6 ms·

The limited, and hence reasonable, effectiveness of mathematics in physics

46 points by discreteevent 11y ago

jonahx 11y ago
Anyone interested in this might also like Lakoff's "Where Mathematics Comes From," which I read after seeing a Bret Victor recommendation of it: http://www.amazon.com/Where-Mathematics-Come-From-Embodied/dp/0465037712 http://www.amazon.com/Where-Mathematics-Come-From-Embodied/d... Absolutely fascinating book.
- ubernostrum 11y ago
  See also Martin Gardner's defenses of mathematical realism for counterpoint.
amelius 11y ago
> There is an objective distinction between past, present, and future. How does that work with relativity, which says that the concept of simultaneity does not even exist in the sense that we understand it? [1] [1] https://en.wikipedia.org/wiki/Simultaneity https://en.wikipedia.org/wiki/Simultaneity
- jerf 11y ago
  There's an objective distinction between "past", "future", and "undefined". ("Present" is sort of more a human concept than a physics one.) Past is in your past light cone, future is in your future light cone, and "undefined" is everything else.
  - gdavisson 11y ago
    I prefer to categorize them as "absolute past", "absolute present", and "absolute elsewhere". This is because for any event in one of your light cones (past or future), there'll be a reference frame that places it at your location (i.e. the same spatial coordinates, just at a different time). But for events outside your light cones, no reference frame will have it at the same location; it's absolutely elsewhere. BTW, if you also want to talk about the present, you can only talk about it at a particular location. If you want, you can define 3 "thin" boundary regions: the surface of the future light cone (boundary between absolute future & absolute elsewhere), the surface of the past light cone (similar), and the "here & now" (the single point in spacetime where all 3 regions meet).
  - DougMerritt 11y ago
    > "Present" is sort of more a human concept than a physics one. Please excuse my quibble, but more precisely it is a classical/Newtonian concept rather than a relativistic concept. "Human concept" might map to "folk physics" (the version of physics assumed by "the common man" humans from current hunter-gatherer societies and older societies etc.) There are illustrations from the middle ages and from ancient Greece, if I'm not mistaken, that show the flight of an arrow to be a straight line up to apex, and then linearly down to target (rather than a realistic parabola) -- and although I've always wondered if an actual bow expert thought that's how it worked (unlikely), but apparently the non-archer philosophers of the day apparently did. But I digress. You are clearly just pointing out the modern understanding.
animefan 11y ago
I think that there are softer versions of Platonism (which in other contexts would not be considered Platonism at all, but are closer to Platonism that Smolin's idea) that are somewhat consistent with naturalism. The softer version is that there appear to be mathematical objects with independent existence, even though there is no a priori philosophical reason to believe in their independent existence, and there are indeed reasons to doubt it (e.g. non-standard models of not only the reals, but also the natural numbers). One reason I find it hard to reject Platonism entirely is that the main alternative in the philosophy of mathematics, Formalism, looks to me like a kind of Platonism. This is because formalism replaces belief in the independent existence of mathematical objects (real numbers, integers) with belief in the independent existence of meta-mathematical objects (theorems, proofs) which most formalists conceive of in mathematical terms. Ultimately I think that there is no a priori limit that we can place on how science is going to pan out, and how this will affect our philosophy. E.g. I find the idea of multiple universes distasteful and nonsensical (e.g. If everything possible happens, then how can we interpret probabilities? Why do we live in a universe where macro observations are consistent with applying the law of large numbers to quantum randomness? I can't think of any anthropic/observer based principle that would enforce this) but I can't yet rule out that that's really how things are.
im3w1l 11y ago
I have heard of two paradigms humanity have used to try to explain the world Mathematics/logics is one of them. It says that the world is guided by principles. "Objects contain different amount of phlogiston and when they burn it is released." Anthropomorphism is the other. I.e. there is one, or many human-like beings with personalities and feelings. And things that happen happen because the beings want them to. "If you offend the fire spirits your house may burn down" Are there more?
- xioxox 11y ago
  Many things happen the way they do just down to luck or chance. If that microscopic quantum fluctuation hadn't been there our galaxy wouldn't have existed. Some people believe in karma. Maybe this is covered as a principle, however, even though it isn't scientific. Perhaps solipsism might also count as an "explanation" of the world. I exist, so that explains the world. Fate is also an explanation for what happens.
lisper 11y ago
"the contents of mathematics is far from arbitrary -- while an infinite number of mathematical objects might potentially be envoked [sic], the few that prove interesting develop a very small number of core concepts. These core concepts are not arbitrary -- they are elaborations of structures which are discovered during the study of nature. There are four of these core concepts: number, geometry, algebra and logic." It could be as well argued that there is only one core concept: computation. (Or maybe "symbol".) The interesting question is: of all the myriad possible things that are computable, how is it that we are able find the unfathomably small subset whose behavior corresponds to (what we perceive as) physical reality? That was the question Wigner was asking, and I don't think Smolin answers it at all.
- amelius 11y ago
  > The interesting question is: of all the myriad possible things that are computable, how is it that we are able find the unfathomably small subset whose behavior corresponds to (what we perceive as) physical reality? Isn't that simply because of the Anthropic principle [1]? We are observing this computation simply because we are part of it. Other creatures may be observing different computations (but we will never be able to ask them because they live in a different reality). [1] https://en.wikipedia.org/wiki/Anthropic_principle https://en.wikipedia.org/wiki/Anthropic_principle
  - lisper 11y ago
    No. The anthropic principle is an answer to a different question, namely, of all the possible universes that could conceivably exist, how is it that we find ourselves in this one, where the fundamental physical constants are (apparently) fine-tuned to allow the formation of atoms and hence life.
    - ubernostrum 11y ago
      Most anthropic arguments are also based on an equivocation. William Lane Craig has some good work on this, using an analogy (which he admits he borrowed): suppose you are tied up and brought before a firing squad consisting of 100 expert marksmen. They all load, aim at point-blank range and fire... and then you open your eyes and see they all missed and you're still alive and unhurt. (1) Anthropically, you should not be "surprised" to observe a world in which 100 expert marksmen all missed from point-blank range, because the very fact of your being able to observe anything at that point makes that world necessary. (2) But you nonetheless should be "surprised" that such a low-probability event has occurred, and you would be justified in wondering whether some as-yet-unaccounted-for factor meant that the true probability was not what you would have estimated it to be. Anthropic arguments, in this view, slyly try to switch between different meanings of "surprised" (or a variant, depending on the level of formality and the particular terminology chosen) to make it look as if (1) actually answers (2) when it doesn't.
skybrian 11y ago
"As we do not believe in timeless Platonic realities, we do not want to say that chess always existed-in our view of the world, chess came into existence at the moment the rules were codified." Okay, but when were the variations of chess invented? For example, instead of playing on an 8x8 board, you could play on an x*y board for any natural numbers x and y. Do we need to set x and y to specific values for that variation to "come into existence". Does it make sense for us to say that a game exists when it's never been played? How specific do we need to be about imagining a game? If we just say that chess can vary in many possible ways by changing any of its rules, does checkers come into existence? Or suppose we write a computer program to generate variations on board games. Do I have to run the program for these games to "come into existence", or just write it, or just think about writing it? If the program has a bug so that it won't generate a particular board game, does that game come into existence when I document the bug, fix the bug, or verify the fix? I don't think any philosophy is all that good at distinguishing between explored and unexplored concepts in the general case. But that's a problem mostly when you try to do philosophy. If you have a practical purpose in mind, it's much easier to come up with a useful working definition for the particular domain you're interested in.
- danbruc 11y ago
  What you are struggling with here is the definition of existence. And Platonism has this problem in general, you rarely get a sharp definition of what it means that something exists. Natural numbers exist. I know what natural numbers refers to but I have no idea what existence refers to. When we talk about things in the physical world I can come up with a definition of existence, say anything I can - at least in principle - interact with. This definition is far from perfect and has quite a few issues itself but it is at least a start. But what does the existence of mathematical objects even mean?
  - skybrian 11y ago
    Yes, exactly. The essay is attempting to tie the moment of creation for a concept to some event in the real world. Certainly there is something physical going on which we call a "discovery" or an "invention", but it gets fuzzier the more you look at it. We can objectively say that 1492 was an important year of first contact and we should talk about what happened. But it wasn't "discovering the New World" from the Native Americans' point of view. Both in science and math, publishing a paper can be an important event and it's a physical process. But the first math paper published may not correspond to the first time someone had an idea or made an observation. If we're going to talk about a concept as a thing that exists in time and space, we need to take into account that ideas may be created, shared, and forgotten many times. Perhaps we can think of Platoism as a way to avoid getting bogged down in the history of an idea by declaring it irrelevant for our purposes.
themgt 11y ago
This is excellent. Interview going over more here: https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-and-the-status-of-modern-physics/ https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-an... "As Roberto Mangabeira Unger and I argue in our new book The Singular Universe, the most important discovery cosmologists have made is that the universe has a history. We argue this has to be extended to the laws themselves. Biology became science when the question switched from listing the species to the dynamical question of how species evolve. Fundamental physics and cosmology have to transform themselves from a search for timeless laws and symmetries to the investigation of hypotheses about how laws evolve." and "Neurosciences are a fabulous area to work in, ripe for great discoveries. I’ve always felt this and indeed the only alternative to a career in physics that ever attracted me was a brief flirtation in college with neuroscience. But that is a field which is as bedeviled by outdated metaphysical baggage as physics is. In particular, the antiquated idea that any physical system that responds to and processes information is isomorphic to a digital programmable computer is holding back progress."
- danbruc 11y ago
  When you listen to some contemporary physicists like Nima Arkani-Hamed they seem to disagree, i.e. they believe that neither time nor space are fundamental properties of our universe but instead emerge from something more fundamental. [1] This leads of course to some really hard problems and especially it is no longer clear what physics is even about. It used to be the science of things moving through space over time, but what if you lose space and time? [1] http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-doomed http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-...
  - DougMerritt 11y ago
    > they believe that neither time nor space are fundamental properties of our universe but instead emerge from something more fundamental. Yes, this has gained increasing acceptance ever since its first (?) suggestion from the late illustrious John Archibald Wheeler in the 1960s, and it is a very compelling idea, although yes, it makes it hard to conceive what we may be talking about. > like Nima Arkani-Hamed they seem to disagree, I missed it, who's disagreeing with whom about what? The parent comment doesn't seem to have content disagreeing, unless I'm missing something.

46 points by discreteevent 11y ago

Anyone interested in this might also like Lakoff's "Where Mathematics Comes From," which I read after seeing a Bret Victor recommendation of it: http://www.amazon.com/Where-Mathematics-Come-From-Embodied/dp/0465037712 http://www.amazon.com/Where-Mathematics-Come-From-Embodied/d... Absolutely fascinating book.

ubernostrum 11y ago
See also Martin Gardner's defenses of mathematical realism for counterpoint.

amelius 11y ago

> There is an objective distinction between past, present, and future. How does that work with relativity, which says that the concept of simultaneity does not even exist in the sense that we understand it? [1] [1] https://en.wikipedia.org/wiki/Simultaneity https://en.wikipedia.org/wiki/Simultaneity

jerf 11y ago
There's an objective distinction between "past", "future", and "undefined". ("Present" is sort of more a human concept than a physics one.) Past is in your past light cone, future is in your future light cone, and "undefined" is everything else.
- gdavisson 11y ago
  I prefer to categorize them as "absolute past", "absolute present", and "absolute elsewhere". This is because for any event in one of your light cones (past or future), there'll be a reference frame that places it at your location (i.e. the same spatial coordinates, just at a different time). But for events outside your light cones, no reference frame will have it at the same location; it's absolutely elsewhere. BTW, if you also want to talk about the present, you can only talk about it at a particular location. If you want, you can define 3 "thin" boundary regions: the surface of the future light cone (boundary between absolute future & absolute elsewhere), the surface of the past light cone (similar), and the "here & now" (the single point in spacetime where all 3 regions meet).
- DougMerritt 11y ago
  > "Present" is sort of more a human concept than a physics one. Please excuse my quibble, but more precisely it is a classical/Newtonian concept rather than a relativistic concept. "Human concept" might map to "folk physics" (the version of physics assumed by "the common man" humans from current hunter-gatherer societies and older societies etc.) There are illustrations from the middle ages and from ancient Greece, if I'm not mistaken, that show the flight of an arrow to be a straight line up to apex, and then linearly down to target (rather than a realistic parabola) -- and although I've always wondered if an actual bow expert thought that's how it worked (unlikely), but apparently the non-archer philosophers of the day apparently did. But I digress. You are clearly just pointing out the modern understanding.

animefan 11y ago

I think that there are softer versions of Platonism (which in other contexts would not be considered Platonism at all, but are closer to Platonism that Smolin's idea) that are somewhat consistent with naturalism. The softer version is that there appear to be mathematical objects with independent existence, even though there is no a priori philosophical reason to believe in their independent existence, and there are indeed reasons to doubt it (e.g. non-standard models of not only the reals, but also the natural numbers). One reason I find it hard to reject Platonism entirely is that the main alternative in the philosophy of mathematics, Formalism, looks to me like a kind of Platonism. This is because formalism replaces belief in the independent existence of mathematical objects (real numbers, integers) with belief in the independent existence of meta-mathematical objects (theorems, proofs) which most formalists conceive of in mathematical terms. Ultimately I think that there is no a priori limit that we can place on how science is going to pan out, and how this will affect our philosophy. E.g. I find the idea of multiple universes distasteful and nonsensical (e.g. If everything possible happens, then how can we interpret probabilities? Why do we live in a universe where macro observations are consistent with applying the law of large numbers to quantum randomness? I can't think of any anthropic/observer based principle that would enforce this) but I can't yet rule out that that's really how things are.

im3w1l 11y ago

I have heard of two paradigms humanity have used to try to explain the world Mathematics/logics is one of them. It says that the world is guided by principles. "Objects contain different amount of phlogiston and when they burn it is released." Anthropomorphism is the other. I.e. there is one, or many human-like beings with personalities and feelings. And things that happen happen because the beings want them to. "If you offend the fire spirits your house may burn down" Are there more?

xioxox 11y ago
Many things happen the way they do just down to luck or chance. If that microscopic quantum fluctuation hadn't been there our galaxy wouldn't have existed. Some people believe in karma. Maybe this is covered as a principle, however, even though it isn't scientific. Perhaps solipsism might also count as an "explanation" of the world. I exist, so that explains the world. Fate is also an explanation for what happens.

lisper 11y ago

"the contents of mathematics is far from arbitrary -- while an infinite number of mathematical objects might potentially be envoked [sic], the few that prove interesting develop a very small number of core concepts. These core concepts are not arbitrary -- they are elaborations of structures which are discovered during the study of nature. There are four of these core concepts: number, geometry, algebra and logic." It could be as well argued that there is only one core concept: computation. (Or maybe "symbol".) The interesting question is: of all the myriad possible things that are computable, how is it that we are able find the unfathomably small subset whose behavior corresponds to (what we perceive as) physical reality? That was the question Wigner was asking, and I don't think Smolin answers it at all.

amelius 11y ago
> The interesting question is: of all the myriad possible things that are computable, how is it that we are able find the unfathomably small subset whose behavior corresponds to (what we perceive as) physical reality? Isn't that simply because of the Anthropic principle [1]? We are observing this computation simply because we are part of it. Other creatures may be observing different computations (but we will never be able to ask them because they live in a different reality). [1] https://en.wikipedia.org/wiki/Anthropic_principle https://en.wikipedia.org/wiki/Anthropic_principle
- lisper 11y ago
  No. The anthropic principle is an answer to a different question, namely, of all the possible universes that could conceivably exist, how is it that we find ourselves in this one, where the fundamental physical constants are (apparently) fine-tuned to allow the formation of atoms and hence life.
  - ubernostrum 11y ago
    Most anthropic arguments are also based on an equivocation. William Lane Craig has some good work on this, using an analogy (which he admits he borrowed): suppose you are tied up and brought before a firing squad consisting of 100 expert marksmen. They all load, aim at point-blank range and fire... and then you open your eyes and see they all missed and you're still alive and unhurt. (1) Anthropically, you should not be "surprised" to observe a world in which 100 expert marksmen all missed from point-blank range, because the very fact of your being able to observe anything at that point makes that world necessary. (2) But you nonetheless should be "surprised" that such a low-probability event has occurred, and you would be justified in wondering whether some as-yet-unaccounted-for factor meant that the true probability was not what you would have estimated it to be. Anthropic arguments, in this view, slyly try to switch between different meanings of "surprised" (or a variant, depending on the level of formality and the particular terminology chosen) to make it look as if (1) actually answers (2) when it doesn't.

skybrian 11y ago

"As we do not believe in timeless Platonic realities, we do not want to say that chess always existed-in our view of the world, chess came into existence at the moment the rules were codified." Okay, but when were the variations of chess invented? For example, instead of playing on an 8x8 board, you could play on an x*y board for any natural numbers x and y. Do we need to set x and y to specific values for that variation to "come into existence". Does it make sense for us to say that a game exists when it's never been played? How specific do we need to be about imagining a game? If we just say that chess can vary in many possible ways by changing any of its rules, does checkers come into existence? Or suppose we write a computer program to generate variations on board games. Do I have to run the program for these games to "come into existence", or just write it, or just think about writing it? If the program has a bug so that it won't generate a particular board game, does that game come into existence when I document the bug, fix the bug, or verify the fix? I don't think any philosophy is all that good at distinguishing between explored and unexplored concepts in the general case. But that's a problem mostly when you try to do philosophy. If you have a practical purpose in mind, it's much easier to come up with a useful working definition for the particular domain you're interested in.

danbruc 11y ago
What you are struggling with here is the definition of existence. And Platonism has this problem in general, you rarely get a sharp definition of what it means that something exists. Natural numbers exist. I know what natural numbers refers to but I have no idea what existence refers to. When we talk about things in the physical world I can come up with a definition of existence, say anything I can - at least in principle - interact with. This definition is far from perfect and has quite a few issues itself but it is at least a start. But what does the existence of mathematical objects even mean?
- skybrian 11y ago
  Yes, exactly. The essay is attempting to tie the moment of creation for a concept to some event in the real world. Certainly there is something physical going on which we call a "discovery" or an "invention", but it gets fuzzier the more you look at it. We can objectively say that 1492 was an important year of first contact and we should talk about what happened. But it wasn't "discovering the New World" from the Native Americans' point of view. Both in science and math, publishing a paper can be an important event and it's a physical process. But the first math paper published may not correspond to the first time someone had an idea or made an observation. If we're going to talk about a concept as a thing that exists in time and space, we need to take into account that ideas may be created, shared, and forgotten many times. Perhaps we can think of Platoism as a way to avoid getting bogged down in the history of an idea by declaring it irrelevant for our purposes.

themgt 11y ago

This is excellent. Interview going over more here: https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-and-the-status-of-modern-physics/ https://scientiasalon.wordpress.com/2015/02/13/lee-smolin-an... "As Roberto Mangabeira Unger and I argue in our new book The Singular Universe, the most important discovery cosmologists have made is that the universe has a history. We argue this has to be extended to the laws themselves. Biology became science when the question switched from listing the species to the dynamical question of how species evolve. Fundamental physics and cosmology have to transform themselves from a search for timeless laws and symmetries to the investigation of hypotheses about how laws evolve." and "Neurosciences are a fabulous area to work in, ripe for great discoveries. I’ve always felt this and indeed the only alternative to a career in physics that ever attracted me was a brief flirtation in college with neuroscience. But that is a field which is as bedeviled by outdated metaphysical baggage as physics is. In particular, the antiquated idea that any physical system that responds to and processes information is isomorphic to a digital programmable computer is holding back progress."

danbruc 11y ago
When you listen to some contemporary physicists like Nima Arkani-Hamed they seem to disagree, i.e. they believe that neither time nor space are fundamental properties of our universe but instead emerge from something more fundamental. [1] This leads of course to some really hard problems and especially it is no longer clear what physics is even about. It used to be the science of things moving through space over time, but what if you lose space and time? [1] http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-doomed http://www.cornell.edu/video/nima-arkani-hamed-spacetime-is-...
- DougMerritt 11y ago
  > they believe that neither time nor space are fundamental properties of our universe but instead emerge from something more fundamental. Yes, this has gained increasing acceptance ever since its first (?) suggestion from the late illustrious John Archibald Wheeler in the 1960s, and it is a very compelling idea, although yes, it makes it hard to conceive what we may be talking about. > like Nima Arkani-Hamed they seem to disagree, I missed it, who's disagreeing with whom about what? The parent comment doesn't seem to have content disagreeing, unless I'm missing something.