The concrete practice of intuitive mathematicians is to introduce the concept of "genus" instead of the concept of set in Cantor's sense. Then Brouwer introduced the concept of "selection sequence" and replaced the concept of rational Cauchy sequence in classical analysis with "rational number selection sequence", which was called "real number generator". Corresponding to the classical analysis, the real number is defined as the equivalent class of rational Cauchy sequence, and the single real number in the sense of construction is defined as the equivalent genus of real number generator. As mentioned above, there is no substantive difficulty in establishing the concept of configurable real numbers, because Cauchy-Weierstrass's whole limit theory is based on the concept of infinite potential. Therefore, the essentially intuitive mathematicians only restate Cauchy series under the requirement of feasibility.
The practice of modern construction mathematicians is that in order to construct a real number, a finite method must be given to transform every positive integer n into a rational number xn', so that x 1', x2', ... becomes a Cauchy sequence and converges to the real number to be constructed. We must also give a clear estimate of the convergence rate of this sequence. It can be seen that modern structural mathematics has gone beyond the concepts that seem to stifle intuitive mathematicians (such as selection sequence and genus concept).
The constructive proof of the basic theorem of algebra.
The classical expression of the basic theorem of algebra is that any non-constant polynomial f with complex coefficients has at least one complex root. ( 1)
The most famous traditional proof of (1) is that if f does not take zero value, the reciprocal of f can be found by using Liuwei theorem, and it is concluded that 1/f is constant, so f is constant, which proves this contradiction.
But structural mathematicians will argue that this proves not the basic theorem, but the following weak conclusion:
Polynomials that do not take zero on complex numbers are constants. (2)
At the same time, the above proof does not suggest the method of finding the root of polynomial.
Brouwer gave a constructive statement of the basic theorem of algebra:
There is a finite method for the non-constant polynomial f with arbitrary complex coefficients, and we can use it to calculate the root of f(3).
Now Brouwer's proof of the algebraic basic theorem of polynomials with the first term coefficient of 1 is given: he first proves that F can be assumed to be a positive polynomial on the Gaussian number field Q [I], and then selects the radius R to be large enough so that f(x) is dominated by its first term. Then, he constructed a theorem by using the fact that the number of circles surrounded by a circle F with O as the center and R as the radius is equal to the order of F. Finally, the complex root of F was constructed by Newton-Raphson iteration.
Comparing the constructive proof with the traditional proof, we can see that although Brouwer's proof is indeed longer than the proof using Liouville theorem, the constructive proof gives much more "information" than the traditional proof. For example, Brouwer's method can find the root of a polynomial whose first term coefficient is 1 for any given positive degree on a complex number. Especially, with his proof method, we can find the root of polynomial of order 100, while the traditional proof does not involve the method of finding the root at all.
Bishop wrote in the book: Every classical theorem poses a challenge: find a constructive statement and give it a constructive proof. But in fact, many classical theorems, such as Porzano-Weisstras theorem and Zon lemma, seem to have no constructive statements and proofs.