He dislikes the idea of presenting infinity as a complete thing. And that one objection leads to a lot of different new directions to pursue. His rational trigonometry does trigonometry entirely without transcendental functions. It can involve square roots at times, but it is kept to a minimum. Everything else is entirely rational based. He has several videos that delve into the problems of each of the standard formulations of real numbers. He also argues for a more practical and computational version of the Fundamental Theorem of Algebra.
An interesting demonstration of the difficulty of real number arithmetic, relevant to some other comments here, is multiplying 1/9 by itself. For fractions, it is trivial as it is 1/81 and this can be converted into a repeating decimal, of course. But try multiplying the decimal form of 1/9 by itself. It is all 1's in the multiplication so it should be easy, right? If you write it down, essentially, the n+1th place is generated by summing n 1s. That is, it is .0123456789(10)(11)(12).... where I put in parentheses the sum of that columns digits. So one has to carry and as it goes further out, one is carrying over many digits; when out a trillion places, we are carrying across 12 places, which is larger than the repeating pattern. Just carrying that first 10 leads to .01234567900(11)(12)... And .012345679 is the basic pattern of 1/81 but it is hard to see feeling confident about that if one only had the infinite decimal to work with. The point is that something with a non-repeating pattern such as computing sqrt(2)pie seems difficult enough that it verges on the vacuous. He does point out the difference for his criticism applying to Pure Mathematics rather than Applied Mathematics. Approximations are fine for applications and what Wildberger is really saying is he wants a Pure Mathematics that really supports that explicitly by focusing on rational numbers as much as possible.
For example, he introduces differential calculus with polynomials by considering transforming p(x) to q(x) = p(x+r), collecting powers of x, and then translating back to p(x) = q(x-r) which is just replacing x with x-r. If expanded out, all the r's cancel, but if one leaves them and then truncates the different powers, one gets the different polynomial approximations. While neat in avoiding limts, the real nice thing is applying this technique to algebriac curves. For example, we can view the unit circle as the solution to 0= p(x,y) = x^2 + y^2 - 1. We can do the same trick above computing p(x+r, y+s), expand, and then retranslate and truncate. This can give us the approximations to the unit circle at a given point on the circle. This sidesteps having to compute the derivative of the square root function to get the tangent lines to the unit circle.
An example of an alternate work flow is multiplying two complex numbers on the unit circle. The traditional approach is to say "compute the angles and then add the angles". But the computing of the angles is impossibly hard to do in a precise fashion (approximate is fine, of course). But there is a perfectly fine accurate procedure. Take the points z and w on the unit circle and draw a line through them. Draw a parallel line through 1. The line will intersect the circle at z*w. As a quick example of this, if you multiply a+bi and -a+bi, this becomes -a^2 -b^2 = -1. Geometrically, the line through these two points is horizontal and the horizontal line through 1 intersects at -1. You can see that with angles, but it feels less intuitive to me that that is how it will work out.
Even the set of Natural Numbers being called infinite is something he questions. He used the term "unending" which I like as well. And by understanding that "most" natural numbers cannot be represented in this universe (assuming it is a finite universe), then it leads to questions such as what numbers can be represented? We have islands of simplicity such as 10^10^10^10^10^10^10 + 23. How dense are they in the larger numbers? Can we do anything useful with those islands? These questions are less prompted when we simply think of the natural numbers as this one big set of sameness. But if we demand that being able to do the computations is actually an important requirement, then we can investigate many more interesting ideas. And Wildberger's point is that this should be in the domain of Pure Mathematics with it being taught to future mathematicians instead of it being relegated to Applied Mathematics.
I have seen some of Wildberger's work, yes. My stance isn't as strong as him on some points - I certainly don't reject the reals outright or take issue with them. I think they're fascinating mathematical objects, I just reject choosing this particular mathematical object as what we mean by a "number".
He dislikes the idea of presenting infinity as a complete thing. And that one objection leads to a lot of different new directions to pursue. His rational trigonometry does trigonometry entirely without transcendental functions. It can involve square roots at times, but it is kept to a minimum. Everything else is entirely rational based. He has several videos that delve into the problems of each of the standard formulations of real numbers. He also argues for a more practical and computational version of the Fundamental Theorem of Algebra.
An interesting demonstration of the difficulty of real number arithmetic, relevant to some other comments here, is multiplying 1/9 by itself. For fractions, it is trivial as it is 1/81 and this can be converted into a repeating decimal, of course. But try multiplying the decimal form of 1/9 by itself. It is all 1's in the multiplication so it should be easy, right? If you write it down, essentially, the n+1th place is generated by summing n 1s. That is, it is .0123456789(10)(11)(12).... where I put in parentheses the sum of that columns digits. So one has to carry and as it goes further out, one is carrying over many digits; when out a trillion places, we are carrying across 12 places, which is larger than the repeating pattern. Just carrying that first 10 leads to .01234567900(11)(12)... And .012345679 is the basic pattern of 1/81 but it is hard to see feeling confident about that if one only had the infinite decimal to work with. The point is that something with a non-repeating pattern such as computing sqrt(2)pie seems difficult enough that it verges on the vacuous. He does point out the difference for his criticism applying to Pure Mathematics rather than Applied Mathematics. Approximations are fine for applications and what Wildberger is really saying is he wants a Pure Mathematics that really supports that explicitly by focusing on rational numbers as much as possible.
For example, he introduces differential calculus with polynomials by considering transforming p(x) to q(x) = p(x+r), collecting powers of x, and then translating back to p(x) = q(x-r) which is just replacing x with x-r. If expanded out, all the r's cancel, but if one leaves them and then truncates the different powers, one gets the different polynomial approximations. While neat in avoiding limts, the real nice thing is applying this technique to algebriac curves. For example, we can view the unit circle as the solution to 0= p(x,y) = x^2 + y^2 - 1. We can do the same trick above computing p(x+r, y+s), expand, and then retranslate and truncate. This can give us the approximations to the unit circle at a given point on the circle. This sidesteps having to compute the derivative of the square root function to get the tangent lines to the unit circle.
An example of an alternate work flow is multiplying two complex numbers on the unit circle. The traditional approach is to say "compute the angles and then add the angles". But the computing of the angles is impossibly hard to do in a precise fashion (approximate is fine, of course). But there is a perfectly fine accurate procedure. Take the points z and w on the unit circle and draw a line through them. Draw a parallel line through 1. The line will intersect the circle at z*w. As a quick example of this, if you multiply a+bi and -a+bi, this becomes -a^2 -b^2 = -1. Geometrically, the line through these two points is horizontal and the horizontal line through 1 intersects at -1. You can see that with angles, but it feels less intuitive to me that that is how it will work out.
Even the set of Natural Numbers being called infinite is something he questions. He used the term "unending" which I like as well. And by understanding that "most" natural numbers cannot be represented in this universe (assuming it is a finite universe), then it leads to questions such as what numbers can be represented? We have islands of simplicity such as 10^10^10^10^10^10^10 + 23. How dense are they in the larger numbers? Can we do anything useful with those islands? These questions are less prompted when we simply think of the natural numbers as this one big set of sameness. But if we demand that being able to do the computations is actually an important requirement, then we can investigate many more interesting ideas. And Wildberger's point is that this should be in the domain of Pure Mathematics with it being taught to future mathematicians instead of it being relegated to Applied Mathematics.