«Once you accept that [Wigner] cherry-picks, the argument becomes more properly represented as saying that mathematics is unreasonably effective at describing the things that we have found it to be effective at describing.»—I wonder. Yes, Wigner does cherry-pick, but it's hard to cherry-pick his main example (i.e. fundamental physics). And he was writing in a time when we seemed to be approaching a Theory of Everything, and when the main components of that theory were suffice to say *extraordinarily beautiful*. Now, I don't think Wigner's address would be quite so convincing in a 21st-century context of stagnant physics and numerically-analyzed natural sciences, but if one puts that aside for the moment to consider the astonishing elegance of special relativity, general relativity, and quantum mechanics his argument makes a bit more sense, I think.
On the flip side, you kind of expect fundamental physics to be comparatively simple due to, you know, being *fundamental.* If anything, the fact that the path-integral already involves mathematical questions that are *still* open in the present day kind of makes one wonder if maybe fundamental physics is all that mathematically simple after all.
I mean, one would *hope* so—and when Wigner gave his address such hopes were running high! And yes, those hopes not having panned out yet definitely provides some counterevidence for Wigner's thesis… although I do console myself that the stagnation doesn't seem to be *confined* to fundamental physics, and so it probably isn't fundamental physics' "fault"…
I am still processing your definition; compared with all other attempts to defining Math I've seen, this seems the best one. Also, very powerful for being very short. Of course there may be pitfalls , and philosophers may have a field day attacking it. Still, quite satisfying for a working mathematician.
On the other hand, Mathematics creates its own contexts. I have lost count of the situations in which the presentation of a definition of an abstract structure is followed by "concrete examples" which may be an infinite dimensional Banach algebra, a skew field with positive characteristic, the circuit space of a planar graph. All of these feel very concrete but from the outside the use of the adjective sounds almost ludicrous.
I don't wish to devalue looking for examples (concrete or not) for definitions. After all, that is certainly my preferred way to teach: show the general definition, and then show many, many ways in which it may come up. None of those examples *is* the thing that we are studying, but nonetheless, they help in understanding it. Context is a tool, and it can be wielded to great benefit.
On the other hand, if you dislike some particular interpretation or proposed intuition for a mathematical structure... you can always discard it. If it gets in the way of your work, then it is no longer serving its function.
The general perception of mathematics being so divorced from what mathematics is actually about is pretty sad. Texts like Euclid make it absolutely clear, what mathematics is about, namely not memorising seemingly arbitrary rules about numbers, but discovering the properties precisely defined systems.
In my reading of his famous article, Eugene Wigner's “unreasonable effectiveness of mathematics” was not referring to the (unsurprising) fact that mathematics is an effective language for describing regularities in nature.
Rather, Wigner was referring more specifically to how mathematical concepts are surprisingly successful in generalizing to universal laws, and with validity surprisingly far beyond the very limited empirical data available when the laws were first introduced.
For example, Newton's law of gravitation was developed based upon parabolic projectile motion on earth and elliptical motions of planets. Newton's law extrapolated far beyond these very specific empirical observations to develop a universally valid law of gravitation. This is what he considers surprising and unreasonably effective.
This reminded me of the kind-of philosophical question whether mathematics is discovered or invented.
I've always been on the camp of "discovered", but it still makes one wonder if the structures that we're studying through math would exist (be inventable/discoverable) if the context associated with them was completely absent from our world.
Thank you for the article! Now I found my favorite definition of mathematics :) The question that comes to my mind is: Why do people have an urge to study structures divorced from their context?
Historically, I think it is fair to say that humans had to be dragged kicking and screaming toward this. Consider, for example, that when Descartes came up with what we now call the Cartesian plane, he only considered points with non-negative coordinates; to cover the entire plane, he broke it up into four quadrants, each with its own coordinate system.
Why do this? Because it allowed him to avoid negative numbers. Why do that? Because mathematicians of that era were deeply uncomfortable with the idea, since they didn't see how it could be interpreted physically. (Even though the concept of loans and banking had already existed for over a millennium!)
This was still a fairly common attitude as late as the middle of the 19th century, and that was over a hundred years after Newton, who had laid all the groundwork for a physical interpretation of negative numbers... if only anyone was willing to venture in that direction.
I think it is only really fair to say that humanity has had an urge to study structures divorced from context since the 1800s, once it had become overwhelmingly apparent that it was an extremely fruitful perspective. Prior to that point, humanity was mostly grasping in the dark, with some rare but extraordinary exceptions.
«Once you accept that [Wigner] cherry-picks, the argument becomes more properly represented as saying that mathematics is unreasonably effective at describing the things that we have found it to be effective at describing.»—I wonder. Yes, Wigner does cherry-pick, but it's hard to cherry-pick his main example (i.e. fundamental physics). And he was writing in a time when we seemed to be approaching a Theory of Everything, and when the main components of that theory were suffice to say *extraordinarily beautiful*. Now, I don't think Wigner's address would be quite so convincing in a 21st-century context of stagnant physics and numerically-analyzed natural sciences, but if one puts that aside for the moment to consider the astonishing elegance of special relativity, general relativity, and quantum mechanics his argument makes a bit more sense, I think.
On the flip side, you kind of expect fundamental physics to be comparatively simple due to, you know, being *fundamental.* If anything, the fact that the path-integral already involves mathematical questions that are *still* open in the present day kind of makes one wonder if maybe fundamental physics is all that mathematically simple after all.
I mean, one would *hope* so—and when Wigner gave his address such hopes were running high! And yes, those hopes not having panned out yet definitely provides some counterevidence for Wigner's thesis… although I do console myself that the stagnation doesn't seem to be *confined* to fundamental physics, and so it probably isn't fundamental physics' "fault"…
I am still processing your definition; compared with all other attempts to defining Math I've seen, this seems the best one. Also, very powerful for being very short. Of course there may be pitfalls , and philosophers may have a field day attacking it. Still, quite satisfying for a working mathematician.
On the other hand, Mathematics creates its own contexts. I have lost count of the situations in which the presentation of a definition of an abstract structure is followed by "concrete examples" which may be an infinite dimensional Banach algebra, a skew field with positive characteristic, the circuit space of a planar graph. All of these feel very concrete but from the outside the use of the adjective sounds almost ludicrous.
I don't wish to devalue looking for examples (concrete or not) for definitions. After all, that is certainly my preferred way to teach: show the general definition, and then show many, many ways in which it may come up. None of those examples *is* the thing that we are studying, but nonetheless, they help in understanding it. Context is a tool, and it can be wielded to great benefit.
On the other hand, if you dislike some particular interpretation or proposed intuition for a mathematical structure... you can always discard it. If it gets in the way of your work, then it is no longer serving its function.
The general perception of mathematics being so divorced from what mathematics is actually about is pretty sad. Texts like Euclid make it absolutely clear, what mathematics is about, namely not memorising seemingly arbitrary rules about numbers, but discovering the properties precisely defined systems.
In my reading of his famous article, Eugene Wigner's “unreasonable effectiveness of mathematics” was not referring to the (unsurprising) fact that mathematics is an effective language for describing regularities in nature.
Rather, Wigner was referring more specifically to how mathematical concepts are surprisingly successful in generalizing to universal laws, and with validity surprisingly far beyond the very limited empirical data available when the laws were first introduced.
For example, Newton's law of gravitation was developed based upon parabolic projectile motion on earth and elliptical motions of planets. Newton's law extrapolated far beyond these very specific empirical observations to develop a universally valid law of gravitation. This is what he considers surprising and unreasonably effective.
This reminded me of the kind-of philosophical question whether mathematics is discovered or invented.
I've always been on the camp of "discovered", but it still makes one wonder if the structures that we're studying through math would exist (be inventable/discoverable) if the context associated with them was completely absent from our world.
Thank you for the article! Now I found my favorite definition of mathematics :) The question that comes to my mind is: Why do people have an urge to study structures divorced from their context?
Historically, I think it is fair to say that humans had to be dragged kicking and screaming toward this. Consider, for example, that when Descartes came up with what we now call the Cartesian plane, he only considered points with non-negative coordinates; to cover the entire plane, he broke it up into four quadrants, each with its own coordinate system.
Why do this? Because it allowed him to avoid negative numbers. Why do that? Because mathematicians of that era were deeply uncomfortable with the idea, since they didn't see how it could be interpreted physically. (Even though the concept of loans and banking had already existed for over a millennium!)
This was still a fairly common attitude as late as the middle of the 19th century, and that was over a hundred years after Newton, who had laid all the groundwork for a physical interpretation of negative numbers... if only anyone was willing to venture in that direction.
I think it is only really fair to say that humanity has had an urge to study structures divorced from context since the 1800s, once it had become overwhelmingly apparent that it was an extremely fruitful perspective. Prior to that point, humanity was mostly grasping in the dark, with some rare but extraordinary exceptions.