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I think Kant may have rendered this article's vector at the question pretty meaningless over 200 years ago. In the Critique of Pure Reason, he posits that spac

I think Kant may have rendered this article's vector at the question pretty meaningless over 200 years ago. In the Critique of Pure Reason, he posits that space and time are the forms of our intuition, and he claims that anything that comes to us does so through these forms. That is to say, we do not know things as they are in themselves--we only know them insofar as they have properties that can be reflected in a form that can be processed by our intuition. We bring space and time with us, as part of our structure, and mathematics is an examination of that structure. So Kant would have it that mathematics is neither a Platonic form nor something we have created-it's an exploration of our capabilities of sensation. This seems to me like a much better explanation of why mathematics are "universal" than the article's approach. It also leaves open the question of whether we are "living in a computer simulation" without suggesting that it's a simple matter (or even necessarily possible) to see out from inside.

evanb 13y ago
I think the supposition that space and time are forms of our intuition is suspect, and is indeed undermined by relativity. The supposition that maybe spacetime are the forms of our intuition also doesn't necessarily withstand scrutiny, because there is at least one way of formulating quantum field theories with no reference to spacetime at all (see, for example, https://news.ycombinator.com/item?id=6403285 https://news.ycombinator.com/item?id=6403285 ).
- discostrings 13y ago
  I don't think relativity or quantum field theory undermine the idea that space and time are our forms of intuition. On the contrary, I feel it may be the first framework that really created a sensible space for those phenomena. Kant's approach holds that our /intuition/ is limited by space and time and that "things in themselves" may in actuality be many different ways. We can only receive and understand things insofar as they exist or can be manifest in space/time. I think his theory very much leaves open that we can better grasp the properties of things by studying the way they appear in space/time and drawing inferences. His very point is that space and time do not exist in themselves as far as we perceive them--they are merely the forms through which we are capable of receiving anything. And thus I would say he created a groundwork of thought that made room for relativity and quantum observations well before they were conceived--in fact, I imagine Kant's groundwork was important in our path to them. [Edited for a slight bit more clarity] [HN meta comment: it's really annoying that any given story that has a lot of upvotes and that is seemingly appropriate for HN can be flagged from the front page. Why spend time writing an insightful comment when any given story could be sent from the top of the front page to the middle of page 3 by the capricious action of a flagger or moderator, meaning no one will ever see the comment? Such unpredictability really discourages quality participation and encourages quick responses--why expend a lot of effort when the chance of anyone reading your work could evaporate so quickly and arbitrarily?]
  - evanb 13y ago
    Space and time are things themselves and do exist independent of our perception. As evidence: all the solutions to general relativity that have no matter content but nonetheless are not flat, because the curvature of spacetime itself has energy that gravitates (the Schwarzschild solution comes to mind). Spacetime has its own real, physical, dynamics.
    - discostrings 13y ago
      Kant agrees that empirically, space and time are things themselves and exist independent of our perception. In fact, for Kant, anything that can come to us or be reasoned by us, and the relations of those things, must comply with space, time, and, therefore, mathematics. I wasn't trying to say that space, time, and mathematics do not exist independent of our perception. Rather, I was refuting the idea that mathematics is either a Platonic form or something we made up. The Kantian idea is: anything we come into contact with or understand must come to us in a way that conforms with mathematics, or else we wouldn't be able to experience it. Therefore, everything we know complies with mathematics--it's not some Platonic form outside us or our creation, but the very structure of our understanding. There might be a superset of laws that things in themselves comply with that we don't have access to. We are, however, limited to the understanding we can achieve through space, time, and mathematics (which do describe everything we can encounter). My initial response wasn't terribly well written and was rambling, so I tried to clean it up a bit. It started to address your point but veered off-topic. There's also an important difference between perception and intuition that I originally didn't express very well. Getting back to what I was originally trying to say, I don't think the nature of mathematics has much to do with whether or not we live in a computer simulation. We could be bound to mathematics for any number of reasons that are beyond our capability for understanding. What I was commenting on was the article's introduction to the question. I suppose, though, that the article itself admits the relation between the introduction and the topic may not be strong ("But one fanciful possibility is that we live in a computer simulation based on the laws of mathematics"). I just don't think "Mathematics applies to everything we can experience, therefore we live in a computer" is a particularly interesting approach. I really appreciate your responses.
nandemo 13y ago
I'm just gonna leave this here: >We can eat oysters only insofar as they are brought under the physiological and chemical conditions which are the presuppositions of the possibility of being eaten. >Therefore, we cannot eat oysters as they are in themselves. http://web.maths.unsw.edu.au/~jim/worst.html http://web.maths.unsw.edu.au/~jim/worst.html

evanb 13y ago

I think the supposition that space and time are forms of our intuition is suspect, and is indeed undermined by relativity. The supposition that maybe spacetime are the forms of our intuition also doesn't necessarily withstand scrutiny, because there is at least one way of formulating quantum field theories with no reference to spacetime at all (see, for example, https://news.ycombinator.com/item?id=6403285 https://news.ycombinator.com/item?id=6403285 ).

discostrings 13y ago
I don't think relativity or quantum field theory undermine the idea that space and time are our forms of intuition. On the contrary, I feel it may be the first framework that really created a sensible space for those phenomena. Kant's approach holds that our /intuition/ is limited by space and time and that "things in themselves" may in actuality be many different ways. We can only receive and understand things insofar as they exist or can be manifest in space/time. I think his theory very much leaves open that we can better grasp the properties of things by studying the way they appear in space/time and drawing inferences. His very point is that space and time do not exist in themselves as far as we perceive them--they are merely the forms through which we are capable of receiving anything. And thus I would say he created a groundwork of thought that made room for relativity and quantum observations well before they were conceived--in fact, I imagine Kant's groundwork was important in our path to them. [Edited for a slight bit more clarity] [HN meta comment: it's really annoying that any given story that has a lot of upvotes and that is seemingly appropriate for HN can be flagged from the front page. Why spend time writing an insightful comment when any given story could be sent from the top of the front page to the middle of page 3 by the capricious action of a flagger or moderator, meaning no one will ever see the comment? Such unpredictability really discourages quality participation and encourages quick responses--why expend a lot of effort when the chance of anyone reading your work could evaporate so quickly and arbitrarily?]
- evanb 13y ago
  Space and time are things themselves and do exist independent of our perception. As evidence: all the solutions to general relativity that have no matter content but nonetheless are not flat, because the curvature of spacetime itself has energy that gravitates (the Schwarzschild solution comes to mind). Spacetime has its own real, physical, dynamics.
  - discostrings 13y ago
    Kant agrees that empirically, space and time are things themselves and exist independent of our perception. In fact, for Kant, anything that can come to us or be reasoned by us, and the relations of those things, must comply with space, time, and, therefore, mathematics. I wasn't trying to say that space, time, and mathematics do not exist independent of our perception. Rather, I was refuting the idea that mathematics is either a Platonic form or something we made up. The Kantian idea is: anything we come into contact with or understand must come to us in a way that conforms with mathematics, or else we wouldn't be able to experience it. Therefore, everything we know complies with mathematics--it's not some Platonic form outside us or our creation, but the very structure of our understanding. There might be a superset of laws that things in themselves comply with that we don't have access to. We are, however, limited to the understanding we can achieve through space, time, and mathematics (which do describe everything we can encounter). My initial response wasn't terribly well written and was rambling, so I tried to clean it up a bit. It started to address your point but veered off-topic. There's also an important difference between perception and intuition that I originally didn't express very well. Getting back to what I was originally trying to say, I don't think the nature of mathematics has much to do with whether or not we live in a computer simulation. We could be bound to mathematics for any number of reasons that are beyond our capability for understanding. What I was commenting on was the article's introduction to the question. I suppose, though, that the article itself admits the relation between the introduction and the topic may not be strong ("But one fanciful possibility is that we live in a computer simulation based on the laws of mathematics"). I just don't think "Mathematics applies to everything we can experience, therefore we live in a computer" is a particularly interesting approach. I really appreciate your responses.

nandemo 13y ago

I'm just gonna leave this here: >We can eat oysters only insofar as they are brought under the physiological and chemical conditions which are the presuppositions of the possibility of being eaten. >Therefore, we cannot eat oysters as they are in themselves. http://web.maths.unsw.edu.au/~jim/worst.html http://web.maths.unsw.edu.au/~jim/worst.html