I agree that mathematicians want understanding, but that's true of all scientists. Physicists or biologists, economists, or sociologists, psychologists or historians want to understand, but they want to understand different things, with different methods. If you want to explain what defines mathematics as a scientific discipline, you need to explain what it is that mathematicians want to understand, and with which methods. But maybe that's not what you want to do?
In the first place what I pointed out in the earlier post was not so much that mathematicians want understanding but rather that they say, repeatedly, that understanding is what they're after, in contrast to AI researchers, who focus on "reasoning." The choice of vocabulary is telling, even if the meaning is never made explicit.
In the second place, natural scientists can say that what they want to understand is the natural world; social scientists and historians acknowledge that "understanding" in their disciplines is loaded with methodological preconceptions and write long treatises about how it can be used to disguise implicit bias. Mathematicians, if they are honest, which they usually are, recognize that it is impossible to give a precise meaning to the word — even if they are platonists — and yet use the word as if it were self-explanatory — as it effectively is in communication within the discipline.
Here and elsewhere I'm pointing to characteristic and often unconscious habits of the profession that are overlooked or don't fit into attempts to define mathematics as a scientific discipline. And is it a scientific discipline? In my book I quote a number of well-known mathematicians who weren't altogether sure.
"And is it a scientific discipline? In my book I quote a number of well-known mathematicians who weren't altogether sure."
Fair enough, I remember Claire Voisin saying that for her, mathematics is closer to literature than to science (maybe not her exact words); and that her husband, who is an applied mathematicians saw things very differently.
But independently of the issue whether math is or isn't a part of science: what would you answer if someone asked you what it is that mathematicians try to understand, and with which methods?
I would have to write a book. But rather than wait they could read Bessis' book. And if they can't read French or Italian they can look forward to reading the English translation in the near future — Jordan Ellenberg tells me the rights have been purchased by Yale University Press.
As for the second part of your question, mathematicians historically take advantage of all available methods. When electronic computing became possible mathematicians used high-speed calculation to formulate conjectures, as I acknowledged in my article for Pour la Science (translated at https://siliconreckoner.substack.com/p/mathematicians-challenged-by-machines). New branches of dynamical systems theory ("chaos theory") developed through work with computer graphics. I have already predicted here that virtual reality will be adapted as a method for mathematical understanding.
But to talk about mathematics historically already amounts to something of a reconstruction. The idea that geometry and arithmetic were part of something called mathematics, in a way that music is not, evolved very gradually and not uniformly across cultures. And I've already written about historical disputes over the legitimacy of certain methods in mathematics (see https://siliconreckoner.substack.com/p/cantors-paradise-is-a-place-where?utm_source=%2Fsearch%2FNaples&utm_medium=reader2). So in fact books have been written about the second part of your question as well, in a way that's highly relevant to the first part.
The Greeks said it was turtles all the way down. Modern mathematicians say it's elephants. Vive les pachyderms.
I agree that mathematicians want understanding, but that's true of all scientists. Physicists or biologists, economists, or sociologists, psychologists or historians want to understand, but they want to understand different things, with different methods. If you want to explain what defines mathematics as a scientific discipline, you need to explain what it is that mathematicians want to understand, and with which methods. But maybe that's not what you want to do?
In the first place what I pointed out in the earlier post was not so much that mathematicians want understanding but rather that they say, repeatedly, that understanding is what they're after, in contrast to AI researchers, who focus on "reasoning." The choice of vocabulary is telling, even if the meaning is never made explicit.
In the second place, natural scientists can say that what they want to understand is the natural world; social scientists and historians acknowledge that "understanding" in their disciplines is loaded with methodological preconceptions and write long treatises about how it can be used to disguise implicit bias. Mathematicians, if they are honest, which they usually are, recognize that it is impossible to give a precise meaning to the word — even if they are platonists — and yet use the word as if it were self-explanatory — as it effectively is in communication within the discipline.
Here and elsewhere I'm pointing to characteristic and often unconscious habits of the profession that are overlooked or don't fit into attempts to define mathematics as a scientific discipline. And is it a scientific discipline? In my book I quote a number of well-known mathematicians who weren't altogether sure.
"And is it a scientific discipline? In my book I quote a number of well-known mathematicians who weren't altogether sure."
Fair enough, I remember Claire Voisin saying that for her, mathematics is closer to literature than to science (maybe not her exact words); and that her husband, who is an applied mathematicians saw things very differently.
But independently of the issue whether math is or isn't a part of science: what would you answer if someone asked you what it is that mathematicians try to understand, and with which methods?
I would have to write a book. But rather than wait they could read Bessis' book. And if they can't read French or Italian they can look forward to reading the English translation in the near future — Jordan Ellenberg tells me the rights have been purchased by Yale University Press.
As for the second part of your question, mathematicians historically take advantage of all available methods. When electronic computing became possible mathematicians used high-speed calculation to formulate conjectures, as I acknowledged in my article for Pour la Science (translated at https://siliconreckoner.substack.com/p/mathematicians-challenged-by-machines). New branches of dynamical systems theory ("chaos theory") developed through work with computer graphics. I have already predicted here that virtual reality will be adapted as a method for mathematical understanding.
But to talk about mathematics historically already amounts to something of a reconstruction. The idea that geometry and arithmetic were part of something called mathematics, in a way that music is not, evolved very gradually and not uniformly across cultures. And I've already written about historical disputes over the legitimacy of certain methods in mathematics (see https://siliconreckoner.substack.com/p/cantors-paradise-is-a-place-where?utm_source=%2Fsearch%2FNaples&utm_medium=reader2). So in fact books have been written about the second part of your question as well, in a way that's highly relevant to the first part.