tag:blogger.com,1999:blog-5867400.post109274390942715252..comments2021-03-02T17:37:58.235+11:00Comments on Journey's End: TumboliaStevehttp://www.blogger.com/profile/08905624839178001596noreply@blogger.comBlogger5125tag:blogger.com,1999:blog-5867400.post-1093414944687220462004-08-25T16:22:00.000+10:002004-08-25T16:22:00.000+10:00This comment has been removed by a blog administrator.Mhttps://www.blogger.com/profile/08539576551592525597noreply@blogger.comtag:blogger.com,1999:blog-5867400.post-1093414902795299632004-08-25T16:21:00.000+10:002004-08-25T16:21:00.000+10:00Steve:
>Hrm, I don't think math is completely ind...Steve:<br /><br />>Hrm, I don't think math is completely independent from our perception of the >world, although the way you expressed is certainly elegant :)<br /><br />I would say that maths itself is independent of our perceptions, but the bits of it that we discover depend on our perceptions. That's why it took so long to discover all the different geometries (other than Euclidean that is): because we perceive the world in a Euclidean way, so anything else is counter-intuitive, so it took longer to discover. Or create. Now there's a conundrum: is maths created or discovered? I would say discovered...since it is abstract concepts and ideas, we don't really 'create' them, we just 'think of' them, i.e. discover. But it's debatable.<br /><br />>For example, a triangle on a sphere does not have an angle sum of 180 >degrees, derivative of e^x doesn't have to be e^x, not if its respective to >y. <br /><br />Ah well, that's not because the maths is wrong: it's just that I was not exact enough in what I said. Let me re-phrase it: The sum of the angles of a triangle in Euclidean space is and always will be 180 degrees. The derivative of e^x with respect to x is and always will be e^x.<br />And they always have been, even before we knew it.<br /><br /><br />>Just as science can't be seperated from the physical world, math can not be >seperated from our perception of the world, and the mental constructs >required to comprehend its concepts. ie. limits of infinity for example, >explain THAT to a fish.. in fact, try some 70% of the population, same result >most likely.<br /><br />That's because there is an isomorphism (or mapping) between maths and the physical world. But, here's a thought: how do you know there is not a whole lot of maths that we just haven't found yet because our perceptions of the physical world are limiting our imagination? Maths that isn't isomorphic to the physical world? And also: maths with no application is not very useful. Interesting, aesthetic, but not useful. We are not as likely to bother finding out something that's not at all useful. So that is why there is a relationship between the mathematics we know and our prceptions: not because our perceptions influence maths, but because our perceptions influence our ability to discover and understand maths - so the mathematics we know is limited to that which we can understand.<br /><br />>I would however concur some axioms of math are constants, 1 apple + 1 apple >always gives you 2 apples, simply conservation of mass, which also happens to >be a constant :) Similarly, speed of light is a constant, just as e is a >constant, and pi.<br /><br />But there is a difference between pi, e, and the speed of light. The speed of light has been *shown* to be a constant by experimentation. Pi and e are constants by *definition*: if they were not constant, they would not be pi and e. Pi, for instance, is the ratio of the diameter of a circle to its circumference (in Euclidean space!). A circle is well defined (see Euclid's Elements). It follows, and can be deduced from, this definition, that pi is constant. Perhaps it follows from the definition of lght that its speed must be a constant...but science is, in a sense, reverse engineering the universe. We do not have the axioms, the definitions. We are trying to find them from the theorems we can work out from our experiments. And so there is no guarantee that the speed of light is in fact constant: it is simply that the evidence presented to date indicates this. Perhaps tomorrow we will discover otherwise. That is why science is uncertain, and maths is not. Once something has been proven in maths, it is proven for all time. Once something has been proven in science, it is proven until someone else proves otherwise and shows why the original proof was wrong. In maths, that can't happen. The only way to make a theorem incorrect is to change the axioms, and in that case you are not disporving the current mathematics, but creating a new branch of maths. As with the different geometries derived from deciding whether or not parallel lines meet: Euclidean versus geometry on a sphere.<br /><br /><br />>I always believed there is an underlying structure in the universe that makes >everything work. Math and science are but the same structure probed by >different methods: one by logic, the other by observation. Yet both are >intrinsically in twined: no math with out basic observation, no science with >out basic logic. <br /><br />I think maths would *exist* without observation, we would just never have found out about it. And the same with science...observation is all very well, but you need logic to interpret your results. As for an underlying structure, well, this is something that intrigues me...why *is* energy conserved? Are there places where it isn't? Why does matter bend spacetime, why is the speed of light constant, why, in short, are things the way they are? And so we wander into the realm of philosophy...<br /><br /><br />>One day, humanity shall evolve to the level, where we can see that structure >for what it is, and it shall be a union of logic and observation, a merger of >mind and matter, for that shall be the day when we see the universe for what >it is: <br /><br />>An engine to drive time to its end.<br /><br />I like that. :)<br /><br />When we know everything, science will be as constant and sure as mathematics. The thing is: how will we know when we get there?<br /><br /><br />M.<br />(who is glad she's not the only one...)Mhttps://www.blogger.com/profile/08539576551592525597noreply@blogger.comtag:blogger.com,1999:blog-5867400.post-1093414368642854972004-08-25T16:12:00.001+10:002004-08-25T16:12:00.001+10:00This comment has been removed by a blog administrator.Mhttps://www.blogger.com/profile/08539576551592525597noreply@blogger.comtag:blogger.com,1999:blog-5867400.post-1093414365596117822004-08-25T16:12:00.000+10:002004-08-25T16:12:00.000+10:00This comment has been removed by a blog administrator.Mhttps://www.blogger.com/profile/08539576551592525597noreply@blogger.comtag:blogger.com,1999:blog-5867400.post-1093363108854798522004-08-25T01:58:00.000+10:002004-08-25T01:58:00.000+10:00Hrm, I don't think math is completely independent ...Hrm, I don't think math is completely independent from our perception of the world, although the way you expressed is certainly elegant :)<br /><br />For example, a triangle on a sphere does not have an angle sum of 180 degrees, derivative of e^x doesn't have to be e^x, not if its respective to y. <br /><br />Just as science can't be seperated from the physical world, math can not be seperated from our perception of the world, and the mental constructs required to comprehend its concepts. ie. limits of infinity for example, explain THAT to a fish.. in fact, try some 70% of the population, same result most likely.<br /><br />I would however concur some axioms of math are constants, 1 apple + 1 apple always gives you 2 apples, simply conservation of mass, which also happens to be a constant :) Similarly, speed of light is a constant, just as e is a constant, and pi.<br /><br />I always believed there is an underlying structure in the universe that makes everything <I>work</I>. Math and science are but the same structure probed by different methods: one by logic, the other by observation. Yet both are intrinsically in twined: no math with out basic observation, no science with out basic logic. <br /><br />One day, humanity shall evolve to the level, where we can <I>see</I> that structure for what it is, and it shall be a union of logic and observation, a merger of mind and matter, for that shall be the day when we see the universe for what it is: <br /><br />An engine to drive time to its end.<br /><br />Steve<br />(who has never thought he or M was strange...)Stevehttps://www.blogger.com/profile/08905624839178001596noreply@blogger.com