How did the shift from the times of Thales, Pythagoras, and even Euclid, where mathematical objects were found, exhibited, or constructed from given entities, to modern mathematics occur? In modern mathematics, entities no longer need to be constructed or computed in order to be named and manipulated; they simply need to exist. What was the transformative change that enabled this level of abstraction to be embraced and widely adopted, namely the distinction between mere existence and actual realization?
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2$\begingroup$ Wikipedia makes the following claim: "Until the end of 19th century, all mathematical proofs were essentially constructive. The first non-constructive constructions appeared with Georg Cantor’s theory of infinite sets, and the formal definition of real numbers." It is not exactly clear what is meant here by non-constructive construction and the words proofs were essentially constructive leave some wiggle room. Certainly the constructivism of Brouwer intuitionalism can be seen as a reaction to Cantor and others. $\endgroup$– nwrCommented Jun 7, 2023 at 23:02
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10$\begingroup$ @nwr In this case, Wikipedia is very wrong. Reductios, hence non-constructive proofs, were used already by Pythagoreans and are common in Euclid's Elements, starting with I.6. They are probably confusing constructiveness of proofs with constructiveness of objects they are about. Euclid did mostly confine his theorems to objects he could construct with straightedge and compass, as did other ancient mathematicians. But even that was not universal. Some irrationals were featured in "exhaustion" proofs without "construction". $\endgroup$– ConifoldCommented Jun 8, 2023 at 0:27
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$\begingroup$ I strongly concur with @Conifold's comment, based on my own observations. $\endgroup$– paul garrettCommented Jun 9, 2023 at 2:05
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$\begingroup$ "entities no longer need to be constructed" does not mean that in modern mathematics, entities are never constructed either. Rather than a "turn" isn't it a case of relying on some type of proof more generously than before? $\endgroup$– FrankCommented Jun 9, 2023 at 15:34
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$\begingroup$ "[E]ntities no longer need to be constructed or computed in order to be named and manipulated; they simply need to exist" – existence is not the only thing to prove, though; the other thing is uniqueness. That is, you need to prove that something exists and is unique – hence it is well-defined, it can be manipulated, you can explore its properties etc., without an explicit construction. $\endgroup$– printfCommented Jun 9, 2023 at 19:44
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4 Answers
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TL;DR It is important to distinguish different senses of "constructiveness", the technical one, as in constructivism, and the one of approach to mathematics as primarily problem-solving. They followed two different lines of development that are often confused (this is where Wikipedia goes wrong), and only the second underwent a shift in 19th century. The first one, on the other hand, shows gradual expansion of construction tools since Euclid and is continuous through the 19th century.
The problem-solving ("constructive" in a different sense) attitude also goes back to Euclid and dominated the field up to the end of 18th century. It gave way to the more meta-theoretical "conceptual" attitude even before Cantor, although he championed it along with Dedekind and Weierstrass, and it was canonized by Hilbert, Zermelo, etc., in the modern axiomatic method. The rise of impossibility and "pure existence" results to prominence accompanied it, but it was not directly related to non-constructivism in the technical sense.
Here is Lützen describing the shift in Why was Wantzel overlooked for a century? to explain the fate of Wantzel's now famous impossibility proofs:
"Indeed, during the 18th century, mathematics was a constructive enterprise consisting of finding solutions to problems, and this view of mathematics was still shared by most of Wantzel’s contemporaries... In the history of the quintic, Abel explicitly tried to change this view of impossibility results. As emphasized by Sørensen, Abel suggested that one ought to rephrase the problem as a problem that always has an answer.
Instead of asking for a solution by radicals, one should first ask if the quintic has a solution of this kind, and only if the answer turns out to be yes would one proceed to the question of finding such a solution constructively. By formulating the existence question as a mathematical problem, Abel had allowed the impossibility result to be a solution of a problem, not just a statement about the impossibility of finding an answer to a problem. Thus impossibility statements became proper theorems in mathematics. Several younger mathematicians continued along the same line as Abel. Evariste Galois is the most obvious example, but also Joseph Liouville’s investigations of integrability of functions and differential equations in finite form follow this lead.
These new tendencies relative to impossibility results went hand in hand with a new focus on existence theorems, and can be viewed as two sides of the same coin. In particular, Cauchy insisted that before finding the sum of an infinite series or the solution of a differential equation, one should prove their existence. And both trends were parts of a change from a constructive to a conceptual paradigm in mathematics. Another instance of this change was the emergence in the 1830s of Sturm–Liouville theory where the (qualitative) properties of solutions to certain differential equations were investigated without solving the equation in analytic form."
The above does not apply to constructive/non-constructive in the technical or even more relaxed sense, which does not necessarily track problem-solving vs theory-building. If we go by the modern definition of constructive proof, which discards proofs by contradiction = reductios ad absurdum, then the non-constructive goes all the way back to ancient Greeks. Euclid employs reductios not only in incommensurability proofs, but even in the most elementary proposition I.6, the isosceles triangle theorem, which surely goes back to Pythagoreans if not Thales.
And even if we focus just on straightedge and compass constructions, what changes over the centuries is mostly the range of construction tools. Euclid's constructionism is not that different in spirit from modern one, where axioms of set theory are used as construction postulates. Most ZFC axioms (specification, pairing, union, powerset, replacement, choice) are even formulated as such postulates. Even employment of the axiom of choice to produce sets to specifications, the very paradigm of non-constructive, is unabashedly called "construction", as in Vitali construction of Lebesgue non-measurable sets, etc. See Rodin's Doing and Showing on structural analogies between Euclidean and set-theoretic constructionism.
Already in Euclid, there are tricky cases where he implicitly relaxes his straightedge and compass strictures. The proof of proposition XII.2, which applies the so-called "method of exhaustion" to the circle quadrature, essentially presents a re-construction of the circle by transcendental means (as a limit of inscribed polygons), and then uses a reductio to prove its equivalence to the elementary compass construction. Archimedes used the "method of exhaustion" on conic sections and even solids, and others happily intersected transcendental curves (quadratrix, spiral) with lines. Unlike the modest (and omitted) completeness assumptions needed in the Elements, these "constructions" implicitly rely on strong forms of completeness for the continuum. See Bos, Construction of equations, 1637-1750 on how transcendental methods were gradually accepted in geometry.
By the 17th century, algebra and number theory also matured, so "constructions" involved infinite decimals (Stevin) and then infinite series and products (Wallis, Newton, Leibniz). Algebraic analysis, where algebra was transferred to functional infinite sums and products, ruled the 18th century. Euler's experiments with infinite "constructions" are well-known, while Fourier's "constructions" with trigonometric series were even seen as "revolutionary" at the turn of 19th century. Many more functions were thereby made "constructible", see Bressoud, Radical Approach to Real Analysis. Both lines of development paved the way to Cantor's innovations, his work on sets was motivated by Fourier series and influenced by the emerging conceptual attitude.
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4$\begingroup$ I recall skimming through a paper on the history of Galois theory that had significant discussions of how research trends leading to its development (from the late 1700s through the mid-to-late 1800s) led to changes in the acceptance of certain types of existence proofs, including specifics about which mathematicians heavily criticized the new methods and which mathematicians advocated acceptance of the new methods. The paper might be The development of Galois Theory from Lagrange to Artin by Kiernan (1971), but I don't have access. $\endgroup$ Commented Jun 8, 2023 at 9:27
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$\begingroup$ @DaveLRenfro I looked through the paper, and it does seem like what you describe. There is an irony in Gauss giving 'pure existence' proof of the fundamental theorem of algebra, but disregarding the idea when it came to the impossibility side of inscribing regular polygons, which implies, in particular, impossibility of trisection. Lützen says "there is little doubt that Gauss knew how to prove it", but he "did not attach much importance to it". Kiernan also names Lagrange as a precursor for "dimly" seeing ""known" as an existence property, and not necessarily a constructive property". $\endgroup$– ConifoldCommented Jun 9, 2023 at 10:53
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$\begingroup$ It turns out that I actually have a (appropriately magnified) photocopy of Kiernan's 1971 paper, made perhaps 15-20 years ago. In fact, it's in its own 3-ring binder (hole puncher used on the photocopied pages) on one of my bookshelves. $\endgroup$ Commented Jul 1, 2023 at 10:34
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This change apparently occurred in the middle of 19th century.
Let me cite Carl Ludwig Siegel about it:
It is completely clear to me which conditions caused the gradual decadence of mathematics, from its high level some 100 years ago, down to the present hopeless nadir. Degeneration of mathematics begins with the ideas of Riemann, Dedekind and Cantor which progressively repressed the reliable genius of Euler, Lagrange and Gauss.
and he continues:
Through the influence of textbooks like those of Hasse, Schreier and van der Waerden, the new generation was seriously harmed, and the work of Bourbaki finally dealt the fatal blow.
C. L. Siegel, Letter to A. Weil, 1959. German original is cited in the lecture of H. Grauert, in: Mathematics and Theoretical Physics, 1993, ed.: Minaketan Behara, Rudolf Fritsch, Rubens G. Lintz, W. de Gruyter, 1995.
Similar views about the influence of Riemann on mathematics were shared by many mathematicians of that time, for example by Chebyshev and Lyapunov.
Bourbaki also acknowledges the role of Riemann in reshaping of mathematics:
Riemann has to be considered the founder of topology as well as of most of the branches of modern mathematics...
N. Bourbaki, Topology I, Historical essay.
answered Jun 8, 2023 at 6:37
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6$\begingroup$ "Siegel" is not enough for the layman to know what you are referring to. Can you give the name of the work? Thx $\endgroup$ Commented Jun 8, 2023 at 7:57
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3$\begingroup$ Riemann repressed the genius of Gauss? I'm writing this one down :-) $\endgroup$– StefCommented Jun 8, 2023 at 10:21
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2$\begingroup$ According to the OP's web-site (C. L. Siegel, Letter to A. Weil, 1959. German original is cited in the lecture of H. Grauert, in: Mathematics and Theoretical Physics, ch 1993, ed.: Minaketan Behara, Rudolf Fritsch, Rubens G. Lintz, W. de Gruyter, 1995.) @PrimeMover $\endgroup$– mdeweyCommented Jun 8, 2023 at 16:12
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2$\begingroup$ If he had been in a position to post that up on social media in our own era he'd be banned from the internet for eternity. $\endgroup$ Commented Jun 8, 2023 at 19:29
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3$\begingroup$ This is hilarious! I had not been aware of this letter of Siegel to Weil! :) I had been aware of Siegel's complaint about S. Lang's "Diophantine Equations", that "the pigs have gotten into the garden"... :) $\endgroup$ Commented Jun 8, 2023 at 20:19
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The change arguably occurred not in the middle of 19th century but rather already in the 17th, with Leibniz's work. Leibniz clearly realized that some of the mathematical entities he was dealing with, such as infinitesimals and "bounded infinite lines" (infinita terminata in Latin) were not constructed but rather postulated. Their mathematical existence was "justified" by a Law of Continuity. He thought of them as useful fictions that have no counterpart in the phenomenal realm (his interpretation of the physical realm). A number of recent articles have dealt with the issue; see here.
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Let me mention a few turning points in this transition from the mathematics of ancient Greece to late modernity circa 2000. None of these insights are due to me. I learned it from many sources over the years, especially from the writings of John Stillwell, Jeremy Gray, Gerhard Wanner, and many other historians of mathematics.
Axiomatics I. Early proofs (2500 BC) of the Thales theorem used all lengths are (rational) numbers. The discovery of irrational numbers via Pythagoras theorem led to the first axiomatization in Mathematics: replacing numbers with lengths obtained by ruler and compass construction (first three axioms of Euclid). There were still numbers like $2^{1/3}$ and $\pi$, etc. that could not be covered here. But the ancients didn't know this.
Algebra Nova. Skipping the development of decimal arithmetic in India and other significant non-Westen contributions to Mathematics, the next major advance came from linguistics: the development of algebraic notation around 1600. Writings can be classified into three basic types: pictographic (e.g., Egyptian Hieroglyphics), Syllabic writing (Cuneiform), and Alphabetic writing (modern English). The latter two are symbolic.
Arithmetica Infinitorum. Calculus ideas developed between 1600 and 1700, culminating in Leibniz's symbolism. Mathematicians were aware of the shortcomings of their approach in terms of rigor (compared with Euclid). Still, the flood of results prevented them from rewriting everything in terms of Euclidean geometry.
1700 - 1750, Introductio in Analysis Infinitorum. The development of algebraic analysis and the theory of trigonometric functions. Again, the mathematicians were aware of the lack of foundations.
1750 - 1800, Algebra Nova II: Lagrange resolvents and solving equations of higher degree. Some debates: logarithms of negative and complex numbers, and several analytical questions surrounding Fourier series.
1801, Disquisitiones Arithmaticae inaugurated two new subjects: number theory and, shyly, a theory of functions (complex variable).
1800 - 1900. In this period, a new crisis arose in the foundations of mathematics that could not ignored using symbols: the discovery of non-Euclidean geometry, continuous functions without derivatives, etc. Geometry could not be trusted as intuitive anymore. Dedekind, Cantor ... were moving towards "arithmetization".
Circa 1900, a crisis in communication. Higher-dimensional geometry, starting, say, with Riemann's works with a mix of intuition and logic, was becoming incomprehensible. The writings were becoming mutually unintelligible. Thus, Hilbert posed his famous problems: Is mathematics complete and consistent? Is mathematics decidable?
1900-1950, Axiomatics II. Emil Post, Kurt Godel, Alan Turing, etc., provided answers to Hilbert's questions. Together with the developments in set theory and logic, this led to an acceptable formal system for Mathematics (e.g., Zermelo-Frankel set theory).
1950+, Bourbaki++. The main novelty in this period is defining things before proving theorems, which was, to some extent, impossible before axiomatization II.
Clearly, many important points are left out.
Siegel's comment on the degeneration of mathematics is not controversial: Mathematicians today agree that he was off the mark. In his (in)famous letter to Mordell, he writes, "I am afraid that mathematics will perish before the end of this century if the present trend for senseless abstraction - as I call it: theory of the empty set - cannot be blocked up."
In other words, Sigel's question is the following: what mathematics of our time (2000+) is comparable to the discovery of algebra nova (~1600), calculus (~1700), complex function theory or Fourier analysis (19th century), Lie groups and representation theory or Theory of computation (20th century)? In other words, what is a new mathematical theory from the 2000s that is simple and universally useful? Maybe we will know in 2100?
