Tag: mathematician

We Are Finite. Does This Affect What We Can Know?

§2. What Can We Know?

The theme of this section is that We Are Finite: while our knowledge of the number 3 is intimately associated with experiences of some kind or other of triples of one or another sort, surely nothing like that is the case for our knowledge of the number ω or, for that matter, the number 10101010.

To avoid misunderstanding, let me emphasize that the issue being addressed is not “How can we have knowledge of mathematical objects despite their abstractness?” but rather “How can we have knowledge of infinite mathematical objects despite their huge size and consequent remoteness from experience?” Here is what I mean by remoteness: Not only do we not have experience of infinite mathematical objects, but we do not have any experience of anything suitably like them. The number 2 bears some relation to pairs of objects. The points and lines of Euclidean geometry bear some relation to pencil points and lines. But there does not seem to be anything suitable to play any analogous role for infinite mathematical objects.

Of course infinite mathematical objects are abstract. The issue of remoteness is an addition to that of abstractness. Both are apparent epistemological difficulties caused by the distance of mathematical objects from experience.

But if we divide the problem of the abstract into two parts, the problem of the finite and the problem of the infinite, it becomes clear that the two have quite distinct features.

The problem of the abstract, in Paul Benacerraf’s words, is this [Ben73, p. 409]: “the concept of mathematical truth, as explicated, must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have. An acceptable semantics for mathematics must fit an acceptable epistemology?” For Benacerraf, an acceptable semantics is a Platonist one, and so his problem is that of the difficulties involved in giving an acceptable account of knowledge of abstract objects. Benacerraf concentrated on a weaker version of the problem: not on accounting for the mathematical knowledge that we have but on accounting for how we can have any mathematical knowledge whatever. “The minimal requirement, then, is that a satisfactory account of mathematical truth must be consistent with the possibility that some such truths be knowable” [Ben73, p. 409]. Philosophers have tended to concentrate on the minimal requirement and to offer up solutions to the problem of the finite to solve it.

Even those skeptical about the existence of any abstract mathematical objects at all want to endorse the truism ‘2 + 2 = 4’ in some way or other that acknowledges that it is better than ‘2 + 2 = 5’ even though they may not grant that it is true. Even without an abstract number 2, one is still faced with explaining the general fact that the members of two nonoverlapping pairs form a quadruple.

Whatever one’s views about mathematical objects, it is necessary to make sense of our counting, computing, and bookkeeping activities. Skepticism about small finite mathematical objects—in particular small natural numbers is just not doubt about the acceptability in some form or other of many of the putative facts about them. Moreover, there are many stories one could tell about a source for genuine knowledge concerning some finite mathematical objects, including, as the most trivial special case, knowledge of their existence. Take, for example, small natural numbers. Various explanations of our knowledge about them might invoke the experience of time, the experience of bunches of physical objects or of patterns exhibited by them, or the sequencing of words in sentences. Other explanations might rely on the exigencies of the construction of theories of the physical world. We are faced with too many ways of accounting for our knowledge of small finite mathematical objects, not too few. For example, Parsons [Par80] showed how to account for such knowledge on the basis of our linguistic capacities, while Maddy [Mad90] showed how to do it on the basis of experiences with medium-sized physical objects.

Benacerraf’s minimal requirement can surely be met with an acceptable solution to the problem of the finite—though I make no claim to know what the actual solution is. That is a question whose answer involves detailed psychological information about how people typically actually acquire knowledge of small finite mathematical objects, and we do not yet have sufficiently detailed information to answer it. Acquisition of the number concept does, however, involve both linguistic components—learning to count aloud—and experience of medium-sized physical objects—counting them using the spoken number sequence. It is therefore likely that the actual solution involves components of both the one proposed by Parsons and the one proposed by Maddy—and probably other components as well.

In sharp contrast to the situation about ‘2 + 2 = 4’, many of those who are skeptical about the existence of infinite combinatorial collections would want to doubt or deny the Axiom of Choice—not only its truth, but its acceptability in any form whatever. General facts about the infinite are not robust in the same way that the facts of counting, computing, and bookkeeping are. Moreover, it is not at all clear what we can fall back on as a source of mathematical knowledge concerning the infinite—what can play the role that bunches and sequences of moments, objects, or words seem so well suited to play for small finite mathematical objects. It is that lack that raises the problem posed by the remoteness of the infinite: it seems that we cannot have grounds to know what we find we actually do know about the infinite.

In Chapter VIII I shall show that as a matter of fact the combinatorial infinite is not remote—it has pretty much the same kinds of ties to experience as do small natural numbers. (That may be a bit misleading see Chapter VIII for a more careful formulation.) That solves the problem of the remoteness of the infinite philosophical problems concerning infinite mathematical objects become just like the familiar ones concerning finite mathematical objects. That is important because the problems concerning finite mathematical objects are not skeptical ones—the genuine doubts about the acceptability of our theory of the infinite are refuted. It also provides the essential missing ingredient for an explanation of the grounds on which mathematicians are entitled to make claims of self-evidence. But before presenting the solution, it is necessary to become clearer on the nature of the problem.

The two Benacerrafian problems—of the finite and the infinite—are both important, and every adequate philosophy of mathematics must be compatible with solutions to them. Nonetheless, the problem of the infinite deserves special emphasis because it is in danger of being lost as the result of the huge amount of attention being devoted to the Benacerrafian problem of the abstract in its simplified guise as the problem of the finite. That loss would be most unfortunate. The problem concerning the infinite was a primary concern of the philosophy of mathematics for many years—as may be seen in the work of Brouwer and that of several philosophers discussed by Benacerraf, such as Hilbert, Gödel, and Quine.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 162-165.

You do not doubt that ‘7 – 7 = 0’, but you cannot say ‘∞ – ∞ = 0’. In math lingo, the latter is indeterminate. Does this mean we are incapable of understanding the indeterminate, or is this just the nature of the universe? Consider Eugene Wigner’s lecture, “The Unreasonable Effectiveness of Mathematics in the Natural Sciences”. He noted, “It is important to point out that the mathematical formulation of the physicist’s often crude experience leads in an uncanny number of cases to an amazingly accurate description of a large class of phenomena.”

Still, complete understanding often seems to elude us.

“Fog Everywhere” Continued

[read the previous essay on this topic]

Consider mathematics and the aspect of fog. Nietzsche argued that the world is not knowable to us as a whole. This includes trying to express it with numbers. A recent example is Max Tegmark’s assertion that the universe is not something explained by mathematics; rather, it is itself mathematics. To contrast, Karl Jaspers, summarizing Nietzsche, wrote:

We cannot say what the world as a whole is. It is false to change all processes into a familiar world of our own, and then say: “All is will (everything wills); all is pleasure or pain (everything endures); all is motion (everything flows); all is tone (everything sounds); all is spirit (everything thinks); all is number (everything reckons).” Nietzsche warns us against all notions of the whole: “Let us guard against thinking that the world is a living being…or that the universe is a machine…Let us guard against saying that there are laws of nature…Let us guard against thinking that the world eternally creates novelties.” All these “shadows of God” darken actuality. We are within the world, and the whole of the world is, as a whole, not accessible to us.

Karl Jaspers, Nietzsche: An Introduction to the Understanding of His Philosophical Activity, Henry Regnery Company, 1966, p. 293

Notice in the paragraph quoted above that Nietzsche specifically excludes reducing everything to numbers, in opposition to Tegmark and the simulation hypothesis.

Robert Kanigel describes Srinivasa Ramanujan as “the man who knew infinity” in his biography of the same title, but mathematicians disagree on the ontological status and utility of infinity. The discussion of applying Ramanujan’s sum to arrive at -1/12 is very intricate and complex.

To make things even more puzzling, math students are confronted with the Gaussian integral, with its positive and negative infinities.

You can gain additional insight into infinity with:

Cauchy and Weierstrass had eliminated infinitely small and infinitely great numbers from analysis and replaced them by limits. But the theory of limits that thereby became so central required a clearer theory of the real line, that is to say, a theory of the irrational numbers. And that theory promptly reintroduced the infinite into analysis. The old infinity of infinitesimal and infinite numbers was simply replaced by the new infinity of infinitely large collections.12

In 1831 Carl Friedrich Gauss said [Kli72, p. 994], “I protest against the use of an infinite quantity as an actual entity; this is never allowed in mathematics. The infinite is only a manner of speaking, in which one properly speaks of limits to which certain ratios can come as near as desired, while others are permitted to increase without bound.” But only 52 years later, we find this in Cantor’s Grundlagen [Can76, p. 75]: “The idea of considering the infinitely large not only in the form of the unlimitedly increasing magnitude and in the closely related form of convergent infinite series…but to also fix it mathematically by numbers in the definite form of the completed infinite was logically forced upon me, almost against my will since it was contrary to traditions which I had come to cherish in the course of many years of scientific effort and investigations.”

12 See Russell’s Principles of Mathematics [Rus03, p. 304] for a related sentiment.

Shaughan Lavine, Understanding the Infinite, Harvard University Press, 1994, pgs. 38-39

In 1961, the great Russian physicist George Gamow’s popular math and physics overview was republished to great acclaim among science and math aficionados. Intriguingly titled One, Two, Three…Infinity, the book gives you a hint that the relationship of numbers and integers to the concept of infinity is still intriguing even after the intricate analyses of Cauchy, Weierstrass et al.

Let us conclude by considering Zeno’s dichotomy paradox of motion, in which “That which is in locomotion must arrive at the half-way stage before it arrives at the goal.” As the goal is divided into halves, one must complete an infinite number of tasks, which Zeno maintains is an impossibility.

[read the previous essay on this topic]

Poly-Awareness and the Year 1900

On the way to the year 1900, we encounter the comment, “In 1890 the stock exchanges of London, Paris, Berlin and New York controlled the economic progress of the whole world.” This marriage of geography and the financial world is very striking, culminating with:

The year 1900 was a wonderful one, when men were proud to be middle-class, and to be Europeans. The fate of the whole world was decided around green baize-covered tables in London, Paris or Berlin. Rubber trees from the Amazons were shipped to Malaya, the vast coal seams of the Upper Hwang-Ho were being exploited at the expense of the wretched labourers, and in the north of the Upper Vaal a mining city sprang up in a few short weeks. Mobilized by steam, the planet’s riches were being shifted ‘from one side of the world to the other’, to quote Le Bateau Ivre, on orders flashed by telegraph in two or three minutes. Decisions reached by boards of directors in London, Paris or Berlin affected the lives of millions of human beings who did not suspect that their right to happiness depended on quotations scribbled on blackboards in three noisy exchanges built like temples, in which raged the battles of unbridled financial ambition. Not a single detail escaped the notice of Europe’s financial capitals: they fixed the price of a tram ticket in Rio de Janeiro, and the working hours of a coolie in Hong Kong. So much power had never before been concentrated in so few hands within so small an area of the globe. It was the age of triumph of the European middle classes.

Charles Morazé, The Triumph of the Middle Classes: A Political and Social History of Europe in the Nineteenth Century, Anchor Books, 1968, page ix.

Morazé adds the following sentence, “The Europe of 1900 knew nothing of the world catastrophes which were to come.” At the core of this is the rise of modern science and technology. Max Planck published the first paper that gave us quantum mechanics. David Hilbert, at the 1900 Paris conference of the International Congress of Mathematicians, presented a collection of 23 problems (later known as Hilbert’s problems). Mathematicians, including Grigori Perelman (famous for his contributions to Riemannian geometry), are still attempting to solve these problems.

Henry Adams, attending the Exposition Universelle (1900), observed the dynamo and wrote the chapter “The Dynamo and the Virgin” in his book The Education of Henry Adams. He thinks of the dynamos as a moral force, much as the early Christians felt the Cross.

Remember that in the world of 1900, in the background to all of this, we have the Boxer Rebellion in China, part of the Chinese century of humiliation (which angers them to this day).

The quest for meta-intelligent understanding (i.e., poly-awareness) involves comparing then and now and how they are connected.

Kierkegaard and Existence

There are various striking intuitions about human existence. For example, in his brilliant memoirs, Speak, Memory, Nabokov begins with the deep reflection where human existence is compared to a baby in a cradle, rocking, completely vulnerable and uncertain. All of this is bracketed by two episodes of infinite darkness. The first episode took place before you were born and the second takes place after you’re gone. Your existence is a temporary flame, like that of a lit match.

A MetaIntelligent comment on this would be that the profound ingenuity of the 19th century mathematicians analyzing the size and nature of infinity (e.g., Richard Dedekind or Georg Cantor) cannot in the last analysis wrestle down human existence into mathematics.

The modern progenitor of this kind of human existence-watching is the Danish genius Søren Kierkegaard. In one of his masterpieces, Concluding Unscientific Postscript to Philosophical Fragments (1846), he makes the claim that knowledge, theory, speculative thinking and infinity-watching à la Dedekind and Cantor, cannot possibly explain human existence, because it subsumes all of these.

In 2025, this would mean that the Kierkegaard sense of things would tell you that neuroscience can never really explain how existence is sensed by a living person.

Kierkegaard writes, “in my view the misfortune of the age was precisely that it had too much knowledge, had forgotten what existence means, and what inwardness signifies.” He continues, “for a knowledge-seeker, when he has finished studying China he can take up Persia; when he has studied French he can begin Italian; and then go on to astronomy, the veterinary sciences, and so forth, and always be sure of a reputation as a tremendous fellow.”

By way of contrast, “inwardness in love does not consist in consummating seven marriages with Danish maidens, then cutting loose on the French, the Italian, and so forth, but consists in loving one and the same woman, and yet being constantly renewed in the same love, making it always new in the luxuriant flowering of the mood.” (Concluding Unscientific Postscript to Philosophical Fragments, page 232.)

Kierkegaard’s kind of existence-watching can be understood as a turning-upside-down of the famous phrase from Descartes, “I think, therefore I am.” For Kierkegaard, “I am, therefore I think.” Notice that “I think” is an epistemological statement or knowledge-watching. “I am” is an ontological statement.

This existentialist tradition of putting ontology before epistemology finds its culmination in Heidegger. As he says in his opus, Being and Time (1927), “human being is ultimately the being for whom being itself is an issue.”

Knot Theory and the Strangeness of Reality

The subfield of “knot theory” in math as a kind of geometry of “twistiness” gives us a deep “meta-intelligent” signal or lesson.

Meta-intelligent means “perspective-challenging” with or without full details of any subfield itself.

Consider this overview or comment on “knot theory” now:

“In mathematical knot theory, you throw everything out that’s related to mechanics,” Dunkel (MIT math professor) says. “You don’t care about whether you have a stiff versus soft fiber—it’s the same knot from a mathematician’s point of view. But we wanted to see if we could add something to the mathematical modeling of knots that accounts for their mechanical properties, to be able to say why one knot is stronger than another.”

But you immediately think: in the real world knots are not only twisted up in mathematically definable ways but are in fact actual shoelaces, neckties, ropes, etc, that have chemical and molecular properties before you describe their twist-and-tighten or slide-and-grip “shapes.”

Which is the real: the math or the “ropiness” of the ropes or the “laciness” of the laces?

The relationship between things and numbers is elusive.

Mathematicians have long been intrigued by knots, so much so that physical knots have inspired an entire subfield of topology known as knot theory—the study of theoretical knots whose ends, unlike actual knots, are joined to form a continuous pattern.

In knot theory, mathematicians seek to describe a knot in mathematical terms, along with all the ways that it can be twisted or deformed while still retaining its topology, or general geometry.

MIT mathematicians and engineers have developed a mathematical model that predicts how stable a knot is, based on several key properties, including the number of crossings involved and the direction in which the rope segments twist as the knot is pulled tight.

“These subtle differences between knots critically determine whether a knot is strong or not,” says Jörn Dunkel, associate professor of mathematics at MIT. “With this model, you should be able to look at two knots that are almost identical, and be able to say which is the better one.”

“Empirical knowledge refined over centuries has crystallized out what the best knots are,” adds Mathias Kolle, the Rockwell International Career Development Associate Professor at MIT. “And now the model shows why.”

As per usual in science, one is dazzled by the ingenuity of the quest and the formulations but puzzled by the larger implications since we can never decide whether math “made” us or we “made” (i.e., invented) math.

Education and “The Three-Body Problem”

The brilliant math-watcher, Ian Stewart, says of this classic physics problem, the Three-Body Problem:

Newton’s Law of Gravity runs into problems with three bodies (earth, moon, sun, say).

In particular, the gravitational interaction of a mere three bodies, assumed to obey Newton’s inverse square law of gravity, stumped the mathematical world for centuries.

It still does, if what you want is a nice formula for the orbits of those bodies. In fact, we now know that three-body dynamics is chaotic–so irregular that is has elements of randomness.

There is no tidy geometric characterization of three-body orbits, not even a formula in coordinate geometry.

Until the late nineteenth century, very little was known about the motion of three celestial bodies, even if one of them were so tiny that its mass could be ignored.

(Visions of Infinity: The Great Mathematical Problems, Ian Stewart, Basic Books, 2014, page 136)

Henri Poincaré, the great mathematician, wrestled with this with tremendous intricacy and ingenuity all his life:

Jules Henri Poincaré was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as “The Last Universalist,” since he excelled in all fields of the discipline as it existed during his lifetime.

Born: April 29, 1854, Nancy, France
Died: July 17, 1912, Paris, France.

We now think of applying in an evocative and not a rigorous mathematical way, the unexpected difficulties of the three-body problem to the n-body (i.e., more than three) problems of sociology or economics or history itself, and sense that social life is always multifactorial and not readily pin-downable, since “everything is causing everything else” and extracting mono-causal explanations must be elusive for all the planetary and Poincaré reasons and beyond.

This suggests to the student that novels are one attempt to say something about n-body human “orbits” based on “n-body” stances and “circumstances” with large amounts of randomness governing the untidy mess that dominates human affairs.

Words are deployed in novels and not numbers as in physics, but the “recalcitrance” of the world, social and physical, remains permanent.

Education and meta-intelligence would be more complete by seeing how the world, as someone put it, “won’t meet us halfway.” Remember Ian Stewart’s warning above:

“There is no tidy geometric characterization of three-body orbits…” and you sense that this must apply to human affairs even more deeply.