Read on about how philosophical doubts expose the effrontery of math textbooks, and how mathematical rules are game-like, contrary to the platonic conceits.
What makes math so fascinating for some people, I think, is the same thing that makes it so boring for others, namely, that the 'rules of the game' can be grasped in a purely abstract, a priori, way. I don't know what the square root of Pi is, but I know that it must be less than Pi.
This self-referential character of math is, like logic's, unique in the sense that the inferences are not arbitrary, unlike the rules of other games. But at the same time, one doesn't really learns anything new.
Granted, the game can get incredible complex and there are unsolvable paradoxes, but in the end, I think it is just reason playing with itself.
Though I like math, I also hated math class more than any other. One of those reasons was indeed the arbitrary & dogmatic manner in which the textbooks defined things. I also couldn't stand the way the books stripped math of all its richness, degrading it into a mere tool for balancing one's budget or calculating a tip. Mind you, I understand that we all need to balance our budgets & tip properly, but focusing exclusively on the practical aspect felt like using a Hatori Hanzo sword to hack down blackberry bushes. Most of the quotidian uses that we apply math to everyday, as you pointed out in your article, can be reduced to linguistic conventions that require no highfalutin mathematical abstractions. On the purely practical level, natural numbers are a mere convenience, an algebraic shorthand that saves us much needless effort.
You mentioned the genetic fallacy. While I agree that we don't need to understand the religious & philosophical underpinnings of numbers use them, I also think it's unwise to omit them from education. Nietzche once quipped that one of the worst sins of philosophers, especially ethicists, is their crass disregard for the history of their subject. I think if we really want to understand math we cannot neglect its history, which includes its roots in Pythagorean & Hindu mysticism. That history embarrasses mathematicians like Martin Gardner who like to see themselves as supremely rational & free of all that superstitious nonsense & so they pretend that their subject is a science rather than a something much closer to an art. But just because a subject takes its inspiration from empirical observations doesn't make it scientific. Art imitates life, but it isn't life itself. Likewise, life also sometimes imitates art, but that doesn't obliterate the distinction between fact & fiction.
Don't get me wrong. I haven't undergone a 180° rotation since our last conversation on this subject. I'm still a thorough going Platonist when it comes to math, but that's only to say that I think it reflects a deeper reality, however darkly. Mystics warn us again & again that, contrary to Anselm's conceit, God is too big to fit inside our imaginations & so it's futile to try to reason Him into existence. I feel the same way about numbers.
The question is whether the history of science matters as much as the history of a subject in the humanities. The conceit is that the more objective a discipline, the less relevant its history is, since that history wouldn't affect the epistemic status of its latest output. The arts are more subjective or expressive of our values, in which case the genetic fallacy is less relevant to assessing them.
The alternative would amount to a postmodern critique of science which would deny the possibility of objectivity and would reduce scientific truth to something like the theory's utility. My pragmatism doesn't go that far.
If math is game-like, the history of math should matter only as much as the history of any other game. You don't have to know the history of a game to play by its rules. However, history can always reveal what we prefer to forget and can deflate our preconceptions.
It might surprise you but the foundations of mathematics are somewhat shaky. In 1901 Bertrand Russell discovered a paradox in set theory. It was about whether the set of all sets which do not contain them-selves contains itself. This led to a crisis. This crisis was 'solved' by making mathematical reasoning more formal and complicated. Since that time mathematicians themselves avoid discussing the foundations of mathematics. There are realist, constructivist and other positions in mathematical philosophy.
There is the mathematical universe hypotheses by Max Tegmark. It is the most extreme form of mathematical realism. The hypotheses states that the physical universe is not merely described by mathematics, but in fact is mathematics. Not all mathematicians and physicist adhere to such extreme and rather absurd ideas.
On the opposite extreme you have the intuitionists. The intuitionists are mathematical anti-realists or constructivists. They believe that mathematics solely consists of mental constructions. In particular, mathematics in their eyes is not the discovery of fundamental principles that exist in some objective reality. As a consequence standard logic is invalid in their eyes. Intuitionist do not believe that the axiom of the excluded middle is valid for mathematical reasoning. Therefore a lot of proofs in standard mathematics are not valid according to intuitionists.
Indeed, I'm aware of the math foundation problem. I wrote something against a view that's similar to Tegmark's, called Pythagorean Illuminism (link below). I believe it's also called structuralism in metaphysics (only structures or abstract relations are ultimately real). Intuitionism derives from Kant, as I recall.
In any case, this unresolved foundations problem feeds into my point in the above article, about the ethical lapse of presenting mathematical systems as facts to be memorized, while ignoring the philosophical questions. The philosophical underpinnings are ignored not so much because they're irrelevant to math itself, but because the underlying subject is in disarray. It's a coverup.
What makes math so fascinating for some people, I think, is the same thing that makes it so boring for others, namely, that the 'rules of the game' can be grasped in a purely abstract, a priori, way. I don't know what the square root of Pi is, but I know that it must be less than Pi.
ReplyDeleteThis self-referential character of math is, like logic's, unique in the sense that the inferences are not arbitrary, unlike the rules of other games. But at the same time, one doesn't really learns anything new.
Granted, the game can get incredible complex and there are unsolvable paradoxes, but in the end, I think it is just reason playing with itself.
I'm sure you're right. A vast series of rules can be intimidating to right-brained folks, and inviting to left-brained ones.
DeleteThough I like math, I also hated math class more than any other. One of those reasons was indeed the arbitrary & dogmatic manner in which the textbooks defined things. I also couldn't stand the way the books stripped math of all its richness, degrading it into a mere tool for balancing one's budget or calculating a tip. Mind you, I understand that we all need to balance our budgets & tip properly, but focusing exclusively on the practical aspect felt like using a Hatori Hanzo sword to hack down blackberry bushes. Most of the quotidian uses that we apply math to everyday, as you pointed out in your article, can be reduced to linguistic conventions that require no highfalutin mathematical abstractions. On the purely practical level, natural numbers are a mere convenience, an algebraic shorthand that saves us much needless effort.
ReplyDeleteYou mentioned the genetic fallacy. While I agree that we don't need to understand the religious & philosophical underpinnings of numbers use them, I also think it's unwise to omit them from education. Nietzche once quipped that one of the worst sins of philosophers, especially ethicists, is their crass disregard for the history of their subject. I think if we really want to understand math we cannot neglect its history, which includes its roots in Pythagorean & Hindu mysticism. That history embarrasses mathematicians like Martin Gardner who like to see themselves as supremely rational & free of all that superstitious nonsense & so they pretend that their subject is a science rather than a something much closer to an art. But just because a subject takes its inspiration from empirical observations doesn't make it scientific. Art imitates life, but it isn't life itself. Likewise, life also sometimes imitates art, but that doesn't obliterate the distinction between fact & fiction.
Don't get me wrong. I haven't undergone a 180° rotation since our last conversation on this subject. I'm still a thorough going Platonist when it comes to math, but that's only to say that I think it reflects a deeper reality, however darkly. Mystics warn us again & again that, contrary to Anselm's conceit, God is too big to fit inside our imaginations & so it's futile to try to reason Him into existence. I feel the same way about numbers.
The question is whether the history of science matters as much as the history of a subject in the humanities. The conceit is that the more objective a discipline, the less relevant its history is, since that history wouldn't affect the epistemic status of its latest output. The arts are more subjective or expressive of our values, in which case the genetic fallacy is less relevant to assessing them.
DeleteThe alternative would amount to a postmodern critique of science which would deny the possibility of objectivity and would reduce scientific truth to something like the theory's utility. My pragmatism doesn't go that far.
If math is game-like, the history of math should matter only as much as the history of any other game. You don't have to know the history of a game to play by its rules. However, history can always reveal what we prefer to forget and can deflate our preconceptions.
Hi,
ReplyDeleteIt might surprise you but the foundations of mathematics are somewhat shaky. In 1901 Bertrand Russell discovered a paradox in set theory. It was about whether the set of all sets which do not contain them-selves contains itself. This led to a crisis. This crisis was 'solved' by making mathematical reasoning more formal and complicated. Since that time mathematicians themselves avoid discussing the foundations of mathematics. There are realist, constructivist and other positions in mathematical philosophy.
There is the mathematical universe hypotheses by Max Tegmark. It is the most extreme form of mathematical realism. The hypotheses states that the physical universe is not merely described by mathematics, but in fact is mathematics. Not all mathematicians and physicist adhere to such extreme and rather absurd ideas.
On the opposite extreme you have the intuitionists. The intuitionists are mathematical anti-realists or constructivists. They believe that mathematics solely consists of mental constructions. In particular, mathematics in their eyes is not the discovery of fundamental principles that exist in some objective reality. As a consequence standard logic is invalid in their eyes. Intuitionist do not believe that the axiom of the excluded middle is valid for mathematical reasoning. Therefore a lot of proofs in standard mathematics are not valid according to intuitionists.
Indeed, I'm aware of the math foundation problem. I wrote something against a view that's similar to Tegmark's, called Pythagorean Illuminism (link below). I believe it's also called structuralism in metaphysics (only structures or abstract relations are ultimately real). Intuitionism derives from Kant, as I recall.
DeleteIn any case, this unresolved foundations problem feeds into my point in the above article, about the ethical lapse of presenting mathematical systems as facts to be memorized, while ignoring the philosophical questions. The philosophical underpinnings are ignored not so much because they're irrelevant to math itself, but because the underlying subject is in disarray. It's a coverup.
https://therabbitisin.com/the-pythagorean-illuminati-and-their-mathematical-reality-a207ee952d30?source=friends_link&sk=7f82d21816eba832d98bb1d3a2692e04