Tag: Platonism

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.

Japanese Philosopher KARATANI Kōjin (柄谷 行人) Awarded the 2022 Berggruen Prize

An expansive thinker who crosses boundaries.

[from Nōema Magazine, by Nathan Gardels, Editor-in-Chief]

KARATANI Kōjin has been named this year’s laureate for the $1 million Berggruen Prize for Culture and Philosophy. An expansive thinker who straddles East and West while crossing disciplinary boundaries, Karatani is not only one of Japan’s most esteemed literary critics, but a highly original mind who has turned key suppositions of Western philosophy on their heads.

In Karatani’s sharpest departure from conventional wisdom, he locates the origins of philosophy not in Athens, but in the earlier Ionian culture that greatly influenced the so-called “pre-Socratic thinkers” such as Heraclitus and Parmenides. Their ideas centered on the flux of constant change, in which “matter moves itself” without the gods, and the oneness of all being—a philosophical outlook closer to Daoist and Buddhist thought than to Plato’s later metaphysics, which posited that, as Karatani puts it, “the soul rules matter.”

In the political realm, Karatani contrasts the form of self-rule from Ionian times based on free and equal reciprocity among all inhabitants — “isonomia” — with what he calls the “degraded democracy” of Athens that rested on slavery and conquest. He considers the former the better foundation for a just polity.

In a novel twist on classical categorizations, Karatani regards Socrates himself as fitting into the pre-Socratic mold. “If one wants to properly consider the pre-Socratics, one must include Socrates in their number,” he writes. “Socrates was the last person to try to re-institute Ionian thought in politics.”

A Degraded Form of Democracy in Athens

For Karatani, Athenian democracy was debased because it was “constrained by the distinctions between public and private, and spiritual and manual labor,” a duality of existence that Socrates and the pre-Socratics sought to dismantle. As a result, by Karatani’s reading, Socrates was both held in contempt by the “aristocratic faction,” which sought to preserve its privileges built on the labor of others, and condemned to death by a narrow-minded mobocracy for his idiosyncratic insistence on autonomy and liberty in pursuit of truth.

Appalled at Socrates’ fate, Plato blamed democracy for giving birth to demagoguery and tyranny, radically rejecting the idea of rule by the masses and proposing instead a political order governed by philosophers. In Karatani’s reckoning, Plato then “took as his life’s work driving out the Ionian spirit that touched off Athenian democracy”—in short, throwing out the baby with the bathwater but maintaining the disassociations, such as citizen and slave, that follow from the distinction between public and private grounded in an apprehension of reality that separates the spiritual from the material.

In order to refute “Platonic metaphysics,” Karatani argues, “it is precisely Socrates that is required.”

Turning Marx On His Head

In his seminal work, The Structure of World History, Karatani flips Marx’s core tenet that the economic “mode of production” is the substructure of society that determines all else. He postulates instead that it is the ever-shifting “modes of exchange” among capital, the state and nation which together shape the social order.

For Karatani, historically cultivated norms and beliefs about fairness and justice, including universal religions, compel the state to regulate inequality within the mythic commonality of the nation, which sees itself as whole people, tempering the logic of the unfettered market. As he sees it, the siren call of reciprocity and equality has remained deeply resonant throughout the ages, drawing history toward a return to the original ideal of isonomia.

Expanding the Space of Civil Society

Not an armchair philosopher, Karatani has actively promoted a modern form of the kind of reciprocity he saw in ancient Ionian culture, which he calls “associationism.” In practical terms in Japan, this entails the activation of civil society, such as through citizens’ assemblies, that would exercise self-rule from the bottom up.

In the wake of the Fukushima nuclear accident in 2011, Karatani famously called for “a society where people demonstrate” that would expand the space of civil society and constrict the collusive power of Japan’s political, bureaucratic and corporate establishment. Like other activists, he blamed this closed system of governance that shuts out the voices of ordinary citizens for fatally mismanaging the nuclear power industry in a country where earthquakes and tsunamis are an ever-present danger.

An Expansive Mind

Along with The Structure of World History (2014) and Isonomia and The Origins of Philosophy (2017), the breadth of Karatani’s interests and erudition are readily evident in the titles of his many other books. These include Nation and Aesthetics: On Kant and Freud (2017), History and Repetition (2011), Transcritique: On Kant and Marx (2003), Architecture As Metaphor: Language, Number, Money (1995) and Origins of Modern Japanese Literature (1993).

The prize ceremony will be held in Tokyo in the spring.

The View From Nowhere as an Additional Problem in “Thinking About Thinking”

The View From Nowhere is a book by philosopher Thomas Nagel.

Published by Oxford University Press in 1986, it contrasts passive and active points of view in how humanity interacts with the world, relying either on a subjective perspective that reflects a point of view or an objective perspective that takes a more detached perspective. Nagel describes the objective perspective as the “view from nowhere,” one where the only valuable ideas are ones derived independently.

Epistemology (what we can know and why) is puzzling to the max if you ponder it for a moment. Think of a painting in a Boston museum. If you walk up to it, you see only the little piece in front of your nose so you back up and try to get an “optimal grip.” (to use Prof. Merleau-Ponty’s language.) If you walk all the way to China and try to see it from there, you will see nothing of it, no matter what telescope you might use. This is sort of what we mean by “the view from nowhere.” You’re way too far.

This brings us to the problem of the “detached observer” (modern versions of which stem from Descartes, who wants to get a bird’s eye view of all other bird’s eye views.  This is tricky and elusive for the obvious reasons. When Richard Feynman or some other physicist theorizes, is he not achieving a view from nowhere or is he? No one will deny a place to theoretical “standpoints” and “viewpoints.” The theoretician is himself a person who breathes, and sneezes, and yawns, and gets hungry and has to stretch his or her legs after too much sitting. One can’t quite “move into one’s own mind” since all theory is “embodied.”

Human beings have the unique ability to view the world in a detached way: 

We can think about the world in terms that “transcend” our own experience or interest, and consider the world from a vantage point that is, in Nagel’s words, “nowhere in particular.”

The strange human situation is seen from the fact that this “view from nowhere,” this “detached observer” theoretical stance, includes the theorist himself, the detachment and the theory as part of the “bird’s eye view” without any particular concrete bird serving as your ambassador or proxy.

“The unifying theme, as Nagel puts it at the beginning, is the problem of how to combine the perspective of a particular person ‘inside the world’ with an objective view of that same world, the person and his viewpoint included.”

(Bernard Williams, 1986 book review, London Review of Books.)

We have already seen the problem of Husserl‘s (died in 1938) “rhomboid” or “matchbox” (i.e., you can’t see the entire matchbox all at once) or Ortega y Gasset‘s “orange” (i.e., you cannot see the back or obverse or reverse of a spherical orange unless you walk around it and lose the first view from the front) and all this “partial viewing” takes place on “Neurath’s boat.” (Where we’re like sailors on a knowledge ship and can’t go back to any origins and can’t discuss Platonism with Plato himself. The Harvard philosopher Quine, among others, mentions this problem.) The ship movies forward and the “matchbox/orange” are viewed in some cabin on the ship (i.e., your field, such as chemistry or history or biology).

Lastly: think of the opening line of Thomas Mann’s (died in 1955) great novel, Joseph and His Brothers: “Deep is the well of the past. Should we not call it bottomless?”

In other words, there is no way for us as “knowledge detectives” to go back to the origins of ourselves or our history since that’s all unrecoverable and lost “in the mist of time.”

A student embarking on a “knowledge quest” (university education) should not dodge these puzzles and mysteries but look at them “unblinkingly.”  A deep education means all the dimensions of the quest are in front of the student and not wished away.  This includes the student’s own danger of being lost as “a leaf in the whirlwind of time.” (Hannah Arendt phrase we have already seen.). Career aside, there are multiple “Rubik’s Cubes” here if the student wants to experience the deep and the wide.