Mathematician Norman Wildberger presents an educational program for a wide audience as Insights into Mathematics connecting to the principles I have followed in Body and Soul and Leibniz World of Math.
A basic concern of Wildberger is how to cope with real numbers underlying analysis or calculus, geometry, algebra and topology, since they appear to require working with aspects of infinities coming with difficulties, which have never been properly resolved, like computing with decimal expansions with infinitely many decimals and no last decimal to start a multiplication. Or the idea of an infinity of real numbers beyond countability.
I share the critique of Wildberger but I take a step further towards a resolution in terms of finite precision computation, which can be seen to be the view of an applied mathematician or engineer. In practice decimal expansions with a finite number of decimals are enough to represent the world and every representation can be supplied with a measure of quality as a certain number of decimals as a certain finite precision. This offers a foundation of mathematics without infinities in the spirit of Aristotle with infinities as never attained potentials representing "modes of speaking" rather than effective realities.
In particular the central concept of "continuum" takes the form of a computational mesh of certain mesh size or finite precision. With this view a "continuum" has no smallest scale yet is finite and there is a hierarchy of continua with variable mesh size.
The difficulty of infinities comes from an idea of exact physical laws and exact solutions to mathematical equations like $x^2=2$ expressed in terms of symbols like $\sqrt{2}$ and $\pi$. But this can be asking for too much, even if it is tempting, and so lead to complications which have to be hidden under the rug creating confusion for students.
A more down-to-earth approach is then to give up exactness and be happy with finite precision not asking for infinities.
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