Tag: infinity

Realms and Domains: Levels and Confusion

Are we governed by words or numbers? Martin Heidegger’s star pupil, Hans-Georg Gadamer, points a penetrating flashlight at this question of words vs. numbers when he writes, “It is obvious that not mathematics but the linguistic nature of people is the basis of human civilization.”

According to Gadamer, our primary way of being is interpretative rather than computative. Our fundamental function is to cope, not to theorize. He argues, we can never finally step outside the traditions and practices of our culture. As one critic stated, “the metaphysical aid of a view from nowhere is seen by Gadamer as a questionable illusion that can have damaging consequences for a culture. It is not that scientific methods are mistaken—he thinks that science is involved an unstoppable dynamic which cannot be halted by philosophical or other objections…Gadamer gives a central role to art in questioning the dominance of the methods of the natural sciences. The artwork is not something to be determined by concepts, but something which ‘happens’ via its reception in real social contexts…”

Think about the interaction between words and numbers in the opening of Vladimir Nabokov’s memoir, Speak, Memory:

The cradle rocks above an abyss, and common sense tells us that our existence is but a brief crack of light between two eternities of darkness. Although the two are identical twins, man, as a rule, views the prenatal abyss with more calm than the one he is heading for (at some forty-five hundred heartbeats an hour).

Vladimir Nabokov, Speak, Memory, Vintage Books, 1989, page 19.

Note how Nabokov describes our existence above. Think about the word “eternities”? It brings to mind infinity. For example, in algebra, 1/x goes to infinity as 1 approaches zero. Nabokov also states that man is doing all this infinity-watching which he describes in heartbeats per hour.

Another issue in this realm and domain confusion is provided by Gabriel Marcel, when he writes, “We must carefully avoid all confusion between the mysterious and the unknowable.” Marcel continues:

A problem is something which I meet, which I find complete before me, but which I can therefore lay siege to and reduce. But a mystery is something in which I myself am involved, and it can therefore only be thought of as “a sphere where the distinction between what is in me and what is before me loses its meaning and its initial validity”. A genuine problem is subject to an appropriate technique by the exercise of which it is defined; whereas a mystery, by definition, transcends every conceivable technique. It is, no doubt, always possible (logically and psychologically) to degrade a mystery so as to turn it into a problem. But this is a fundamentally vicious proceeding, whose springs might perhaps be discovered in a kind of corruption of the intelligence. The problem of evil, as the philosophers have called it, supplies us with a particularly instructive example of this degradation.

Just because it is the essence of mystery to be recognized or capable of recognition, it may also be ignored and actively denied. It then becomes reduced to something I have “heard talked about” but which I refuse as only “being for other people”; and that in virtue of an illusion which these “others” are deceived by, but which I myself claim to have detected.

We must carefully avoid all confusion between the mysterious and the unknowable. The unknowable is in fact only the limiting case of the problematic, which cannot be actualized without contradiction. The recognition of mystery, on the contrary, is an essentially positive act of the mind, the supremely positive act in virtue of which all positivity may perhaps be strictly defined. In this sphere everything seems to go on as if I found myself acting on an intuition which I possess without immediately knowing myself to possess it— an intuition which cannot be, strictly speaking, self-conscious and which can grasp itself only through the modes of experience in which its image is reflected, and which it lights up by being thus reflected in them.

Gabriel Marcel, The Mystery of Being, Vol. 1: Reflection & Mystery, Harper Torchbooks, 1965, page 260-261.

A final profound confusion is the body as a physical item vs. a means of expression. Picture Fred Astaire dancing opposite Ginger Rogers. You have both the movements of his dance and what he conveys through body language. In order to dance, you have the biochemical fuel (food) to enable the biomechanical movement of the dance. The courtship expressed through his movements is something different. Marcel describes it thus:

We should recall, at this point, what we said in an earlier lecture about the body; the latter is not merely an instrument, it presents us with a kind of reality which is quite different from the reality of any sort of apparatus, in so far as it, my body, is also my way of being in the world.

Gabriel Marcel, page 257.

See also “Existence and the Problem of Separability”, “Is It Good to Be a Detached Observer?” and “Arguments Without End: A Few Simple Examples” which also reference Marcel.

Existence and the Problem of Separability

In contracts and legal documents, there is a concept of “separability” which means that clause A is connected to clause B in a way that cannot be undone.

The great twentieth century French philosopher, Gabriel Marcel, has a profound non-separability analysis of existence itself. Marcel wrote:

…We cannot really separate:—

  1. Existence
  2. Consciousness of self as existing
  3. Consciousness of self as bound to a body, as incarnate.
Gabriel Marcel, Being and Having: An Existentialist Diary, Harper Torchbooks, 1965, page 10.

This leads to saying things that are very deep aspects of existential thinking governed by the points above. For example, Marcel wrote, “As I have said elsewhere, the moment I treat my body as an object of scientific knowledge, I banish myself to infinity.” Consider the end of the previous phrase. The very notion that someone, within themselves could go down a route to “banish themselves” in this way, shows you a door into very deep psychological problems. In other words, Marcel is telling you that some psychological problems could have a philosophical or existential root cause and are a type of blunder.

This is not to say that one should avoid scientific thought. Taking a drop of your blood and viewing it under a microscope is not a blunder. However, if you take this to the extreme of trying to analyze everything scientifically, to the exclusion of philosophy in daily life, this is Marcel’s warning.

Marcel’s context for this thinking is given here:

Notes for a Paper to the Philosophical Society

Undated, written in 1927 or 19281

When I affirm that something exists, I always mean that I consider this something as connected with my body, as able to be put in contact with it, however indirect this contact may be. But note must be taken that the priority I thus ascribe to my body depends on the fact that my body is given to me in a way that is not exclusively objective, i.e. on the fact that it is my body. This character, at once mysterious and intimate, of the bond between me and my body (I purposely avoid the word relation) does in fact colour all existential judgments.


What it comes to is this. We cannot really separate:—

  1. Existence
  2. Consciousness of self as existing
  3. Consciousness of self as bound to a body, as incarnate.

From this several important conclusions would seem to follow:

  1. In the first place, the existential point of view about reality cannot, it seems, be other than that of an incarnate personality. In so far as we can imagine a pure understanding, there is, for such an understanding, no possibility of considering things as existent or non-existent.
  2. On the one hand, the problem of the existence of the external world is now changed and perhaps even loses its meaning; I cannot in fact without contradiction think of my body as non-existent, since it is in connection with it (in so far as it is my body) that every existing thing is defined and placed. On the other hand, we ought to ask whether there are valid reasons for giving my body a privileged metaphysical status in comparison with other things.
  3. If this is so, it is permissible to ask whether the union of the soul and body is, in essence, really different from the union between the soul and other existing things. In other words, does not a certain experience of the self, as tied up with the universe, underlie all affirmation of existence?
  4. Inquire whether such an interpretation of the existential leads towards subjectivism.
  5. Shew how idealism tends inevitably to eliminate all existential considerations in view of the fundamental unintelligibility of existence. Idealism versus metaphysics. Values detached from existence: too real to exist.

Existential and personalist interests closely linked. The problem of the immortality of the soul is the pivot of metaphysic.

Every existent is thought of like an obstacle by which we take our bearings—like something we could collide with in certain circumstances—resistent, impenetrable. We think of this impenetrability, no doubt, but we think of it as not completely thinkable.2 Just as my body is thought of in so far as it is a body, but my thought collides with the fact that it is my body.

To say that something exists is not only to say that it belongs to the same system as my body (that it is bound to it by certain connections which reason can define), it is also to say that it is in some way united to me as my body is.

Incarnation—the central ‘given’ of metaphysic. Incarnation is the situation of a being who appears to himself to be, as it were, bound to a body. This ‘given’ is opaque to itself: opposition to the cogito. Of this body, I can neither say that it is I, nor that it is not I, nor that it is for me (object). The opposition of subject and object is found to be transcended from the start. Inversely, if I start from the opposition, treating it as fundamental, I shall find no trick of logical sleight of hand which lets me get back to the original experience, which will inevitably be either eluded or (which comes to the same thing) refused. We are not to object that this experience shews a contingent character: in point of fact, all metaphysical enquiry requires a starting-point of this kind. It can only start from a situation which is mirrored but cannot be understood.

Inquire if incarnation is a fact; it does not seem so to me, it is the ‘given’ starting from which a fact is possible (which is not true of the cogito).

A fundamental predicament which cannot be in a strict sense mastered or analysed. It is exactly this impossibility which is being stated when I declare, confusedly, that I am my body; i.e. I cannot quite treat myself as a term distinct from my body, a term which would be in a definable connection with it. As I have said else-where, the moment I treat my body as an object of scientific know-ledge, I banish myself to infinity.

This is the reason why I cannot think of my death, but only of the standstill of that machine (illam, not banc). It would perhaps be more accurate to say that I cannot anticipate my death, that is, I cannot ask myself what will become of me when the machine is no longer working.3

[1] This paper was never delivered.

[2] It is thought of, but it is never resolved. The opacity of the world is in a certain sense insoluble. The link between opacity and Meinbeit. My idea is opaque to me personally in so far as it is mine. We think of it as an adherence. (Note written Feb. 24th, 1929.)

[3] ‘To be involved.’ (idée d’un engagement) Try to shew in what sense this implies the impossibility (or absolute non-validity of my representing my death. In trying to think of my death I break the rules of the game. But it is radically illegitimate to convert this impossibility into a dogmatic negation. (Note written Feb. 24th, 1929.)

It is evident that this whole train of thought is at the root of le Gouvernail: the first notes on the theme of le Gouvernail were composed a few days after these. (Note written April 13th, 1934.)

Gabriel Marcel, Being and Having: An Existentialist Diary, pages 10-12

The reader should not be afraid of converting this level of discussion to everyday life and everyday slang. For example, when Marcel uses phrases like, “I banish myself to infinity” that should resonate with phrases like “flipping out.”

Take this example from Kierkegaard. There’s a book-seller in Copenhagen and he greets a customer who has just entered his shop. The book-seller turns to his wife and says, “It is I who am speaking, isn’t it?” Kierkegaard’s book-seller seems to have “banished himself” and cannot return.

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.”

Education and Spontaneous Learning

We give you examples of being receptive to the world around you and learning to see and hear as a form of education:

There is a show on PBS called Stories from the Stage. People come forward to a microphone on a stage and tell personal stories from their past, stories that they consider important, informative, educational (in the widest sense), and usable by the listener. One of the early “people at the mic” on stage is a teenage girl who says something, in a plaintive sorrowful voice, like: “I have been waiting far too long…to wait for someone…to see me.”

This perplexed girl is unwittingly raising the question of a deep human hunger: the hunger for “personhood.” At a young age, this primordial hunger expresses itself as somebody befriending me (i.e., the speaker needs a real friend) so that the befriended person comes into clearer focus to themselves, achieving personhood.

Very intelligent philosophers like Emmanuel Levinas of France have spent their entire lives trying to understand the connections between countenances (how a person “wears a face”), personhood, interactive life, etc.

In his book, The Face of the Other (the girl wants somebody to notice her and her face and like her and “smile upon her”) Levinas has a deep analysis of all these human yearnings and self-definitional journeys and quests:

“The Face of the Other” is an evocative phrase used by Emmanuel Levinas, an important twentieth-century philosopher.

  • “Other” (sometimes capitalized, sometimes not) usually translates the French word autrui, which means “the other person” or “someone else” (other than oneself). It is thus the personal other, the other person, whoever it is, that each of us encounters directly, or experiences the traces of, every day. Of course, we encounter a multiplicity of others, but Levinas more often uses the singular “other” to emphasize that we encounter others one at a time, face to face.
  • By “face,” Levinas means the human face (or in French, visage), but not thought of or experienced as a physical or aesthetic object. Rather, the first, usual, unreflective encounter with the face is the living presence of another person.

Thus, when we come “face to face” with another person, the experience is a social and ethical one (rather than intellectual, aesthetic, or merely physical). “Living presence,” for Levinas, would imply that the other person (as someone genuinely other than myself) is exposed to me—that is, is vulnerably present—and expresses him or herself simply by being there as an undeniable reality that I cannot reduce to images or ideas in my head.

This impossibility of capturing the other conceptually or otherwise reveals the other’s “infinity” (i.e., irreducibility to a finite [bounded] entity over which I can have power).

The other person is, of course, exposed and expressive in other ways than through the literal face (e.g., through speech, gesture, action, and bodily presence generally), but the face is the most exposed, most vulnerable, and most expressive aspect of the other’s presence.

Thus, a student could be channel surfing on TV, observe this young girl saying these things on Stories from the Stage, and expand one’s understanding of this entire set of hungers and self-identity efforts and go (say) from the moment of TV watching to reading Levinas.

This is a simple example from the current world of TV where a certain particular “cri de coeur” (French: “cry from the heart”) of a girl you don’t know at all could deepen and widen your understanding by following the thread to Levinas and other profound people. The girl’s plaint where she’s “waiting for someone to see me” becomes much deeper and can be understood on a larger canvas which is exactly what we want.

Many experiences from daily life, from walking around, from moments on TV, from tiny incidents, can be pathways to higher understanding and learning if you can see and hear “with the third eye and the third ear.” (Theodore Reik talks about “listening with the third ear.”)

Education is a kind of “applied awareness.”

Education and Word and Number Hidden Vagueness

These mini-essays help students of any age to re-understand education in a deeper and more connected way.

They look for “circum-spective” intelligence. (Not in the sense of prudential or cautious but in the sense of “around-looking.”)

One of the things to begin to see is that explaining things in schools is misleading “ab initio” (i.e., from the beginning).

Let’s do an example:

In basic algebra, you’re asked: what happens to (x2 – 1)/(x – 1) as x “goes to” (i.e., becomes) 1.

If you look at the numerator (thing on top), x2 is also 1 (since 1 times 1 is 1) and (1 – 1) is zero. The denominator is also (1 – 1) and zero.

Thus you get 0 divided by 0.

You’re then told that’s a no-no and that’s because zeros and infinities lead to all kinds of arithmetic “bad behavior” or singularities.

You’re then supposed to see that x2 – 1 can be re-written as (x – 1)(x + 1) and since “like cancels like,” you cancel the x – 1 is the numerator and denominator and “get rid” of it.

This leaves simply x + 1. So, as x goes to 1, x + 1 goes to 2 and you have a “legitimate” answer and have bypassed the impasse of 0 acting badly (i.e., zero divided by zero).

If you re-understand all this more slowly you’ll see that there are endless potential confusions:

For example: you cannot say that (x2 – 1)/(x – 1) = x + 1 since looking at the two sides of the equal sign shows different expressions which are not equal.

They’re also not really equivalent.

You could say that coming up with x + 1 is a workaround or a “reduced form” or a “downstream rewrite” of (x2 – 1)/(x – 1).

This reminds us of the endless confusions in high school science: if you combine hydrogen gas (H2) with oxygen gas (O2) you don’t get water (H2O). Water is the result of a chemical reaction giving you a compound.

A mixture is not a compound. Chemistry is based on this distinction.

Math and science for that matter, are based on taking a formula or expression (like the one we saw above) and “de-cluttering” it or “shaking loose” a variant form which is not identical and not the same but functionally equivalent in a restricted way.

A lot of students who fail to follow high school or college science sense these and other “language and number” problems of hidden vagueness.
School courses punish students who “muse” to themselves about hidden vagueness. This behavior is pre-defined as “bad woolgathering” but we turn this upside down and claim it is potentially “good woolgathering” and might lead to enchantment which then underlies progress in getting past one’s fear of something like math or science or anything else.

One is surrounded by this layer of reality on all sides, what Wittgenstein calls “philosophy problems which are really language games.”

Think of daily life: you say to someone: “you can count one me.” You mean trust, rely on, depend on, where count on is a “set phrase.” (The origin of the phrase and how it became a set phrase is probably unknowable and lost in the mists of time.)

“You can count on me” does not mean you can stand on me and then count something…one, two, three.

In other words in all kinds of language (English, say, or math as a language) one is constantly “skating over” such logic-and-nuance-and-meaning issues.

The genius Kurt Gödel (Einstein’s walk around buddy at Princeton) saw this in a deep way and said that it’s deeply surprising that languages work at all (spoken, written or mathematical) since the bilateral sharing of these ambiguities would seem deadly to any clarity at all and communication itself would seem a rather unlikely outcome.

You could also say that drama giants of the twentieth century like Pinter, Ionesco and Beckett, intuit these difficulties which then underlie their plays.

All of this together gives you a more “composite” “circum-spective” view of what is really happening in knowledge acquisition.

Then and Now Thinking: Facile Comparisons Lead to “Concept-Fraud”

The economist Arthur Laffer recently received an award from President Trump. Laffer wants to deceptively “cartoonize” reality by arguing that as taxes “go to 100%” (i.e., confiscation), output will go to zero and conversely as taxes “go to zero” output will go to “infinity.”

This is an example of playing with “bad infinities.”

This Laffer argument has been naively compared to David Hume’s economics:

“Back in the eighteenth century, the wise Scot David Hume anticipated David Hume in these 1756 words of sooth:

“‘Exorbitant taxes, like extreme necessity, destroy industry by producing despair; and even before they reach this pitch, they raise raise the wages of the laborer and manufacturer, and heighten the price of all commodities. An attentive disinterested legislature will observe the point where the emolument ceases and the prejudice begins.’”

(David Hume, Writings on Economics, ed. Eugene Rotwein, Edinburgh, Thomas Nelson and Sons, 1955, page 87)

(quoted in Greed is Not Enough: Reaganomics, Robert Lekachman, Pantheon Books, 1982, page 49)

Reaganomics and Laffer-nomics have nothing to do with David Hume and facile “then-and-now” comparisons, all of which are false since the “anarcho-capitalism” of Reagan/Thatcher views has noting to do with Hume. 

Thatcher said: “properly speaking, there is no such thing as society. There are only individuals.”

But Hume believes the exact opposite as a socially conscious brand of conservative:

“Hume cherished the structures that sustain our social life. He was in this respect deeply conservative, in the good sense of the conservationist of the shapes and forms which these institutions have taken.

“And of course he was deeply mistrustful of any scatterbrained project of doing better, by promoting anarchism or society without government or law, or dismantling the institutions of contract or private property. 

“He would have had absolutely no patience whatsoever with the contemporary takeover of social ideals by monetary and market values.

“When free-marketeers say that there is no such thing as society, they are denying the very arches needed to sustain contracts, law, government, and markets in the first place, and then knavery loses its stigma, and we may well expect the worst, as their practice becomes ‘answerable’  to their ‘speculation.’”

(quoted in How to Read Hume, Simon Blackburn, Granta, 2008, page 70)

Deceivers make duplicitous linkages between hallowed names and ideas of the past and the dangerously “tricky” present.

Thus, Hume-to-Laffer linkages and trajectories makes no sense whatsoever. This is an example of “then-and-now thinking” used for “concept-fraud.”

Unnoticed Dimensions of Knowledge

Let’s “get down to cases” right now:

  1. You learn decimals and fractions in school. You see that 1/2 can be written as 0.5 or 0.50 or with as many zeros as you like. That seems “clean.”

But 1/3 is equal to something more complex (i.e., 0.3 recurring or repeating, like 0.3333 and so on infinitely).  If you divide 1 by three you keep getting three.

Imagine you want to experiment a bit, and multiply the fraction 1/3 by three and the 0.3 recurring by three, thus not affecting things since you’re doing the same thing to both sides of the equation.

You get:  1 = 0.9 recurring or repeating.

You’re suddenly puzzled: How can 1 be obtained by adding “slices of 9 fractions” (i.e., 9/10 + 9/100 + 9/1000) to infinity. How do you get to the end? What end? 

It turns out that it’s not that simple to get a grip on all this.  A person who allowed themselves to become fascinated by this specific conundrum would enter a “beautiful ocean” of mathematics beginning with so elementary a phenomenon.

This shows you a deep connection between a part (e.g., the fraction and decimal 1/3 and 0.3 recurring) and the wider world or domain or universe of numbers.

How can it be that such a simple elementary “thing” becomes so intricate, deep and elusive?

  1. Let’s jump over to an entirely different kind of example. Think of Dinesen’s novel Out of Africa. Remember the movie with Meryl Streep and Robert Redford.

Suppose you turn the movie “inside out” and “upside down” and ask: is this movie about coffee and coffee bushes, coffee markets and coffee growing, in a colonial context?  The coffee plantation is near Nairobi (today’s Kenya) and involves plantation economics, colonial relations with Kikuyu peoples, German-British colonial tensions around World War I.

Suppose I take the “backstory” and make that the “frontstory”.

The story of “economic botany” (coffee growing is one case) and colonial tensions between and among Europeans as well as Europeans and Africans is the deeper and larger story while the “musical beds” of the Westerners is a colorful footnote.   

We have the perennial question of “parts and wholes” which is one theme of this book.       

  1. Why does science “orbit” some numbers such as π (pi) (i.e., 22/7)?

You learn in school that there’s a ratio called π (pi) which is 22/7. Think of π (pi) as some kind of essence of circularity. Remember πr2 and 2πr in grade school.

Why does it keep appearing in almost every equation of physics? Why would “circleness” “haunt” science and math? Probability and statistical theory are dependent on π (pi) as a variable. Why?

You could peruse:

A History of Pi is a 1970 non-fiction book by Petr Beckmann that presents a layman’s introduction to the concept of the mathematical constant π (pi). 

Why does science “orbit” some numbers such as π (pi)?

This is an example of this quest for connectedness.