Month: July 2026

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]

Japan-Watching: Economics in the Age of AI

A Thought Experiment Envisioning A “Fully Automated Society”

from REITI, by IKEUCHI Kenta [池内 健太], Senior Fellow (Policy Economist)

In a world where human labor has become unnecessary because of AI, what should be the focus of the study of economics? In the future, if AI and robots have become capable of producing all goods and services necessary for our society, what kind of economic problems could remain?

Of course, a fully automated society is not predicted for the near future. This article imagines such an extreme type of society as a thought experiment designed to consider economic systems in the age of AI. Here, a fully automated society refers not only to one in which corporate production is automated, but also to one in which the legislative, administrative, and judicial functions of government are substantially supported—and in some cases automated—by AI.

I started thinking about this matter when I heard from a researcher acquaintance that a paper concerning AI’s impact on employment had caused quite a stir on X, and I decided to read it out of curiosity. The paper discussed the possibility that AI-driven job cuts could reduce workers’ incomes, weaken consumer demand, and ultimately backfire on firms themselves (Falk and Tsoukalas 2026). Companies may earn higher profits in the short term by taking advantage of AI to cut back on personnel costs. However, if most companies follow that same approach, overall market demand would weaken because workers are also consumers.

The purpose of this article is not to question the validity of that paper’s argument. Rather, the focus is on what economic problems would remain in a future society if AI not only partially replaces human labor but also produces most goods and services.

Scarcity Will Continue to Be a Problem

There has already been extensive research regarding the impact of AI on employment. For example, Acemoğlu and Restrepo (2019) argued that while automation may replace existing human jobs, it may also create new ones. Moreover, the possibility that technological advances could free humans from labor and greatly alleviate economic problems was discussed long ago by Keynes (1930).

If AI and robots become capable of producing goods and services on a sufficiently large scale, would economic problems disappear? That would not necessarily be the case. Even in a fully automated society, scarce resources such as land, location, natural environments, energy, and rare metals will remain limited. Moreover, social status, influence, and political decision-making power are deeply connected to human intentions and perception, and cannot simply be delegated to AI. More land will not become available simply because more people wish to live in convenient urban locations. Quiet natural environments, advanced healthcare resources, social attention, and political influence also cannot be maximized for everyone at the same time.

Therefore, even in a fully automated society, some economic problems will remain. However, the main focus will shift from the “problem of insufficient ordinary commodities” to “how to allocate fundamentally scarce resources.” As production capacity increases, the value of true scarcity only becomes clearer.

In that case, the roles of markets and prices will still exist. Prices are not merely figures that allow for corporate earnings; they convey information about which resources are scarce and to what extent, how much demand there is for those resources, and what supply constraints there are. That perspective connects to a classic argument made by Hayek (1945), who argued that prices function as a mechanism through which dispersed information can be aggregated.

However, if AI becomes deeply integrated into the market, the concept of price itself may change. At present, prices serve as one-dimensional signals representing the levels of various factors, such as scarcity, quality, demand, supply, environmental impact, and future risks, expressed in terms of a single metric, that is, monetary value. If AI agents become capable of processing large volumes of information on behalf of consumers and companies, it is possible that multi-dimensional market signals that convey information concerning all those various factors, including quality, environmental impact, congestion, delivery time, reliability and social impact may come into use. Narita (2025) also discussed the possibility that the roles of money and prices may change, with more diverse evaluation standards becoming involved in economic coordination.

For People to Enjoy Affluence

In a fully automated society, how people participate in the market and society will become more important than ever. If the premise that people earn income through labor becomes obsolete, it will be necessary to develop a mechanism whereby purchasing power is distributed to everyone. In this context, universal basic income (UBI) may be reframed not only as relief for the unemployed, but as a form of fundamental purchasing power used by people to express their preferences. Managi [馬奈木 俊介] (2025) also pointed out that governance over the equitable distribution of the benefits of AI is essential.

Moreover, UBI may not be limited to simple monetary payments. In the future, UBI may take the form of a system combining other benefits as well, including energy use quotas, rights of access to basic healthcare services and education, and rights to refuse or control the use of personal data. In a fully automated society, UBI would therefore be a matter not only of how much to provide, but also of what kinds of access to guarantee and over what time horizon.

Another important issue is whether it is appropriate to treat people merely as consumers. In a fully automated society, the need for people to work for a living may diminish. However, even without such a necessity, humans will likely still possess the desire to create or to be creative. It is human nature to try new things and to try to surprise or impress other people. The spirit of fun and curiosity, a desire for self-expression, an inquisitive mind, and an appetite for challenges are deeply and fundamentally connected to human nature. Therefore, when designing a future UBI system, it will be important to treat people not merely as consumers but as agents who can participate in creation and exploration.

Additionally, the question of who owns and controls AI systems, robots, foundation models, and computing infrastructure is also a major issue. Even if a certain level of income is distributed to everyone, there may remain a power gap between those who control AI systems and robots and those who merely have access to them, in place of the income gap that currently exists in society.

All of the above-mentioned points for debate are relevant to the study of economics. How scarce resources should be allocated, how to guarantee people’s range of choices, and how to design ownership and controlling rights are problems central to economics. A fully automated society is not a near-future prediction. However, this extreme thought experiment serves as a useful guide for considering economic systems in the age of AI. Economics in the age of AI is not about discarding the intellectual legacy of economics, but about inheriting it and extending it toward a new society.

References

June 12, 2026

“Fog everywhere” and Other Confusions

Charles Dickens gives us wonderful sociopolitical insight in his serial Bleak House, with his imagery of impenetrable fog in the second paragraph of chapter 1:

Fog everywhere. Fog up the river, where it flows among green aits and meadows; fog down the river, where it rolls deified among the tiers of shipping and the waterside pollutions of a great (and dirty) city. Fog on the Essex marshes, fog on the Kentish heights. Fog creeping into the cabooses of collier–brigs; fog lying out on the yards and hovering in the rigging of great ships; fog drooping on the gunwales of barges and small boats. Fog in the eyes and throats of ancient Greenwich pensioners, wheezing by the firesides of their wards; fog in the stem and bowl of the afternoon pipe of the wrathful skipper, down in his close cabin; fog cruelly pinching the toes and fingers of his shivering little ’prentice boy on deck. Chance people on the bridges peeping over the parapets into a nether sky of fog, with fog all round them, as if they were up in a balloon and hanging in the misty clouds.

Charles Dickens, Bleak House. Bradbury & Evans, 1852-1853.

Think of the Trump–Epstein cover-up and the machinations of the U.S. government to attempt to conceal everything in a similar fog. We discussed another dimension of our ignorance with Friedrich Nietzsche’s assertion that we “are unknown to ourselves”.

Let’s consider a third level of our confusion and how we attempt to extricate ourselves by expressing the world around us through mathematics. For example, the square root of -1 is i. ii is approximately 0.208. To a student encountering this concept for the first time, it can be inscrutable that an imaginary number to the power of itself results in a real number. To quote Wikipedia:

In electrical engineering, signal processing, and similar fields, signals that vary periodically over time are often described as a combination of sinusoidal functions (see Fourier analysis), and these are more conveniently expressed as the sum of exponential functions with imaginary exponents, using Euler’s formula.

Wikipedia, Euler’s formula [with added links]

The esteemed physicist Roger Penrose has said on multiple occasions that he believes the realm of imaginary numbers or complex analysis will be more informative in physics than real numbers. (See his classic book, The Road to Reality: A Complete Guide to the Laws of the Universe.)

Let’s conclude with our inevitable mortality as living beings as another source of perplexity and confusion. As we grow older, half of our mind is fixated on the enjoyment we get from life, while the other half is focused on the anxiety that it is not forever. We attempt to sidestep this anxiety by clinging to the escapist thought that “Besides, it’s always the others who die.”