Tag: epistemology

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.

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.”

COVID-19 and “Naïve Probabilism”

[from the London Mathematical Laboratory]

In the early weeks of the 2020 U.S. COVID-19 outbreak, guidance from the scientific establishment and government agencies included a number of dubious claims—masks don’t work, there’s no evidence of human-to-human transmission, and the risk to the public is low. These statements were backed by health authorities, as well as public intellectuals, but were later disavowed or disproven, and the initial under-reaction was followed by an equal overreaction and imposition of draconian restrictions on human social activities.

In a recent paper, LML Fellow Harry Crane examines how these early mis-steps ultimately contributed to higher death tolls, prolonged lockdowns, and diminished trust in science and government leadership. Even so, the organizations and individuals most responsible for misleading the public suffered little or no consequences, or even benefited from their mistakes. As he discusses, this perverse outcome can be seen as the result of authorities applying a formulaic procedure of “naïve probabilism” in facing highly uncertain and complex problems, and largely assuming that decision-making under uncertainty boils down to probability calculations and statistical analysis.

This attitude, he suggests, might be captured in a few simple “axioms of naïve probabilism”:

Axiom 1: more complex the problem, the more complicated the solution.

This idea is a hallmark of naïve decision making. The COVID-19 outbreak was highly complex, being a novel virus of uncertain origins, and spreading through the interconnected global society. But the potential usefulness of masks was not one of these complexities. The mask mistake was consequential not because masks were the antidote to COVID-19, but because they were a low cost measure the effect of which would be neutral at worst; wearing a mask can’t hurt in reducing the spread of a virus.

Yet the experts neglected common sense in favor of a more “scientific response” based on rigorous peer review and sufficient data. Two months after the initial U.S. outbreak, a study confirmed the obvious, and masks went from being strongly discouraged to being mandated by law. Precious time had been wasted, many lives lost, and the economy stalled.

Crane also considers another rule of naïve probabilism:

Axiom 2: Until proven otherwise, assume that the future will resemble the past.

In the COVID-19 pandemic, of course, there was at first no data that masks work, no data that travel restrictions work, no data of human-to-human transmission. How could there be? Yet some naïve experts took this as a reason to maintain the status quo. Indeed, many universities refused to do anything in preparation until a few cases had been detected on campus—at which point they had some data, as well as hundreds or thousands of other as yet undetected infections.

Crane touches on some of the more extreme examples of his kind of thinking, which assumes that whatever can’t be explained in terms of something that happened in the past is speculative, non-scientific and unjustifiable:

“This argument was put forward by John Ioannidis in mid-March 2020, as the pandemic outbreak was already spiralling out of control. Ioannidis wrote that COVID-19 wasn’t a ‘once-in-a-century pandemic,’ as many were saying, but rather a ‘once-in-a-century data-fiasco’. Ioannidis’s main argument was that we knew very little about the disease, its fatality rate, and the overall risks it poses to public health; and that in face of this uncertainty, we should seek data-driven policy decisions. Until the data was available, we should assume COVID-19 acts as a typical strain of the flu (a different disease entirely).”

Unfortunately, waiting for the data also means waiting too long, if it turns out that the virus turns out to be more serious. This is like waiting to hit the tree before accepting that the available data indeed supports wearing a seatbelt. Moreover, in the pandemic example, this “lack of evidence” argument ignores other evidence from before the virus entered the United States. China had locked down a city of 10 million; Italy had locked down its entire northern region, with the entire country soon to follow. There was worldwide consensus that the virus was novel, the virus was spreading fast and medical communities had no idea how to treat it. That’s data, and plenty of information to act on.

Crane goes on to consider a 3rd axiom of naïve probabilism, which aims to turn ignorance into a strength. Overall, he argues, these axioms, despite being widely used by many prominent authorities and academic experts, actually capture a set of dangerous fallacies for action in the real world.

In reality, complex problems call for simple, actionable solutions; the past doesn’t repeat indefinitely (i.e., COVID-19 was never the flu); and ignorance is not a form of wisdom. The Naïve Probabilist’s primary objective is to be accurate with high probability rather than to protect against high-consequence, low-probability outcomes. This goes against common sense principles of decision making in uncertain environments with potentially very severe consequences.

Importantly, Crane emphasizes, the hallmark of Naïve Probabilism is naïveté, not ignorance, stupidity, crudeness or other such base qualities. The typical Naïve Probabilist lacks not knowledge or refinement, but the experience and good judgment that comes from making real decisions with real consequences in the real world. The most prominent naïve probabilists are recognized (academic) experts in mathematical probability, or relatedly statistics, physics, psychology, economics, epistemology, medicine or so-called decision sciences. Moreover, and worryingly, the best known naïve probabilists are quite sophisticated, skilled in the art of influencing public policy decisions without suffering from the risks those policies impose on the rest of society.

Read the paper. [Archived PDF]

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.

What We Mean by “Epochal Waters”

We sometimes use the phrase “epochal waters” to refer to the deepest layers of the past which we “swimmers” at the surface of the ocean don’t see or know. “Epochal waters” are latent, currents are closer to the surface.

There’s a similar idea from the French philosopher Michel Foucault who died in 1984. In his The Order of Things, classic from 1966, he talks about the “episteme” (as in epistemology) that frames everything from deep down. (The Greeks distinguished between “techne” (arts, crafts, practical skills and “episteme” (theory, overview).

“In essence, Les mots et les choses (Foucault’s The Order of Things) maintains that every period is characterized by an underground configuration that delineates its culture, a grid of knowledge making possible every scientific discourse, every production of statements. Foucault designates this historical a priori as an episteme, deeply basic to defining and limiting what any period can—or cannot—think.

Each science develops within the framework of an episteme, and therefore is linked in part with other sciences contemporary with it.

(Didier Eribon, Michel Foucault, Harvard University Press,  1991, page 158)

Take a simple example. A discussion comes up about what man is or does or thinks or knows. In today’s episteme or pre-definition, one thinks immediately not of man in terms of language or the invention of gods, but in terms of computational genomics, big data, bipedalism (walking upright on two legs). Its assumed in advance via an invisible episteme, that science and technology. physics, genetics, big data, chemistry and biology hold the answer and the rest is sort of outdated. This feeling is automatic and reflexive like breathing and might be called “mental breathing.”

One’s thoughts are immediately sent in certain directions or grooves, a process  that is automatic and more like a “mental reflex” than a freely chosen “analytical frame.” The thinker has been “trained” in advance and the episteme pre-decides what is thinkable and what is not.

There are deep episteme that underlie all analyses: for example, in the Anglo-American tradition of looking at things, the phrase “human nature” inevitably comes in as a deus ex machina (i.e., sudden way of clinching an argument, the “magic factor” that has been there all along). If you ask why are you suddenly “importing” the concept of “human nature,” the person who uses the phrase has no idea. It’s in the “epochal water” or Foucault’s episteme, and it suddenly swims up from below at the sea floor.

Another quick example: In the Anglo-American mind, there’s a belief from “way down and far away” that failure in life is mostly about individual behavior (laziness, alcoholism, etc.) and personal “stances” while “circum-stances” are an excuse. This way of sequencing acceptable explanations is deeply pre-established in a way that is itself hard to explain. It serves to “frame the picture” in advance. These are all “epochal water“ or episteme phenomena.