It is often suggested that the human mind is incapable of conceiving of the concept of infinity. I would submit firstly that in many ways the concept of a finite universe is actually as hard if not harder to conceive of and, further, that looking deeper into the question of what it means to 'conceive of a concept' demonstrates that the concept of infinity does not deserve any special status.

We should note first that infinity, while a global property that describes the entirety of a space, is actually tested locally. Consider the natural numbers 1,2,3 and the like. The collection, or set, of all such numbers is said to be (countably) infinite, by definition, because it has the property such that if you give me any number N (say, a million), I can give you the number N+1 and we agree that it too is a natural number and so in our collection. If, at all points, this property holds that for every N in the set, N+1 also is in the set, then we define it as infinite. The key point here is that when we are testing for infinity we are looking quite locally just around any given number N. Now consider a finite set such as the numbers one through ten. At some points, such as the number five, the N to N+1 property holds and at others, such a ten, it does not. In both cases, it is actually the same basic test that we are conducting to see if it fails somewhere. An infinite set is one where this property never fails and a finite set is one where we can isolate some point where it finally stops.

In everyday life we have a lot of experience with finite things, such as a table of finite length. However, we experience it embedded in a somewhat larger space. There is a room beyond the table. Beyond the walls of the finite room there is more space. The analogue of the above paragraph (which applies for discrete, ordered spaces like numbers) for continuous spaces like a line is that no matter how far along one goes, you can always go just a little bit further. In other words, every thing we have ever experienced when it comes to our spatial universe, upon which we can base our intuition, is that we CAN go just a bit further at every point. In other words, our intuition is for the infinite case and not the finite case.

I want to consider now the notions of a finite or an infinite universe and see how well they correlate with our real world intuition. Imagine now, if you will, what it would mean to have a finite universe. For technical purposes, let us assume the topology (or shape) of three dimensional real space which corresponds with our intuition from day to day life. Were this universe to be finite then there would be some stopping point, some edge to the universe. However, intuition begs the questions "what is beyond the edge?". In a finite universe, there is nothing beyond the edge, no concept of "beyond" even. Because every edge of which we are aware of, such as the edge of a table or a wall, is embedded in a larger three dimensional space we have a very strong notion that the ambient space has the property that one can always continue a bit further. This difficulty in imagining truly nothing beyond the edge - something which our intuition gives us no guide - is at least as difficult if not harder than imagining the universe going on for ever.

A similar difficulty arises with the concept of time. The big bang model of the universe postulates not just that the big bang occurred some fourteen billion years ago but that time itself, as a concept, began at that moment. We are led by our intuition (which observes previous and subsequence moments of time for any finite time interval such as an hour) to ask "what happened before the big bang?". This kinda of thinking which assumes the property that one can move locally forward and backward from any point is true at all points - i.e., our intuition corresponds to an infinite time scale not a finite one.

Leaving this specific issue aside, one can ask what it means to conceive of a concept at all? In general we can do this at best very loosely. For example, we can all picture a horse in our head but this picture is of a very low resolution. We probably don't even have the broad anatomical features correct and certainly can't conceive of it at the level of organs, cells, atoms and the like. Now I agree that the human mind isn't capable of an authentic visualization of, say, an infinite line. But we certainly can come up with reasonable representations of it, discuss it mathematically, prove all sorts of things about it, and so on. Just in the same way we can not authentically visualize a horse in its entirety - but can visualize it well enough to be practical - so too can we not visualize infinity in its entirety but can well enough to be useful. What infinity does not deserve is a meme as being uniquely difficult to conceive of.

The same basic story repeats itself with regards to multiple dimensions. We live in a three dimensional word and so our intuition is certainly inclined to think three dimensionally. That said, as a PhD student in mathematics, I can assure you that thinking in multiple dimensions is quite possible. I have a friend who is very good at doing Rubiks cube type puzzles in higher dimensions than three. Personally, my work involves infinite dimensional spaces (each dimension being infinite in length). These problems are not trivially easy, but they are also not intractably difficult and almost anybody can learn cool things about multiple dimensions and infinity.

We should note first that infinity, while a global property that describes the entirety of a space, is actually tested locally. Consider the natural numbers 1,2,3 and the like. The collection, or set, of all such numbers is said to be (countably) infinite, by definition, because it has the property such that if you give me any number N (say, a million), I can give you the number N+1 and we agree that it too is a natural number and so in our collection. If, at all points, this property holds that for every N in the set, N+1 also is in the set, then we define it as infinite. The key point here is that when we are testing for infinity we are looking quite locally just around any given number N. Now consider a finite set such as the numbers one through ten. At some points, such as the number five, the N to N+1 property holds and at others, such a ten, it does not. In both cases, it is actually the same basic test that we are conducting to see if it fails somewhere. An infinite set is one where this property never fails and a finite set is one where we can isolate some point where it finally stops.

In everyday life we have a lot of experience with finite things, such as a table of finite length. However, we experience it embedded in a somewhat larger space. There is a room beyond the table. Beyond the walls of the finite room there is more space. The analogue of the above paragraph (which applies for discrete, ordered spaces like numbers) for continuous spaces like a line is that no matter how far along one goes, you can always go just a little bit further. In other words, every thing we have ever experienced when it comes to our spatial universe, upon which we can base our intuition, is that we CAN go just a bit further at every point. In other words, our intuition is for the infinite case and not the finite case.

I want to consider now the notions of a finite or an infinite universe and see how well they correlate with our real world intuition. Imagine now, if you will, what it would mean to have a finite universe. For technical purposes, let us assume the topology (or shape) of three dimensional real space which corresponds with our intuition from day to day life. Were this universe to be finite then there would be some stopping point, some edge to the universe. However, intuition begs the questions "what is beyond the edge?". In a finite universe, there is nothing beyond the edge, no concept of "beyond" even. Because every edge of which we are aware of, such as the edge of a table or a wall, is embedded in a larger three dimensional space we have a very strong notion that the ambient space has the property that one can always continue a bit further. This difficulty in imagining truly nothing beyond the edge - something which our intuition gives us no guide - is at least as difficult if not harder than imagining the universe going on for ever.

A similar difficulty arises with the concept of time. The big bang model of the universe postulates not just that the big bang occurred some fourteen billion years ago but that time itself, as a concept, began at that moment. We are led by our intuition (which observes previous and subsequence moments of time for any finite time interval such as an hour) to ask "what happened before the big bang?". This kinda of thinking which assumes the property that one can move locally forward and backward from any point is true at all points - i.e., our intuition corresponds to an infinite time scale not a finite one.

Leaving this specific issue aside, one can ask what it means to conceive of a concept at all? In general we can do this at best very loosely. For example, we can all picture a horse in our head but this picture is of a very low resolution. We probably don't even have the broad anatomical features correct and certainly can't conceive of it at the level of organs, cells, atoms and the like. Now I agree that the human mind isn't capable of an authentic visualization of, say, an infinite line. But we certainly can come up with reasonable representations of it, discuss it mathematically, prove all sorts of things about it, and so on. Just in the same way we can not authentically visualize a horse in its entirety - but can visualize it well enough to be practical - so too can we not visualize infinity in its entirety but can well enough to be useful. What infinity does not deserve is a meme as being uniquely difficult to conceive of.

The same basic story repeats itself with regards to multiple dimensions. We live in a three dimensional word and so our intuition is certainly inclined to think three dimensionally. That said, as a PhD student in mathematics, I can assure you that thinking in multiple dimensions is quite possible. I have a friend who is very good at doing Rubiks cube type puzzles in higher dimensions than three. Personally, my work involves infinite dimensional spaces (each dimension being infinite in length). These problems are not trivially easy, but they are also not intractably difficult and almost anybody can learn cool things about multiple dimensions and infinity.

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I enjoyed this one. :)

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