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July 23 - August 26, 2025
Perhaps because human evolution took place in a challenging environment of essentially game-theoretic human choices, with consequences for strategic failures, we seem to have an innate capacity for the strategic reasoning underlying these complex, alternating-quantifier mathematical assertions. I find it remarkable how we can leverage our human experience in this way for mathematical insight.
In algebra, it is equal, equal, equal. But in analysis, it is less-than-or-equal, less-than-or-equal, less-than-or-equal.
that the continuous functions are those for which small changes in input cause only small changes in the output, and the derivative of a function at a point is obtained from the average rate of change of the function over increasingly tiny intervals surrounding that point.
In particular, with ordinary continuity, the value δ chosen for continuity at c can depend not only on ε, but also on c. With uniform continuity, in contrast, the quantity δ may depend only on ε. The same δ must work uniformly with every x and y (the number y in effect plays the role of c here).
Consider the function f(x) = x2, a simple parabola, on the domain of all real numbers. This function is continuous, to be sure, but it is not uniformly continuous on this domain, because it becomes as steep as one likes as one moves to large values of x.
A function f on the real numbers is continuous when whereas f is uniformly continuous when
Errett Bishop (1977) describes the epsilon-delta notion of continuity as “common sense,” and I am inclined to agree. To my way of thinking, however, the fact that minor syntactic variations in the continuity statement lead to vastly different concepts seems to underline a fundamental subtlety and fragility of the epsilon-delta account of continuity, and the delicate nature of the concept may indicate how greatly refined it is.
Suppose we have a sequence of continuous functions f0, f1, f2, …, and they happen to converge pointwise to a limit function fn(x) → f(x). Must the limit function also be continuous? Cauchy made a mistake about this, claiming that a convergent series of continuous functions is continuous. But this turns out to be incorrect. For a counterexample, consider the functions x, x2, x3, …on the unit interval, as pictured here in blue. These functions are each continuous, individually, but as the exponent grows, they become increasingly flat on most of the interval, spiking to 1 at the right. The limit
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In a sense, the least-upper-bound principle is to real analysis what the induction principle is to number theory.
Heine-Borel theorem, which asserts that the unit interval [0, 1] is compact.
Mathematicians are sometimes surprised by the principle of continuous induction—by the idea that there is a principle of induction on the real numbers using the real-number order—because there is a widely held view, or even an entrenched expectation, that induction is fundamentally discrete and sensible only with well-orders. Yet here we are with an induction principle on the real numbers based on the continuous order.
This function was defined by Cantor using his middle-thirds set, now known as the Cantor set, and serves as a counterexample to what might have been a certain natural extension of the fundamental theorem of calculus, since it is a continuous function from the unit interval to itself, which has a zero derivative at almost every point with respect to the Lebesgue measure, yet it rises from 0 to 1 during the interval. This is how the Devil ascends from 0 to 1 while remaining almost always motionless.
A curve is a continuous function from the one-dimensional unit interval into a space, such as the plane, a continuous function c: [0, 1] → ℝ2
function f is not determined merely by specifying its domain X, and the function values f(x) for each point in that domain x ∈ X. Rather, one must also specify what is called the codomain of the function, the space Y of intended target values for the function f: X → Y. The codomain is not the same as the range of the function, because not every y ∈ Y needs to be realized as a value f(x).
What an astounding development it must have been in 1961, when Abraham Robinson introduced his theory of nonstandard analysis. This theory, arising from ideas in mathematical logic and based on what are now called the hyperreal numbers ℝ*, provides a rigorous method of handling infinitesimals, having many parallels to the early work in calculus. I look upon this development as a kind of joke that mathematical reality has played on both the history and philosophy of mathematics.
Argue that f is continuous at c if and only if the defender has a winning strategy in the continuity game. Does your argument use the axiom of choice in order to prove the forward implication—that is, in order that the strategy would play a particular δ when faced with a particular ε? Can you eliminate the need for AC in this argument?
Design a uniform continuity game, between defender and challenger, and argue that f is uniformly continuous if and only if the defender has a winning strategy in this game.
In what way does fictionalism amount to a retreat from the object theory into the metatheory?
The arguments that we gave for Hilbert’s hotel show that the union of two countable sets is countable, since you can correspond one of them with even numbers and the other with odd numbers. The integers ℤ, for example, form a countable set, for they are the union of two countably infinite sets, the positive integers and the nonpositive integers.
two sets A and B are equinumerous if their elements can be placed in a one-to-one correspondence with each other. The set of people in this room, for example, is equinumerous with the set of noses, simply because we each have a nose and nobody has two
Cantor’s profound discovery was that the set of real numbers is uncountable. The real numbers cannot be placed in a one-to-one correspondence with the natural numbers. In particular, this shows that there are different sizes of infinity, for the infinity of the real numbers is a larger infinity than the infinity of the natural numbers.
in mathematics we seem to have many natural instances of countably infinite sets: The set of natural numbers ℕ; the set of integers ℤ; the set of rational numbers ℚ; the set of finite binary sequences 2<ℕ; the set of integer polynomials ℤ[x]. We seem also to have many natural instances of sets of size continuum: The set of real numbers ℝ; the set of complex numbers ℂ; Cantor space (infinite binary sequences) 2ℕ; the power set of the natural numbers P(ℕ); the space of continuous functions f: ℝ → ℝ. But we seem to have no sets provably of intermediate cardinality between ℕ and ℝ.
The continuum hypothesis (CH), formulated by Cantor, is the assertion that indeed there are no infinities between the countable and the continuum, between the natural numbers ℕ and the real numbers ℝ. Is it true? Is it refutable?
Gödel proved in 1938 that if the axioms of ZF set theory are consistent, then they are also consistent with the continuum hypothesis and the axiom of choice. Thus, one cannot expect to refute these principles, and in this sense, it is safe to assume that they are true. This result explains why Cantor was not able to find a definitive set of intermediate cardinality between ℕ and ℝ, since it is consistent with set theory that there is no such set.
The large cardinal axioms are the strongest-known axioms in mathematics.
Euclid gives us ten axioms—five essentially algebraic “common notions” and five essentially geometric “postulates”—and proceeds to build a monumental edifice, proving one statement and then another and another, each subsequently available for use in further proofs.
Felix Klein (1872), in his professorial dissertation, established what has become known as the Erlangen program (named after the University of Erlangen-Nürnberg, where he worked), by which we are to understand and analyze a geometric space by means of its group of transformations and related invariants.
Which real numbers are constructible? Is every real number constructible? The answer is no. As we have mentioned, there are only countably many constructible numbers, but by Cantor’s theorem, there are uncountably many real numbers. So there must be real numbers, points on the number line, that we cannot construct by straightedge and compass.
Doubling the cube For instance, we cannot construct the number , since this number has degree 3, because it is a root of the irreducible polynomial x3 − 2. This fact is pertinent for the confounding classical problem known as doubling the cube. The problem is, given a side length AB of a cube, to construct a side length CD for a cube of twice the volume. If one regards the original length AB as a unit length, then the first cube will have volume 1, and so the doubled cube will have volume 2, and consequently, side length . Ultimately, therefore, the problem of doubling the cube is exactly the
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How surprising it must have been, after two thousand years of straightedge-and-compass constructions, to learn that any point constructible by straightedge and compass can also be constructed with a compass alone. You do not need the straightedge. This is the content of the Mohr–Moscheroni theorem, proved by Georg Mohr in 1672 and rediscovered by Lorenzo Moscheroni in 1797.
The Poncelet-Steiner theorem, conjectured by Jean Victor Poncelet in 1822 and proved by Jakob Steiner in 1833, asserts that indeed, any point constructible by straightedge and compass can be constructed by straightedge alone, given such an initial circle.
Another geometric construction method arises from the sublime Japanese paper-folding art of origami. In skilled hands, a single square of paper is folded successively to produce surprising works of exquisite beauty. In addition to this artistic aspect of origami, the process of folding is inherently geometrical, and one may take origami folding as an alternative method of geometric construction.
Robert Geretschlager (1995) explains the equivalence of straightedge-and-compass construction with an axiomatic list of seven fundamental origami folds, making reductions in both directions. He also proves, however, that certain more complex origami folds go strictly beyond Euclidean constructibility. One kind of fold, for example, can be described as allowing one to fold along the common tangent to two parabolas, as specified by given focal points and vertex lines (one can in effect achieve this with actual paper). With the addition of this folding construction, it turns out, one can solve
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Geometry is a science which determines the properties of space synthetically, and yet a priori. What, then, must be our representation of space, in order that such knowledge of it may be possible? It must in its origin be intuition. (1781, B40)
The single figure [a triangle] which we draw is empirical, and yet it serves to express the concept, without impairing its universality. For in this empirical intuition we consider only the act whereby we construct the concept, and abstract from the many determinations (for instance, the magnitude of the sides and of the angles), which are quite indifferent, as not altering the concept “triangle.” (1781, B741–742)
According to Hume, our geometric insight—that two distinct lines can have no segment in common, for example—is not based upon any figure that we might draw or even imagine, as any such figure will some angle of inclination, whereas other lines might have a much smaller inclination. Algebra and arithmetic, meanwhile, lead to a more certain form of knowledge than geometry:
The early work tended to proceed from a perspective of Euclidean geometry as the one true geometry, with the goal of proving the parallel postulate from the other axioms. Eventually, this perspective began to shift. János Bolyai had aimed to prove the parallel postulate by contradiction. He accordingly assumed the negation of the parallel postulate and proceeded to make geometric deductions. But rather than finding the sought-after contradiction, what he found developing instead was a beautiful new geometric theory—a theory that led him to strange new geometric conclusions, many of which stood
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But one should not imagine non-Euclidean geometry only as a simulation within Euclidean geometry, for this would miss out on the view of non-Euclidean geometry as its own robust independent geometric concept; it has its own essential geometric nature, with its own geometric character that is fundamentally different from classical Euclidean geometry in important respects. One should imagine inhabiting hyperbolic space as it is, not merely as it is simulated inside Euclidean space. Conversely, one may also find simulations of Euclidean geometry inside non-Euclidean geometry. Imagine an alien
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A statement is true when the idea it expresses is the case. (There is, to be sure, a vibrant philosophical debate on the precise nature of truth, explicating what it might mean to have a truth predicate or to make a truth assertion.)
To prove a statement, in contrast, means to provide a reason why it is true, a reason for believing it, a justification that the statement is indeed the case.
Proof and truth, therefore, lie on opposite sides of the syntax-semantics divide, for at bottom, a proof is a kind of argument, a collection of assertions structured syntactically in some way, while the truth of an assertion is grounded in deeply semantic issues concerning the way things are.
On the semantic side, we are concerned with meaning, truth, existence, and validity.
Pressed for a precise definition, many mathematicians might have difficulty saying exactly what a proof is. In mathematical practice, a proof is any sufficiently detailed convincing mathematical argument that logically establishes the conclusion of a theorem from its premises. Successful proofs of new theorems often succeed by introducing new mathematical ideas or methods, and one gains mathematical insight from the proof beyond merely learning the truth of the theorem itself. We value such proofs because we might use the new methods to answer other questions that intrigue us. To prove a
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In Prover-Skeptic terms, we may say that a proof has been certified if it has been examined by a sufficient number of suitable Skeptics, and none of them has found errors in it.
But Thurston does not find mathematics unreliable. Rather, his point is that the reliability of mathematics arises not from formal proof, but from mathematical understanding: Mathematicians can and do fill in gaps, correct errors, and supply more detail and more careful scholarship when they are called on or motivated to do so. Our system is quite good at producing reliable theorems that can be solidly backed up. It is just that the reliability does not primarily come from mathematicians formally checking formal arguments; it comes from mathematicians thinking carefully and critically about
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Part of Shulman’s point is that formal mathematical statements may not actually capture the intended mathematical insight, and mathematicians accordingly may be less interested in the true/false or proved/refuted status of formal assertions, giving preference and priority to the mathematical insight and understanding that they might achieve without formalization.
Formal proof is exacting and precise, written in a formal language and according to rigid rules concerning the allowed axioms and rules of inference.
Alan Turing, who in 1936, while a student at Cambridge, wrote a groundbreaking paper that overcame Gödel’s diagonalization obstacle and provided a durable foundational concept of computability, while answering fundamental questions of Hilbert and Ackermann. His ideas ultimately led to the contemporary computer era.
arrive at his concept of computability, Turing reflected philosophically on what a person was doing when undertaking computation; he distilled computation to its essence.
which has the meaning: When in state q and reading symbol a, then change to state r, write symbol b, and move one cell in direction d. A Turing-machine program is simply a list of finitely many such instructions.
