Tag: mathematics

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]

“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.”

Is There a Scheme of Things Underlying Everything?

In The Thibaults (the novel sequence for which Roger Martin du Gard won the 1937 Nobel Prize in Literature), a fundamental motif is the question of whether the universe is coherent. Roughly speaking, there are three competing schools of coherence—science, religion and art. Antoine Thibault discusses with the Abbé (French title for abbot):

Antoine did not seem to hear him. “Just think,” he exclaimed, “what it means to a youngster, when he’s turned loose, by gradual stages, on mathematics, physics, chemistry! Suddenly he discovers that he has all space, the universe, for his playground. And after that, religion strikes him as not only cramped, but false, illogical. Untrustworthy.”

Roger Martin du Gard, The Thibaults, translated by Philip Thody & Ellen Kennedy, Bantam Modern Classic Edition, Viking Press, 1968. page 762.

The climax of this debate appears when Antoine says:

“…I talked just now about Universal Order and a Scheme of Things; but that was merely to talk like everyone else. Actually it seems to me that we’ve as many reasons to question the existence of a Scheme of Things as to take it for granted. From his actual viewpoint the human animal I am observes an immense tangle of conflicting forces. But do these forces obey a universal law outside themselves, distinct from them? Or do they, rather, obey—so to speak—internal laws, each atom being a law unto itself, that compels it to work out a kind of ‘personal’ destiny? I see these forces obeying laws which do not control them from outside, but join up with them, which do nothing more than in some way stimulate them.…And anyhow, what a jumble it is, the course of natural phenomena! I’d just as soon believe that causes spring from each other ad infinitum, each cause being the effect of another cause, and each effect the cause of other effects. Why should one want to assume at all costs a Scheme of Things?…”

Roger Martin du Gard, The Thibaults, translated by Philip Thody & Ellen Kennedy, Bantam Modern Classic Edition, Viking Press, 1968. page 768.

The topic of an underlying scheme of things is close to the central question of Western civilization. In Plato’s Republic, we have the allegory of the dark cave occupied by humanity. It looks at shadows dancing on the wall, projected by a fire. Liberating humanity requires leaving the cave and climbing to the surface of the earth, glimpsing the sun for the first time. From here, with the help of philosophy, humanity flies off and encounters the Logos and the Eidos. Mathematical truth crowns this journey.

Aristotle, Plato’s star pupil and later rival, takes this quest and focuses on the biological. The Athenian tradition invents theory, a pillar of Western tradition, culminating in modern science. Jerusalem’s competing tradition, the Judeo-Christian worldview, derives its scheme of things from divinity before biology and mathematics. Thus, the novel’s debate is ultimately the struggle between Athens and Jerusalem.

Being at Home in the World/Universe

The French philosopher Maurice Merleau-Ponty provided an introduction to the problem of being at home when he wrote:

“The world is not what I think, but what I live through. I am open to the world, I have no doubt that I am in communication with it, but I do not possess it; it is inexhaustible. ‘There is a world’, or rather: “There is the world’; I can never completely account for this ever-reiterated assertion in my life.”

Joseph J. Kockelmans (Editor), Phenomenology: The Philosophy of Edmund Husserl and Its Interpretation, Anchor Books Edition, 1967, page 369.

Remy C. Kwant, in his essay “Merleau-Ponty and Phenomenology”, commented:

For, according to him, the original lies buried in a dimension of darkness in such a way that it cannot be brought to light. Our existence is interwoven with the world, is a dialogue with the world. This dialogue reaches its most profound point there where the first and most original meaning arises, a meaning that is pre-conscious and pre-personal. Whatever is in our consciousness, whatever comes to light, becomes lucid, originates also in this darkness. As we have seen, man is able to obtain a measure of knowledge regarding this dark depth. He is able to divine something about the mysterious dialogue between the body-subject and the world. However, according to Merleau-Ponty, an absolute illumination of the phenomenal field is in principle impossible. All man can do is to erect some pointers in a darkness which resists full illumination.

Joseph J. Kockelmans (Editor), Phenomenology: The Philosophy of Edmund Husserl and Its Interpretation, Anchor Books Edition, 1967, page 390-391.

We sense that the interaction between ourselves and the world at every level may not be explainable. Therefore, we seek emotional or psychological shelter. The three levels of shelter are:

  1. hearth and home
  2. a sense of belonging
  3. gods

Think of the song, “A House Is Not a Home”, sung by Dionne Warwick. “A chair is still a chair / Even though there’s no one sitting there … But a room is not a house
/ And a house is not a home” depicts the human longing for shelter via hearth and home. The French philosopher Bruno Latour referred to this as a “parliament of things.”

Consider “Gimme Shelter” by The Rolling Stones, as well as the novel (and later film) The Sheltering Sky by Paul Bowles. Both of these cover the deep issue of shelter.

Heidegger’s essay “Building Dwelling Thinking” (German: Bauen Wohnen Denken) states:

In what follows we shall try to think about dwelling and building. This thinking about building does not presume to discover architectural ideas, let alone to give rules for building. This venture in thought does not view building as an art or as a technique of construction; rather it traces building back into that domain to which everything that is belongs. We ask:

  1.   What is it to dwell?
  2.   How does building belong to dwelling?
I

We attain to dwelling, so it seems, only by means of building. The latter, building, has the former, dwelling, as its goal. Still, not every building is a dwelling. Bridges and hangars, stadiums and power stations are buildings but not dwellings; railway stations and highways, dams and market halls are built, but they are not dwelling places. Even so, these buildings are in the domain of our dwelling. That domain extends over these buildings and yet is not limited to the dwelling place. The truck driver is at home on the highway, but he does not have his shelter there; the working woman is at home in the spinning mill, but does not have her dwelling place there; the chief engineer is at home in the power station, but he does not dwell there. These buildings house man. He inhabits them and yet does not dwell in them, when to dwell means merely that we take shelter in them. In today’s housing shortage even this much is reassuring and to the good; residential buildings do indeed provide shelter; today’s houses may even be well planned, easy to keep, attractively cheap, open to air, light, and sun, but—do the houses in themselves hold any guarantee that dwelling occurs in them? Yet those buildings that are not dwelling places remain in turn determined by dwelling insofar as they serve man’s dwelling. Thus dwelling would in any case be the end that presides over all building. Dwelling and building are related as end and means. However, as long as this is all we have in mind, we take dwelling and building as two separate activities, an idea that has something correct in it. Yet at the same time by the means-end schema we block our view of the essential relations. For building is not merely a means and a way toward dwelling—to build is in itself already to dwell. Who tells us this? Who gives us a standard at all by which we can take the measure of the nature of dwelling and building?

Martin Heidegger, Poetry, Language, Thought, (translated by Albert Hofstadter), Harper & Row, 1975, pages 145-146.

Stuart Kauffman comes at this from a different angle:

Who are we? Where did we come from? Why are we here? Did Neanderthal, Homo habilis, or Homo erectus ask? Around which fire in the past 3 million years of hominid evolution did these questions first arise? Who knows.

Somewhere along our path, paradise has been lost, lost to the Western mind, and in the spreading world civilization, lost to our collective mind. John Milton must have been the last superb poet of Western civilization who could have sought to justify the ways of God to man in those early years foreshadowing the modern era. Paradise has been lost, not to sin, but to science. Once, a scant few centuries ago, we of the West believed ourselves the chosen of God, made in his image, keeping his word in a creation wrought by his love for us. Now, only 400 years later, we find ourselves on a tiny planet, on the edge of a humdrum galaxy among billions like it scattered across vast megaparsecs, around the curvature of space-time back to the Big Bang. We are but accidents, we’re told. Purpose and value are ours alone to make. Without Satan and God, the universe now appears the neutral home of matter, dark and light, and is utterly indifferent. We bustle, but are no longer at home in the ancient sense.

Stuart Kauffman, At Home in the Universe: The Search for the Laws of Self-Organization and Complexity, Oxford University Press, 1995, page 4.

Kauffman comes to grips with this problem with the final line above. He continues:


In this new view of life, organisms are not merely tinkered-together contraptions, bricolage, in Jacob’s phrase. Evolution is not merely “chance caught on the wing,” in Monod’s evocative image. The history of life captures the natural order, on which selection is privileged to act. If this idea is true, many features of organisms are not merely historical accidents, but also reflections of the profound order that evolution has further molded. If true, we are at home in the universe in ways not imagined since Darwin stood natural theology on is head with his blind watchmaker.

Stuart Kauffman, At Home in the Universe: The Search for the Laws of Self-Organization and Complexity, Oxford University Press, 1995, pages 25-26.

Kauffman wants to complete the Darwinian revolution by adding self-organization and complexity to natural selection. In his vision, this will begin to produce a holistic picture of who we are. This will perhaps allow us to feel “We are all at home in the universe, poised to sanctify by our best, brief, only stay.” [page 30.]

Zooming out from this, we can see a meta-intelligent sense in which science believes it can convert mysteries into problems using math. In contrast to this, philosophers believe the opposite, that the problems are becoming more mysterious.

Why Is Technological History So Misleading?

We are conditioned to think of technological history in a very binary way. For thousands of years before motorized transportation, we think of horses and wind-powered ships. We also sense that if we brought great historical minds from before the industrial revolution to a modern city, most likely they would be stunned by the technology surrounding them. Think of a world of medical science before anesthesia and germ theory.

Let’s modify this binary view of human history. David F. Noble gives us a more accurate view:

Augustine, the chief author of Christian orthodoxy, wrote in The City of God, “there have been discovered and perfected, by the natural genius of man, innumerable arts and skills which minister not only to the necessities of life but also to human enjoyment.” Augustine recognized the “astonishing achievements” that had taken place in cloth-making, navigation, architecture, agriculture, ceramics, medicine, weaponry and fortification, animal husbandry, and food preparation; in mathematics, astronomy, and philosophy; as well as in language, writing, music, theater, painting, and sculpture. But he emphasized again that “in saying this, of course, I am thinking only of the nature of the human mind as a glory of this mortal life, not of faith and the way of truth that leads to eternal life… And, remember, all these favors taken together are but the fragmentary solace allowed us in a life condemned to misery.”5

5 St. Augustine, The City of God (Garden City, N.Y.: Doubleday, 1958), pp. 526, 527.

David F. Noble, The Religion of Technology: The Divinity of Man and the Spirit of Invention, Penguin Books, 1999 (originally 1997), pages 11-12.

Note that Augustine wrote The City of God in 426 AD, meaning that even 1600 years ago, they had already made colossal advances. The prejudice that we have, given our scientific training, is utterly misleading. Rather than being blinded by Biblical explanations of how the world came to be, Augustine lauded these scientific advancements. We think of Thomas Edison and the lightbulb, rather than, “Let there be light.”

There are various levels of empirical and artisanal knowledge. In cooking, we rarely worry about molecules that make up ingredients. All these daily life pillars Augustine lists cannot be overlooked, even as we unlock the submicroscopic world of quantum mechanics.

Is It Good to Be a Detached Observer?

The famous Dutch historian, Pieter Geyl, in his Napoleon, for and against (Dutch, Napoleon: voor en tegen in de Franse geschiedschrijving) teaches us that there are “arguments without end.” One example is the question surrounding the concept of detachment. Aristotle, in his Nicomachean Ethics, proposes “eudaimonia,” a Greek word literally translating to the state or condition of good spirit coming from imperturbability. This sense of things is all over the Western tradition. Think of the line from the British poet, Alexander Pope, “For Fools rush in where Angels fear to tread.” (An Essay on Criticism, 1711). You see from this that fools lack detachment and act on impulse.

We get a confirmation of Geyl’s arguments without end when we remember that almost every love song recommends the opposite. For example, “Fools Rush In (Where Angels Fear to Tread)” originally made famous by Frank Sinatra and later Elvis Presley, offers us the line “But wise men never fall in love / So how are they to know.” From this, we can interpret that wise men can be foolish and foolish people can be wise. You may also have in the back of your mind Tennyson’s “Tis better to have loved and lost / Than never to have loved at all.” It is not wise to be careful always.

We get a twist on this in the Rodgers & Hammerstein musical South Pacific. Think of “Some Enchanted Evening”:

Who can explain it?
Who can tell you why?
Fools give you reasons—
Wise men never try.

Fools give you reasons because they think everything can be explained, where wise men realize this is not always true. The larger point, from existential thinker Gabriel Marcel, is that all the phenomena of life that are explainable are themselves wrapped up in a larger mystery. He discusses the question of detachment in Being and Having: An Existentialist Diary, which we covered in “Existence and the Problem of Separability” and “Is the World Broken?”.

Marcel says:

March 8th [1929]

I am more and more struck by the difference between the two modes of detachment: the one is that of the spectator, the other of the saint. The detachment of the saint springs, as one might say, from the very core of reality; it completely excludes curiosity about the universe. This detachment is the highest form of participation. The detachment of the spectator is just the opposite, it is desertion, not only in thought but in act. Herein, I think, lies the kind of fatality which seems to weigh on all ancient philosophy—it is essentially the philosophy of the spectator.

But one thing must be noted: the belief that one can escape pure spectatorship by devotion to a practical science, which cannot quite clearly formulate it as yet. I should express it by saying that the modifications which such a science imposes on reality have no other result (metaphysically of course than of making that science in some sense a stranger to reality. The word ‘alienation’ exactly expresses what I mean. ‘I am not watching a show’—I will repeat these words to myself every day. A fundamental spiritual fact.

The interdependence of spiritual destinies, the plan of salvation; for me, that is the sublime and unique feature of Catholicism.

I was just thinking a moment ago that the spectator-attitude corresponds to a form of lust; and more than that, it corresponds to the act by which the subject appropriates the world for himself. And I now perceive the deep truth of Bérulle’s theocentrism. We are here to serve; yes, the idea of service, in every sense, must be thoroughly examined.

Also perceived this morning, but still in a confused way, that there is profane knowledge and sacred knowledge (whereas previously I have wrongly tended to assert that all knowledge was pro-fane. It isn’t true, profane is a supremely informative word). Inquire on what conditions knowledge ceases to be profane.

Incredible how thronged these days are spiritually! My life is being illuminated right into the depths of the past, and not my life only.

Every time we give way to ourselves we may unawares be laying an additional limitation on ourselves, forging our own chain. That is the metaphysical justification for asceticism. I never understood that till now.

Reality as mystery, intelligible solely as mystery. This also applies to myself.

Gabriel Marcel, Being and Having: An Existentialist Diary, Harper Torchbooks, 1965, pages 20-21.

Notice this discussion starts by analyzing modes of detachment and concludes with Marcel talking about reality and himself as mystery. This brings us full circle to Geyl and his concept of arguments without end because trying to define pros and cons of detachment and what is a mystery is ultimately undecidable. This may remind you of Gödel’s incompleteness theorems, that finding a complete and consistent set of axioms for all mathematics is impossible.

Economics-Watching: Second-Quarter GDP Growth Estimate Unchanged

[from the Federal Reserve Bank of Atlanta]

The growth rate of real gross domestic product (GDP) is a key indicator of economic activity, but the official estimate is released with a delay. The Federal Reserve Bank of Atlanta’s GDPNow forecasting model provides a “nowcast” of the official estimate prior to its release by estimating GDP growth using a methodology similar to the one used by the U.S. Bureau of Economic Analysis.

GDPNow is not an official forecast of the Atlanta Fed. Rather, it is best viewed as a running estimate of real GDP growth based on available economic data for the current measured quarter. There are no subjective adjustments made to GDPNow—the estimate is based solely on the mathematical results of the model.

Recent forecasts for the GDPNow model are available here [archived PDF]. More extensive numerical details—including underlying source data, forecasts, and model parameters—are available as a separate spreadsheet [archived XLSX]. You can also view an archive of recent commentaries from GDPNow estimates.

Please note that the Atlanta Fed no longer supports the GDPNow app. Download the EconomyNow app to get the latest GDP nowcast and more economic data.

Latest estimate: 2.4 percent — July 25, 2025

The GDPNow model estimate for real GDP growth (seasonally adjusted annual rate) in the second quarter of 2025 is 2.4 percent on July 25, unchanged from July 18 after rounding. The forecasts of the major GDP subcomponents were all unchanged or little changed from their July 18 values after this week’s releases from the U.S. Census Bureau and the National Association of Realtors.

The growth rate of real gross domestic product (GDP) measured by the U.S. Bureau of Economic Analysis (BEA) is a key metric of the pace of economic activity. It is one of the four variables included in the economic projections of Federal Reserve Board members and Bank presidents for every other Federal Open Market Committee (FOMC) meeting. As with many economic statistics, GDP estimates are released with a lag whose timing can be important for policymakers. In preparation for FOMC meetings, policymakers have the Fed Board staff projection of this “advance” estimate at their disposal. These projections—available through 2008 at the Philadelphia Fed’s Real Time Data Center—have generally been more accurate than forecasts from simple statistical models. As stated by economists Jon Faust and Jonathan H. Wright in a 2009 paper, “by mirroring key elements of the data construction machinery of the Bureau of Economic Analysis, the Fed staff forms a relatively precise estimate of what BEA will announce for the previous quarter’s GDP even before it is announced.”

The Atlanta Fed GDPNow model also mimics the methods used by the BEA to estimate real GDP growth. The GDPNow forecast is constructed by aggregating statistical model forecasts of 13 subcomponents that comprise GDP. Other private forecasters use similar approaches to “nowcast” GDP growth. However, these forecasts are not updated more than once a month or quarter, are not publicly available, or do not have forecasts of the subcomponents of GDP that add “color” to the top-line number. The Atlanta Fed GDPNow model fills these three voids.

The BEA’s advance estimates of the subcomponents of GDP use publicly released data from the U.S. Census Bureau, U.S. Bureau of Labor Statistics, and other sources. Much of this data is displayed in the BEA’s Key Source Data and Assumptions table that accompanies the “advance” GDP estimate. GDPNow relates these source data to their corresponding GDP subcomponents using a “bridge equation” approach similar to the one described in a Minneapolis Fed [archived PDF] study by Preston J. Miller and Daniel M. Chin. Whenever the monthly source data is not available, the missing values are forecasted using econometric techniques similar to those described in papers by James H. Stock and Mark W. Watson and Domenico Giannone, Lucrezia Reichlin, and David Small. A detailed description of the data sources and methods used in the GDPNow model is provided in an accompanying Atlanta Fed working paper [archived PDF].

As more monthly source data becomes available, the GDPNow forecast for a particular quarter evolves and generally becomes more accurate. That said, the forecasting error can still be substantial just prior to the “advance” GDP estimate release. It is important to emphasize that the Atlanta Fed GDPNow forecast is a model projection not subject to judgmental adjustments. It is not an official forecast of the Federal Reserve Bank of Atlanta, its president, the Federal Reserve System, or the FOMC.

Wrestling with History: Alexis de Tocqueville

Alexis de Tocqueville, a brilliant French historian, wrote Democracy in America. This book is a supreme example of U.S.-watching.

Another book of his, Recollections, shows him wrestling with history itself. If we remember that Clio is the muse of history, then we might say that Recollections is the chronicle of de Tocqueville’s encounter with her.

The question of human history and what de Tocqueville called “the world’s destiny” are described as follows:

l wrote histories without taking part in public affairs, and politicians whose only concern was to control events without a thought of describing them. And I have invariably noticed that the former see gen­eral causes everywhere, whereas the latter, spend­ing their lives amid the disconnected events of each day, freely attribute everything to particular incidents and think that all the little strings their hands are busy pulling daily are those that control the world’s destiny. Probably both of them are mistaken.

For my part I hate all those absolute systems that make all the events of history depend on great first causes linked together by the chain of fate and thus succeed, so to speak, in banishing men from the history of the human race. Their boasted breadth seems to me narrow, and their mathematical exactness false. I believe, pace the writers who find these sublime theories to feed their vanity and lighten their labours, that many important historical facts can be explained only by accidental circumstances, while many others are inexplicable. Finally, that chance, or rather the concatenation of secondary causes, which we call by that name because we can’t sort them all out, is a very important element in all that we see taking place in the world’s theatre. But I am firmly convinced that chance can do nothing unless the ground has been prepared in advance. Antecedent facts, the nature of institutions, turns of mind and the state of mores are the materials from which chance composes those impromptu events that surprise and terrify us.

Alexis de Tocqueville, Recollections, 1893, Anchor Books, page 78.

De Tocqueville warns us that the world’s destiny is always murky and what he calls a labyrinth and a whirlwind. He says:

Mentally I reviewed the history of our last sixty years and smiled bitterly to myself as I thought of the illusions cherished at the end of each phase of this long revolution; the theories feeding these illusions; our historians’ learned daydreams, and all the ingenious false systems by which men sought to explain a present still unclearly seen and to foresee the unseen future.

Recollections, page 83.

He continues:

Shall we reach, as other prophets as vain perhaps as their predecessors assure us, a more complete and profound social transformation than our fathers ever foresaw or desired, and which we ourselves cannot yet conceive; or may we not simply end up in that intermittent anarchy which is well known to be the chronic incurable disease of old peoples? I cannot tell, and do not know when this long voyage will end; I am tired of mistaking deceptive mists for the bank. And I often wonder whether that solid land we have sought for so long actually exists, and whether it is not our fate the rove the seas forever!

Recollections, pages 83-84.

And yet, with all that profound uncertainty, he offers a very sweeping interpretation of French history from the French Revolution (1789) to the French Revolution of 1848. The famous painting by Eugène Delacroix, Liberty Leading the People (French: La Liberté guidant le peuple), commemorating the July Revolution of 1830, falls in between.

Despite de Tocqueville’s warnings about the slipperiness of historical judgement, he arrives at an extremely precise interpretation of his own:

Seen as a whole from a distance, our history from 1789 to 1830 appears to be forty-one years of deadly struggle between the Ancien Régime with its traditions, memories, hopes and men (i.e. the aristocrats), and the new France led by the middle class. 1830 would seem to have ended the first period of our revolutions, or rather, of our revolution, for it was always one and the same, through its various fortunes and passions, whose beginning our fathers saw and whose end we shall in all probability not see. All that remained of the Ancien Régime was destroyed forever. In 1830 the triumph of the middle class was decisive and so complete that the narrow limits of the bourgeoisie encompassed all political powers, franchises, prerogatives, indeed the whole government, to the exclusion, in law, of all beneath it and, in fact, of all that had once been above it. Thus the bourgeoisie became not only the sole director of society, but also, one might say, its cultivator. It settled into every office, prodigiously increased the number of offices, and made a habit of living off the public Treasury almost as much as from its own industry.

Recollections, page 5.

Reviewing the first sentence from the quote above, one can see a deep characterization of an era, with the conclusion “in 1830 the triumph of the middle class was decisive…” Notice the profound paradox that on one hand de Tocqueville spoke of the elusiveness of history despite providing the definite description of this period. Contrast “seen as a whole from a distance” with one of the themes of his recollections, that it is not given to us to understand history.

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.”