>There is no hope of accurately visualising [Weierstraß's] function
I think there is. Of course, you dont see all the detail at once, but you can imagine what it would look like and what you would see as you keep zooming in. In the same way, we might imagine the "movement" of a parametric curve by putting tiny arrowheads on the line, with the spacing indicating speed, and becoming smaller and more frequent as we zoom in. (The first is literally how I imagine it; for the curve, the speed annotation doesnt "look like" anything in particular, but is there anyway. I think we can natively add speed to still images.)
This might seem like a stretch, but I think its necessary even for more normal visualisations. After all, a true function graph is a line of zero thickness, which you couldnt see; you instead imagine it with a lineweight constant relative to your zoom level. And real physical objects can have infinite detail, too, which doesnt prevent visualsing them.
There are functions that go beyond even this, but theyre a lot uglier than that. For example: for numbers with finite binary expansion, its SHA hash, else 0.
That’s a really good point. I wonder, though, whether the distance between the visualisations and the actual concepts is enough to vindicate the Cartesian view against the Kantian one?
Great stuff Alex - this is right up my alley. I am currently working on Kant on this stuff, especially genetic definition being extended from geometry to metaphysics, and thinking that he got the similar idea of function from Euler rather than Spinoza (oddly, I've just been working out how Kant also has a non-physical notion of motion at play also). I take it that for Kant the appeal to intuition serves not for sensory visualization but the same idea that you are talking about, thought's perception, so maybe they are not separated on that point. For various reasons Kant thinks that it only kicks in when the functions are interpreted/applied by being given a determinate value, and then we see the universal in the particular; whereas maybe for Descartes, Spinoza, and Leibniz it is that the real objects of thought's perception are the more general uninterpreted formulae. Maybe the model of thought's perception is determined by what we think the real objects of thought are, such that the mode of thinking is apt for them....
Thank you! It makes sense that Kant wouldn’t have thought like 19-20C British Kantians, come to think of it.
That’s a very good point about what the objects of thought-perception are. Descartes and Spinoza both answer: the true essences of things, but that’s not so helpful, certainly not in the mathematical case. And I’ve been writing about how very bizarre Spinoza’s theory of essences is.
Nice article. It has left me with a slew of questions about how to think of this work.
What do you think the relationship is between the epistemology and ontology of Euclid's elements and the epistemology of Spinoza's Ethics?
You suggest that an untenable epistemology of the Elements, at least seen untenable in the Early Modern period, should be avoiding also for the Ethics. Is it untenable for the Ethics because Spinoza knew it was untenable for the Elements?
Why should we consider epistemologies of ancient geometry that came about after Spinoza was alive? Maybe you are using the various famous positions about the epistemology of geometry as just possible ways of understanding Spinoza. But I could also imagine that someone would argue that the formal similarity of the works does not mean anything for the epistemology and ontology because the subjects are so different.
Thank you. I can’t do this comment justice right now, except to say that you’re absolutely right: many people think the formal similarity between the Elements and the Ethics means very little.
I get that and I didn't mean to ask an annoyingly huge question. But you think that they are related somehow and I think you think that Spinoza thinks so too.
Fascinating discussion! It’s always remarkable to me to learn about how people in the past conceived of mathematical concepts. On a more general level, is there a book or two that you might recommend on history/philosophy of mathematics, for the mathematically-trained lay person?
>There is no hope of accurately visualising [Weierstraß's] function
I think there is. Of course, you dont see all the detail at once, but you can imagine what it would look like and what you would see as you keep zooming in. In the same way, we might imagine the "movement" of a parametric curve by putting tiny arrowheads on the line, with the spacing indicating speed, and becoming smaller and more frequent as we zoom in. (The first is literally how I imagine it; for the curve, the speed annotation doesnt "look like" anything in particular, but is there anyway. I think we can natively add speed to still images.)
This might seem like a stretch, but I think its necessary even for more normal visualisations. After all, a true function graph is a line of zero thickness, which you couldnt see; you instead imagine it with a lineweight constant relative to your zoom level. And real physical objects can have infinite detail, too, which doesnt prevent visualsing them.
There are functions that go beyond even this, but theyre a lot uglier than that. For example: for numbers with finite binary expansion, its SHA hash, else 0.
That’s a really good point. I wonder, though, whether the distance between the visualisations and the actual concepts is enough to vindicate the Cartesian view against the Kantian one?
Great stuff Alex - this is right up my alley. I am currently working on Kant on this stuff, especially genetic definition being extended from geometry to metaphysics, and thinking that he got the similar idea of function from Euler rather than Spinoza (oddly, I've just been working out how Kant also has a non-physical notion of motion at play also). I take it that for Kant the appeal to intuition serves not for sensory visualization but the same idea that you are talking about, thought's perception, so maybe they are not separated on that point. For various reasons Kant thinks that it only kicks in when the functions are interpreted/applied by being given a determinate value, and then we see the universal in the particular; whereas maybe for Descartes, Spinoza, and Leibniz it is that the real objects of thought's perception are the more general uninterpreted formulae. Maybe the model of thought's perception is determined by what we think the real objects of thought are, such that the mode of thinking is apt for them....
Thank you! It makes sense that Kant wouldn’t have thought like 19-20C British Kantians, come to think of it.
That’s a very good point about what the objects of thought-perception are. Descartes and Spinoza both answer: the true essences of things, but that’s not so helpful, certainly not in the mathematical case. And I’ve been writing about how very bizarre Spinoza’s theory of essences is.
Nice article. It has left me with a slew of questions about how to think of this work.
What do you think the relationship is between the epistemology and ontology of Euclid's elements and the epistemology of Spinoza's Ethics?
You suggest that an untenable epistemology of the Elements, at least seen untenable in the Early Modern period, should be avoiding also for the Ethics. Is it untenable for the Ethics because Spinoza knew it was untenable for the Elements?
Why should we consider epistemologies of ancient geometry that came about after Spinoza was alive? Maybe you are using the various famous positions about the epistemology of geometry as just possible ways of understanding Spinoza. But I could also imagine that someone would argue that the formal similarity of the works does not mean anything for the epistemology and ontology because the subjects are so different.
Thank you. I can’t do this comment justice right now, except to say that you’re absolutely right: many people think the formal similarity between the Elements and the Ethics means very little.
I get that and I didn't mean to ask an annoyingly huge question. But you think that they are related somehow and I think you think that Spinoza thinks so too.
Not annoying at all! I do think they’re related, but I need to think more about just how. It’s a good question!
Fascinating discussion! It’s always remarkable to me to learn about how people in the past conceived of mathematical concepts. On a more general level, is there a book or two that you might recommend on history/philosophy of mathematics, for the mathematically-trained lay person?