"18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent.
The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most.
The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.
who was part of the Bourbaki group and asked him why on earth he insisted on inflicting raw dry theory to the world with no intuition , when his day job involved drawing ideas all day long !
Edit: interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
The ‘make thier own mental models’ vs sharing/providing full information is difficult.
‘Make thier own model’ of the domain can lead to deeper understanding but takes time and may lead to different (possibly incorrect) understanding of the issues and complexities. If not reviewed with others.
Providing full information upfront to a person can be quicker but lead to a superficial knowledge.
I think that it comes down to whether that deeper knowledge is directly needed for the main task. Can I get by with an superficial (leaky) abstraction and concentrate on the main job.
> interestingly enough, one of my colleagues thinks very strongly that intuition should not be shared, and the path to intuition should be walked by everyone so that they ´ Make their own mental images ´. I guess that there’s a tradeoff between making things accessible, and deeply understood, but I don’t know what to make of his opinion.
If the objective is to advance mathematics instead of making it accessible, then this is a somewhat reasonable position. The mathematical statements that a person can come up with is often a direct product of their mental image. If everyone has the same image, everyone comes up with similar mathematical statements. For this reason you want to avoid that everyone has the same picture. Forcing everyone to start with a clean canvas increases the chance that there is diversity in the images. Maybe someone finds a new image, that leads to new mathematical statements. At least that's the idea. One could also argue that it just leads to blank canvases everywhere.
On foundations we believe in the reality of mathematics, but of course, when philosophers attack us with their paradoxes, we rush to hide behind formalism and say 'mathematics is just a combination of meaningless symbols,'... Finally we are left in peace to go back to our mathematics and do it as we have always done, with the feeling each mathematician has that he is working with something real. The sensation is probably an illusion, but it is very convenient.
Pretty much. According to someone on Stack Exchange, "The most serious difficulties with Euclid from the modern point of view is that he did not realize that an axiom was needed for congruence of triangles, Euclids proof by superposition is not considered as a valid proof." But making mistakes in a formal treatment of a subject does not negate the fact that it is a formal treatment of the subject IMHO, and AFAICR none of the theorems in Elements is wrong; i.e., Elements is formal enough to have avoided a mistake even though the axioms listed weren't all the axiom that are actually needed to support the theorems.
18th Century European math was much more potent than ancient Greek math, and although parts of it like algebra and geometry were, for a long time, most of it was not understood at a formal or rigorous level for a long time even if we accept the level of rigor found in Elements.
Isn't how formal or rigorous something is just a social convention? Grammer Nazi's used to make online speech be formal with perfect rigor. Isn't it all relative to what your society defines?
No. That's the colloquial definition of formal. In mathematics, the word formal refers to something more specific: one or more statements written using a set of symbols which have fully-defined rules for mechanically transforming them into another form.
A formal proof is then one which proceeds by a series of these mechanical steps beginning with one or more premises and ending with a conclusion (or goal).
If you're a formalist in philosophy of math, then math is neither true nor false, it's merely a bunch of meaningless symbols you transform via mechanical rules.
To an extent. A truly completely formal proof, as in symbol manipulation according to the rules of some formal system, no. It's valid or it isn't.
But no one actually works like this. There are varying degrees of "semiformality" and what is and isn't acceptable is ultimately a convention, and varies between subfields - but even the laxest mathematicians are still about as careful as the most rigorous physicists.
Elements is mostly formal, but it's also concrete and visual.
Euclid developed arithmetic and algebra through constructive geometry, which relies on our visual intuition to solve problems. Non-concrete problems were totally out of scope. Even curved surfaces (denying the parallel postulate) were byond Euclid.
Notably, Elements didn't have imaginary or transcendental numbers. Euclid made no attempt to unify line lengths and arc lengths, and had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers.
> Euclid […] had nothing to say about what fills the gaps between the algebraically (geometrically!) constructible numbers
Did he even know those gaps existed? Euclid lived around 300BC. The problem of squaring the circle had been proposed around two centuries before that (https://en.wikipedia.org/wiki/Anaxagoras#Mathematics), but I don’t think people even considered it to be impossible by that time.
"18th Century" mathematics was intiuitive and informal, to the point that it was inconsistent.
The 19th and 20th Centuries added rigor and formalism (and elitism) and devalued intuition, to the point that it begame uninterpretable to most.
The 21st Century's major contribution to mathematics (including YouTube! and conversational style writing) was to bring back intuition, with the backing of formal foundations.