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Could there (is it conceivable/possible) be an alternate reality/universe (a

Could there (is it conceivable/possible) be an alternate reality/universe (a rich complex universe) which was such that mathematics could not provide any (or say very little) description of it?

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.

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 .

Goldbach's conjecture states that every even integer greater than two can be

Goldbach's conjecture states that every even integer greater than two can be expressed as a sum of two primes. There is no formal proof of this conjecture. However, every even integer greater than two has been shown to be a sum of two primes once we started looking. Is this acceptable justification for believing Goldbach's conjecture? Can we determine mathematical theorems based on observational evidence?

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 numbers we're looking for are both odd. There is only one case where this is not true -- 2+2=4 -- but we can just put this case aside as settled.

Now examining any even number involves writing it, in as many ways as is possible, as the sum of two odd numbers and then looking through all these sums to find one where both of the odd numbers are primes. Obviously, the larger even numbers get, the more different options there are of writing this number as the sum of two odd numbers. When the number in question is 2n, then we have about n/2-1 such options. I say "about", because the number of options is the same for any number divisible by 4 and the even number just below it. Thus we have 2 distinct options of writing 12 as the sum of two odd numbers (3+9, 5+7) and 2 distinct options of writing 10 as the sum of two odd numbers (3+7, 5+5). For any large even number, the number of decomposition options is about one quarter of this number. So it looks like, as we get to larger and larger even numbers, the probability that each can be written as the sum of two primes gets larger and larger, because there are more and more options. This probabilistic reasoning assumes that the probability that any given way of writing some even number as the sum of two odd numbers (E=O1+O2) is such that O1 and O2 are both primes can be estimated as the square of the probability that any odd number smaller than E is prime.

To illustrate, let the even number be 100. There are 50 odd numbers below 100 and of these 24 are prime. So the probability that any randomly selected odd number below 100 is prime is 0.48. And the probability that any two randomly selected odd numbers below 100 are both prime is then 0.48^2 or about 0.23. So the probability that any two randomly selected odd numbers below 100 fail to be both prime is about 0.77. But this substantial probability now gets whittled down by the fact that there are many options for writing 100 as the sum of two odd numbers -- in fact, there are 25 such options. And we may then surmise that the probability that none of these options involves a pair of primes is about 0.77^25, or about one in 700, or about one seventh of 1%, or about 0.0014.

Let's repeat the exercise for E=1000. There are 500 odd numbers below 1000 and of these 167 are prime. So the probability that any randomly selected odd number below 1000 is prime is about one third. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about one ninth. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about eight ninth or 0.89. But there are 250 options for writing 1000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.89^250. This amounts to one chance in 6 trillion, or a probability of about 0.00000000000016.

We see here that, as we go from E=100 to E=1000, the probability of finding no decomposition into two primes falls off very steeply: going from 100 to 1000, the probability was cut by a factor of about 10 billion. Just to be sure, let's repeat the exercise one more time for E=10000. There are 5000 odd numbers below 10000 and of these 1228 are prime. So the probability that any randomly selected odd number below 10000 is prime is about one quarter. And the probability that any two randomly selected odd numbers below 1000 are both prime is then about 6% or 0.06. So the probability that any two randomly selected odd numbers below 1000 fail to be both prime is about 0.94. But there are 2500 options for writing 10000 as the sum of two odd numbers. And the surmised probability that none of these options involves a pair of primes is then about 0.94^2500. This amounts to one chance in 15 times 10^66, or to a probability of about 0.000000000000000000000000000000000000000000000000000000000000000000066.

Note that, as we went from E=1000 to E=10000, the probability of finding no decomposition into two primes has fallen off even much more steeply (than in the earlier move from E=100 to E=1000): going from 1000 to 10000, the probability was cut by a factor of about 2.5 times 10^54!

Now we must also consider one other factor here. Let's think of E=100, E=1000, E=10000, and so on as centers of "neighborhoods". So defined, these neighborhoods become larger as we go up. There are about 140 even numbers in the E=100 neighborhood (those between 31 and 310, roughly) and about 1400 even numbers in the E=1000 neighborhood (those between 310 and 3100, roughly), and so on. How does this factor affect the calculation? Again, estimating very roughly, if the chance that any given even number around 100 is decomposable into two primes is about 0.9986, then the chance that all 140 of them are so decomposable is about 82% or 0.82. If the chance that any given even number around 1o00 is decomposable into two primes is about 0.99999999999984, then the chance that all 1400 of them are so decomposable is about 0.99999999999984^1400, or about 99.999999977%. And if the chance that any given even number around 10000 is decomposable into two primes is about 0.999999999999999999999999999999999999999999999999999999999999999999934, then the chance that all 14000 of them are so decomposable is about 0.999999999999999999999999999999999999999999999999999999999999999999934^14000, or about 0.999999999999999999999999999999999999999999999999999999999999999.

As these very rough calculations show, the probability of finding a counterexample declines very (and increasingly) steeply as we get to higher neighborhoods. To be sure, there is a non-zero probability in each neighborhood, and there are infinitely many neighborhoods. But this still does not add up to a substantial probability -- much like the infinite series of $1 + $0.1 + $0.01 + $0.001 + ... does not add up to a fortune.

I conclude then that, if there's no counter-example to Goldbach's conjecture in the lower neighborhoods, then this conjecture is extremely likely to be correct. My support for this conclusion rest on a kind of (quick and dirty) probabilistic reasoning that perhaps fits what you have in mind when you speak of "observational evidence" (I "observed" how many prime numbers there are in this and that neighborhood, and so on). I find this reasoning compelling, so am starting herewith to believe that Goldberg's conjecture is true. But I would not be surprised at all if mathematicians were entirely unmoved by this sort of reasoning.

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...

In a right angled isosceles triangle with equal sides of 1 unit and 1 unit, the

In a right angled isosceles triangle with equal sides of 1 unit and 1 unit, the third side will be sqroot(2) according to Pythagoras theorem. But sqroot(2)= 1.414213562373095... It is never ending. So theoretically we cannot determine its exact length. But physically it should have a definite length! The side is touching the other two sides of the triangle, so how can the length be theoretically indeterminate but physically determinate ? Does this mean the human understanding is limited and we cannot fully understand the mind of god ? Can you resolve this dilemma ?

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 your problem by pointing to some physical object and then asserting that we cannot determine its precise length. No matter how many digits we may manage to add (through clever measurement) behind the decimal point, there will still remain many further such digits unknown. Leaving modern physics aside, I agree with this but still see no dilemma, no tension with you exclamation that "physically it should have a definite length". Yes, it should, and it can have a definite length even if we cannot possibly ascertain what this length is with perfect precision.

Our human understanding is limited here, but fortunately our human curiosity is limited as well: beyond a few dozen digits after the decimal point, even the nanotechies lose interest in greater precision.

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...

Is there any number larger than all other numbers? George Cantor proved that

Is there any number larger than all other numbers? George Cantor proved that that even infinite quantities may be smaller than other infinities. Still, might there be some infinite number that is greater than all other infinite numbers?

What Prof Pogge has said represents one perspective on this issue, but it involves assumptions that can be rejected. The central issue is whether you are prepared to speak of "how many sets there are". If so, then let Fred be how many sets there are, that is, the number of things that are sets. It is sufficiently clear that Fred is the largest number.

In standard set theory, by which I mean Zermelo-Frankel set theory (ZF) and its extensions, there is no such thing as the number of things that are sets. There just isn't such a number. But there are other set theories in which there is such a number, and one can, in fact, consistently add to ZF an axiom known as HP which allows us to speak of (cardinal) numbers in a way different from how ZF by itself allows us to speak of them. And then there is a number of all the sets there are, and it is again the biggest number.

How can the question how many sets there are simply fail to have an answer? The idea to which Prof Pogge is giving expression is similar to what is known as "indefinite extensibility", the idea being that, in some sense, the universe of sets is never completed but is always in process. If something like that were true, then one can see why there might not be a number of all the sets there are. But this point of view is not wildly popular. Most people—and especially most people who work on set theory—would suppose that sets and numbers, infinite and otherwise, are no more "constructed" by us than are photons, quarks, or electromagnetic fields. As Frege famously wrote, "...[T]he mathematician cannot create things at will, any more than the geographer can; he too can only discover what is there and give it a name" (Foundations of Arithmetic, section 96).

So I am a little puzzled what it means to say that "infinite numbers are not found in nature". I wouldn't have thought any numbers were found in nature. Birds, trees, rocks, stars, photons, and the like, yes, but not 2 and not 1.765 and not π. (If 2 is in nature, where is it?) Perhaps what is meant is that infinite sets are not found in nature, whereas finite sets are. But then I'd insist that there are no sets to be found in nature, period. The elements of the set {Chicago, Dallas} are to be found in Illinois and Texas, respectively, but the set itself is not anywhere. So perhaps the question is whether infinity is in some sense "witnessed" by nature. That depends, among other things, upon whether space is continuous. If so, then there are infinitely many points of space anywhere you care to look. And, according to the fundamental physics of our day, highly complicated mathematical structures are in the same sense "witnessed" by nature, and these structures are very often "infinite" in some sense or other. But however that question is answered, it isn't at all clear what its significance is supposed to be. If the power set of the power set of the power set of the set of real numbers is not witnessed by nature, what of it?

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...

Is there a difference between a number as an abstract object and as a metric

Is there a difference between a number as an abstract object and as a metric unit used to measure things?

I would put the question slightly differently, if I understand it right: The question is whether the cardinal number 3, used to say how many of something there are, is the same or different from the real number 3, which is used to report the results of measurement. There is of course a different between the cardinal number 3 and a length of three meters, but the question is whether, when one says, "There are three apples" and "This board is three meters long", we refer to the same number three both times.

Mathematicians and people who work on foundations of mathematics tend to have different views about this, at least in practice. The way one defines the cardinal numbers in set theory, for example, is very different from how one defines the reals. But working mathematicians will often speak of "identifying" the cardinal with the real and often seem impatient with such niceties as whether they are really the same.

A more difficult question, I think, concerns cardinals and ordinals. Where cardinals are used to say how many, ordinals are used to pick out an object by its position in a sequence: first, second, third, and so forth. So here's a somewhat more contentious question: Is the cardinal number one the same as or different from the ordinal number first? In the usual set theoretic constructions, these would be the same; but in some other ways of doing the construction, they would come out different, and for good reason. And in so-called neo-logicist developments of mathematics, they again come out different, though neo-logicist treatments of the ordinals are a bit lacking, so it's hard to say exactly how it is all supposed to go.

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?

My mathematics teacher says that a line is an infinite sum of points. I disagree

My mathematics teacher says that a line is an infinite sum of points. I disagree and I think that she must not have thought it through very deeply. I argue that instead that though a line can be theoretically be described as a sum of smaller lines that in no way can a line be said to be described as a continuity of points because a point is not in any way extended. If a line has an atomic unit then that unit must have the same properties as the line itself and a point has an altogether different property than a line. (That you can fit a point inside a line only shows their common property of spaciality, it does not demonstrate that a line is in any way composed of points) I hope you understand what I am saying. Do you think I am right?

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 is the extension of the resulting segments? In particular, is their extension zero or is it greater than zero?

Your answer is clear. The extension of these segments cannot be zero. For, it were zero, then these segments could not form the original line. Even infinitely many points of zero extension would not add up to anything of greater-than-zero extension. Therefore the extension of each of these segments must be greater than zero.

Your teacher, I imagine, might say this. Suppose that, after infinitely many divisions, the extension of the segments were indeed greater than zero. Then, lining up these infinitely many segments would get us an infinitely long line. (Multiplied by infinity, even the tiniest non-zero quantity becomes infinite.) But this is not the finite line we started with. Therefore the extension of each of these segments cannot be greater than zero. Since it cannot be smaller than zero either, it has to be zero.

Now it seems we are in real trouble. It seems that neither answer can be right. But is there any third possibility? Can we perhaps stipulate that there is some "infinitesimal" length that is greater than zero and also smaller than any finite length -- and then assert that points actually have infinitesimal extension rather than zero extension? Sure, we can say this. But even this "solution" leaves us with a difficult question. How long is a line that consists of infinitely many infinitesimal segments? Some might be tempted to answer: "32 inches." But what if our infinite division is imagined on a 17-inch long line? Are we to postulate many -- infinitely many -- different infinitesimalities, each yielding a different finite result when multiplied with infinity?

I leave things here, in this sorry state, but let me reiterate that infinities are pretty weird, and what we saw here isn't unusual at all. Thus consider the comparison of two infinities in the apparently simple question of whether there are more positive integers (1,2,3,4,...) than even positive integers (2,4,6,8,...). Here as well there seem to be compelling arguments on both sides. Those who say "yes" can say that there are additional integers, the odd ones, included in the first set and not in the second, and nothing included in the second that's not also in the first. Those who say "no" can point to a one-to-one mapping that will find a matching even positive integer for every positive integer (e.g., 1 is matched by 2, 2 is matched by 4, and so on). In this case, mathematicians have decided to take the latter argument to be decisive. But must we go that way?

One lovely book discussing many of these problems is Shaughan Lavine: Understanding the Infinite (Harvard U.P.). You will also find some fascinating thoughts on mathematical arguments in Ludwig Wittgenstein: Remarks on the Foundations of Mathematics.

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...

Suppose there is an infinitely long ladder in front of me. I do not know that

Suppose there is an infinitely long ladder in front of me. I do not know that this ladder is infinitely long, only that it is either a very long (but finitely long) ladder, or an infinitely long ladder. What kind of evidence would I need to give me reasonable assurance (I don't need absolute certainty) that this ladder is indeed infinitely long? I could walk a mile along the ladder and see that it still shows no signs of stopping soon. But the finitely long ladder would still be a better hypothesis in this case, because it explains the same data with a more conservative hypothesis. If I walk two miles, the finitely long hypothesis is still better for the same reasons. No matter what test I perform, the finitely long hypothesis will still better explain the results. Does this mean that, even if infinite objects exist, empirical evidence will never provide reasonable assurance that they exist?

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."

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."

Parallel Lines:

Parallel Lines: 1) I've been told that parallel lines never meet - except at infinity. 2) Also that a straight line is a circle of infinite radius. 3) Surely if you get two infinitely large circles such that they don't overlap, at their closest point they are straight (as per 2) and parallel yet must both meet (by 1) and not as per 3) - not overlapping. Any suggestions? (I'm confused!)

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.

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.

In my class we had a discussion about the logic behind mathematics today.

In my class we had a discussion about the logic behind mathematics today. Unfortunately we didn't manage to come up with a solution to the question about which the discussion was. The question was: From the beginning of human kind we always used a logical counting pattern (today expressed as 1,2,3); do you believe that if at the beginning of human kind our logical thinking had lacked the idea of counting, maths would have turned out to be something completely different or would it even exist?

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 mathematics (geometry, for instance). But perhaps I am just lacking imagination here about alternate ways of life.

If you are interested in thinking more about our peculiar way of doing mathematics, I would recommend a look at Ludwig Wittgenstein's Remarks on the Foundations of Mathematics. It was not published in his lifetime, so is not a systematic book organized by its author himself. But it contains a lot of wonderful questions and insights.

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...

Are there formal ways (outside of mathematics) in which axioms are chosen? Can

Are there formal ways (outside of mathematics) in which axioms are chosen? Can you give guidelines in constructing axioms? Must axioms base themselves in sensory awareness?

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 further "upstream," but by their implications. Relatedly, axioms may not be especially evident or intuitive. In fact, an axiom may be quite unituitive. What matters is that is "works": that it, together with the other axioms chosen, allows us to derive, without contradiction, what we take to be true.

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...

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