositions we hold true antecedently (including negations of propositions we hold false antecedently). Of course, such an axiomatization is successful only if the set of chosen axioms does not permit derivation of a contradiction. Somewhat paradoxically, axioms are then justified not by appeal to something...