"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.
I was schooled in abstract 20th century math - indeed YouTube is the opposite, and it’s a good thing.
One of my math teachers was once talking to Jean Dieudonné https://en.wikipedia.org/wiki/Jean_Dieudonn%C3%A9
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.
‘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.
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.
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.
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).
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.
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.
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.
Elements has pi.*
*It proves the ratio of a circle's area to the square of its diameter is constant.