I think it’s dangerously easy to get lost in contemplating the “true nature” of mathematics. Math gives some very strong subjective impressions about its nature, such as that its truths are eternal and universal. And like any strong subjective impression, this feeling lends itself to the mind projection fallacy. That isn’t to say that these impressions are wrong, but that even if and when they’re right we tend to trust them for the wrong reasons. And, thus, we don’t notice when those impressions really are wrong.
I don’t claim to have a complete answer to this conundrum. I do, however, see many key pieces that seem to go a long way to dispelling this confusion.
First, as just an empirical observation, it seems that mathematical objects are reifications. If you watch little kids learning how to add, they go through a predictable sequence of development. First they count out objects, put them together, and then count the whole collection:
Here’s one, two, three, four, five. And here’s one, two three. That’s, um, one, two three, four, five, six, seven, eight!
After a while of doing this—and “a while” can be a surprisingly long time—they realize that they can compress the first quantity by jumping to the end:
Here’s one, two, three, four, five. And here’s one, two, three. So that one is five, and then six, seven, eight!
After doing this for a while, they start to think about the process of counting “one, two, three, four, five” in terms of the final state (“five”). This lets them manipulate the process as an object.
Ah, but once this happens, this triggers the parietal cortex to apply the idea of object permanence to “five”. Suddenly there’s this sense that “five” is there even when the child doesn’t see it. And behold, the eternal entity 5 as a mathematical object is born in the child’s mind.
We’re so used to thinking this way that we don’t really see it in ourselves anymore. But it’s still there and shows up in oddities in how we think about even basic math. For instance, what does it mean to add 5 and 3? You put 5 and 3 together… somehow… and suddenly an 8 pops out of nowhere. What happened to 5 and 3? If we pause and think about it, we can make sense of it with visualizations or other mental tricks, but there’s this slight-of-hand we do to ourselves before we pause to think about that process in which we treat 5 and 3 as objects but don’t think to ask how they combine to create the object 8. They just “merge” somehow, and a whole entity—not just a composite, but something thought of as an object—appears.
What really seems to be going on is that we have a built-in capacity from birth to subitize quantities less than 4, and then we build on those in order to perform rituals of ordered synchronization of movement and speech. This is why children find it so important to actually touch the objects they’re counting as they’re speaking the magic words “one, two, three...”. This is also the best current explanation I’m aware of for the seemingly unrelated symptoms of Gerstmann syndrome: when people can no longer distinguish between their fingers, they don’t have the proprioceptive bind to the verbal counting ritual that’s needed to understand numbers greater than three. After a while, the parietal cortex provides a shortcut to dealing with familiar processes by treating the end-state as an object that can stand in for having done the process.
So it might very well be that mathematical truths are not so much “encoded in reality” as that our descriptions of these truths are embodied characterizations of the world. It might be that they seem eternal as an accidental side-effect of our using our parietal cortices to simplify computations. They’re seemingly universal because the universe we’re capable of experiencing is the one in which our bodies work—and notice that in places where our bodies do not work normally (e.g. dreams), mathematics doesn’t seem to work quite so well either.
I’m taking the time to point this out because it’s way too easy to waste tremendous amounts of time wondering about where mathematics “is”. Even if there’s some objective essence of math that is somehow lurking within and guiding the physical world unseen, the question remains as to how we, with our physical brains and bodies, can come to understand those truths. We can’t understand some semi-Platonic Idea in its raw form; we have to use the material tools from which we are constructed in order to model those ideas. Therefore, the only mathematics we can ever possibly know about is that which is governed by the structure of our minds. This makes the origin of mathematics really a question of psychology, not philosophy—which is thankful because psychology has the blessing of being empirical!
I think it’s dangerously easy to get lost in contemplating the “true nature” of mathematics. Math gives some very strong subjective impressions about its nature, such as that its truths are eternal and universal. And like any strong subjective impression, this feeling lends itself to the mind projection fallacy. That isn’t to say that these impressions are wrong, but that even if and when they’re right we tend to trust them for the wrong reasons. And, thus, we don’t notice when those impressions really are wrong.
I don’t claim to have a complete answer to this conundrum. I do, however, see many key pieces that seem to go a long way to dispelling this confusion.
First, as just an empirical observation, it seems that mathematical objects are reifications. If you watch little kids learning how to add, they go through a predictable sequence of development. First they count out objects, put them together, and then count the whole collection:
After a while of doing this—and “a while” can be a surprisingly long time—they realize that they can compress the first quantity by jumping to the end:
After doing this for a while, they start to think about the process of counting “one, two, three, four, five” in terms of the final state (“five”). This lets them manipulate the process as an object.
Ah, but once this happens, this triggers the parietal cortex to apply the idea of object permanence to “five”. Suddenly there’s this sense that “five” is there even when the child doesn’t see it. And behold, the eternal entity 5 as a mathematical object is born in the child’s mind.
We’re so used to thinking this way that we don’t really see it in ourselves anymore. But it’s still there and shows up in oddities in how we think about even basic math. For instance, what does it mean to add 5 and 3? You put 5 and 3 together… somehow… and suddenly an 8 pops out of nowhere. What happened to 5 and 3? If we pause and think about it, we can make sense of it with visualizations or other mental tricks, but there’s this slight-of-hand we do to ourselves before we pause to think about that process in which we treat 5 and 3 as objects but don’t think to ask how they combine to create the object 8. They just “merge” somehow, and a whole entity—not just a composite, but something thought of as an object—appears.
What really seems to be going on is that we have a built-in capacity from birth to subitize quantities less than 4, and then we build on those in order to perform rituals of ordered synchronization of movement and speech. This is why children find it so important to actually touch the objects they’re counting as they’re speaking the magic words “one, two, three...”. This is also the best current explanation I’m aware of for the seemingly unrelated symptoms of Gerstmann syndrome: when people can no longer distinguish between their fingers, they don’t have the proprioceptive bind to the verbal counting ritual that’s needed to understand numbers greater than three. After a while, the parietal cortex provides a shortcut to dealing with familiar processes by treating the end-state as an object that can stand in for having done the process.
So it might very well be that mathematical truths are not so much “encoded in reality” as that our descriptions of these truths are embodied characterizations of the world. It might be that they seem eternal as an accidental side-effect of our using our parietal cortices to simplify computations. They’re seemingly universal because the universe we’re capable of experiencing is the one in which our bodies work—and notice that in places where our bodies do not work normally (e.g. dreams), mathematics doesn’t seem to work quite so well either.
I’m taking the time to point this out because it’s way too easy to waste tremendous amounts of time wondering about where mathematics “is”. Even if there’s some objective essence of math that is somehow lurking within and guiding the physical world unseen, the question remains as to how we, with our physical brains and bodies, can come to understand those truths. We can’t understand some semi-Platonic Idea in its raw form; we have to use the material tools from which we are constructed in order to model those ideas. Therefore, the only mathematics we can ever possibly know about is that which is governed by the structure of our minds. This makes the origin of mathematics really a question of psychology, not philosophy—which is thankful because psychology has the blessing of being empirical!