[I came across this vital question on quora short while back. The question is as follows: "Given an infinite amount of time and an infinite lifespan, will a person be able to discover all of mathematics? Why?" I found the question to be highly complex, and enjoyed very much the framing of the answer. The answer is almost essay-length, which is why I feel it is appropriate to commit it to this page. And yes, here's the link.]
I find this one of the most complex questions that I have seen so far on quora (and it hasn't been long). Before I begin framing an answer, let me make the following disclaimers:
A) Its going to be lengthy and read like a whole lot of gibberish,
B) I may not reach a conclusion,
C) I am not a mathematician - I did study mathematics in college as part of computer science degree course, but that's about it.
D) And no, I am not stoned and haven't been smoking anything :D
I think the answer to your question may not even be mathematical - rather, it would probably be biological, or philosophical, or even metaphysical? The answer may not be about the infiniteness of the number theory or such like, either, because once you know how to generate the next and the previous number for any given number, you've probably figured it all out already. Also, if you have proved that something cannot be solved, then one way of looking at it would suggest that you have exhausted all information about it already. Rather, it is about the definition of knowledge and our perception.
At present, anyone digging popular science seriously might know that certain familiar Euclidian concepts of geometry are no longer valid ever since General Relativity came to be. Instead, Euclidian comes off as a special case of the more complicated Reimannian Geometry, which is supposed to paint a truer picture of our universe. Take, for example, the intuitive understanding of the straightness of a straight line. With General Relative declaring the fabric of space as being curved, you cannot picture your straight line to continue in its path all the way to infinity without bending. You might use your imagination to make a hole in the space fabric and push the line through the whole, but it will be meaningless and incorrect, because you cannot imagine the other side of the space fabric, because it has absolutely no meaning. It's calls for a major shift in the thought framework.
Similar with all knowledge, even. I am inclined to think all knowledge is not discovered, but rather invented. What defines a thing as a mathematical law is how interesting you perceive it to be. For example, a formula to find out the n-th even number is an interesting piece of knowledge, as is the proof that the system of even numbers is infinite. Discovering the individual even numbers, say, the 18,015th even number, or the 120,109,752th or each even number in between may not be mathematically interesting at all. There may be no inherent and absolute property attached to a question in each category; rather, it is our perception that introduces the necessary distinction.
To answer your question: whether a person with an infinite lifespan and infinite time may discover all of mathematics, the fast and loose answer would be yes, because it would seem that the mind is not infinitely powerful, and not with an infinite memory. There is only a finite amount of stuff that he would be able to classify as 'mathematically interesting'. It is like asking, can a computer with a finite amount of RAM but an infinite amount of secondary storage run an infinite number of programs concurrently without swapping any out (no it cannot!)? So, as our knowledge grows, the number of independent parameters needed to ask new questions that would grow so large that they would no longer fit the mind in one go. So, we would cease to ask new questions and when we have no more questions, that will be the end of all knowledge (I am assuming we can say we have 'discovered' all of mathematics when we have no further question about anything mathematical).
But, it's not that simple. Some of the key things to consider are:
a) Does greater mathematical knowledge seem to imply knowing more unrelated things or less? For example, in physics, more knowledge could possibly mean less independent things to know. Like, at one time light, matter and electricity would have been perceived as three different things altogether, then someone comes across and discovers numerous subatomic particles and then we immediately eliminate them as mutually independent things. Then we stumble upon quarks, and they show us how subatomic particles are not 'fundamental', and then finally, we now discover strings, which is even more fundamental to a quark! It does seem like the growth of knowledge in physics is essentially a contraction, and might even have a visible end? It may not be the same for math, though, and more knowledge could well mean a genuine expansion of knowledge. Let us hope for some enlightenment here :)
b) Given a set of mathematically interesting (and possibly unrelated?) pieces of information, is it possible to always derive from them another mathematically interesting question? I have no idea how to go about proving this :D but I am going to give it a try someday. If the answer is no, then it might well mean that the end is once more in sight (limited by the finiteness of our own mind).
One other interesting question (or corollary) that crops up here is whether it is possible to measure the capacity of one's own mind. I feel it is somewhat akin to trying to paint the floor of a room without stepping outside of the room and yet not treading wet paint - it cannot be done! However, that doesn't mean that you cannot discover the boundaries of someone else's mind. Maybe then you could discover all of mathematical knowledge for that person (it should vary between people) ahead of said person in a finite time!
P.s. I realize my answer is a response to some inner narcissistic craving, and as such things go, full of holes. In which case, I would very much appreciate should someone set me on the right path :)
Yours sincerely,
Jude
I find this one of the most complex questions that I have seen so far on quora (and it hasn't been long). Before I begin framing an answer, let me make the following disclaimers:
A) Its going to be lengthy and read like a whole lot of gibberish,
B) I may not reach a conclusion,
C) I am not a mathematician - I did study mathematics in college as part of computer science degree course, but that's about it.
D) And no, I am not stoned and haven't been smoking anything :D
I think the answer to your question may not even be mathematical - rather, it would probably be biological, or philosophical, or even metaphysical? The answer may not be about the infiniteness of the number theory or such like, either, because once you know how to generate the next and the previous number for any given number, you've probably figured it all out already. Also, if you have proved that something cannot be solved, then one way of looking at it would suggest that you have exhausted all information about it already. Rather, it is about the definition of knowledge and our perception.
At present, anyone digging popular science seriously might know that certain familiar Euclidian concepts of geometry are no longer valid ever since General Relativity came to be. Instead, Euclidian comes off as a special case of the more complicated Reimannian Geometry, which is supposed to paint a truer picture of our universe. Take, for example, the intuitive understanding of the straightness of a straight line. With General Relative declaring the fabric of space as being curved, you cannot picture your straight line to continue in its path all the way to infinity without bending. You might use your imagination to make a hole in the space fabric and push the line through the whole, but it will be meaningless and incorrect, because you cannot imagine the other side of the space fabric, because it has absolutely no meaning. It's calls for a major shift in the thought framework.
Similar with all knowledge, even. I am inclined to think all knowledge is not discovered, but rather invented. What defines a thing as a mathematical law is how interesting you perceive it to be. For example, a formula to find out the n-th even number is an interesting piece of knowledge, as is the proof that the system of even numbers is infinite. Discovering the individual even numbers, say, the 18,015th even number, or the 120,109,752th or each even number in between may not be mathematically interesting at all. There may be no inherent and absolute property attached to a question in each category; rather, it is our perception that introduces the necessary distinction.
To answer your question: whether a person with an infinite lifespan and infinite time may discover all of mathematics, the fast and loose answer would be yes, because it would seem that the mind is not infinitely powerful, and not with an infinite memory. There is only a finite amount of stuff that he would be able to classify as 'mathematically interesting'. It is like asking, can a computer with a finite amount of RAM but an infinite amount of secondary storage run an infinite number of programs concurrently without swapping any out (no it cannot!)? So, as our knowledge grows, the number of independent parameters needed to ask new questions that would grow so large that they would no longer fit the mind in one go. So, we would cease to ask new questions and when we have no more questions, that will be the end of all knowledge (I am assuming we can say we have 'discovered' all of mathematics when we have no further question about anything mathematical).
But, it's not that simple. Some of the key things to consider are:
a) Does greater mathematical knowledge seem to imply knowing more unrelated things or less? For example, in physics, more knowledge could possibly mean less independent things to know. Like, at one time light, matter and electricity would have been perceived as three different things altogether, then someone comes across and discovers numerous subatomic particles and then we immediately eliminate them as mutually independent things. Then we stumble upon quarks, and they show us how subatomic particles are not 'fundamental', and then finally, we now discover strings, which is even more fundamental to a quark! It does seem like the growth of knowledge in physics is essentially a contraction, and might even have a visible end? It may not be the same for math, though, and more knowledge could well mean a genuine expansion of knowledge. Let us hope for some enlightenment here :)
b) Given a set of mathematically interesting (and possibly unrelated?) pieces of information, is it possible to always derive from them another mathematically interesting question? I have no idea how to go about proving this :D but I am going to give it a try someday. If the answer is no, then it might well mean that the end is once more in sight (limited by the finiteness of our own mind).
One other interesting question (or corollary) that crops up here is whether it is possible to measure the capacity of one's own mind. I feel it is somewhat akin to trying to paint the floor of a room without stepping outside of the room and yet not treading wet paint - it cannot be done! However, that doesn't mean that you cannot discover the boundaries of someone else's mind. Maybe then you could discover all of mathematical knowledge for that person (it should vary between people) ahead of said person in a finite time!
P.s. I realize my answer is a response to some inner narcissistic craving, and as such things go, full of holes. In which case, I would very much appreciate should someone set me on the right path :)
Yours sincerely,
Jude
