What is the experience of mathematical motion?

(This is not so much an argument or a claim as an attempt to describe an experience.)

Imagine any simple motion of a mathematical thing: a point going across a line, a secant becoming a tangent, a rotation making a circle, etc. Question: how long does that take you? Is there a minimum time that you need for that, or can you see that any time T is always one you could better?  This seems to be a classic limit argument – we can make the time it takes less than any given time. We almost immediately get to the point where we are imagining a flicker of a speed that we understand through numbers that we can increase indefinitely.

Now so far as we are really demanding that the motion go across some distance, we cannot make it instantaneous; but so far (1) as the amount of time it takes can be made less than any given time (2) it is not meaningful to speak about the speed of the motion (3) since the motion just is the representation of a numerical value that can be increased indefinitely, the motion is instantaneous.

(I’ve wondered if we could make a similar point from the fact that there need be no time factor for taking an integral or a derivative.)

Minimally, this might show a difference between physical and mathematical motion, but this leave the question as a mere negation. It might be an insight into that mysterious reality that St. Thomas claim constitutes mathematical reality: intelligible matter.

Is the whole question simpler? Just as we can’t meaningfully ask “how long does that take you?” It seems equally meaningless to say “how long is the line you imagine? or “How large is the circle you are demonstrating  property X about?” It either has no length, or its length is the one you say it represents. Either way, this is some mysterious length!

I wonder how this might compare to the motion of the angels: which is both discrete and yet covering all points between its termini. Is mathematics the angelic intuitions that animals have? This might be another approach to hylemorphism, or a metaphysical account of number lines or the system of all numbers – which both have points only potentially, and yet in such a way that the mind passing over these points does not require the imperfection of time or local separation from a term.


-Science as propaganda for those allergic to ultimates.

-Large ideas vs. subconscious ultimates; i.e. large ideas as opposed to infinite ones.

-Poincare on Euclid’s common notions: “I regard them as analytic a priori, and think of them no further.”

-We begin in media res, philosophy tracing backwards to more distinctly discern what we already know, the sciences striving to find conclusions.

-The indifference to first things. A shrug with an “I don’t get it, it works,”  Bohm vs. Feynman. (a sign of who won: spellcheck recognizes Feynman)

-Possibility: the first things demand that we accept them as invisible. False, though the most successful and dominant ideas are the invisible ones.

-Einstein understood Bohr as ruining the reality of the universe and even its intelligibility. Bohr might even have wanted to be taken that way. One wonders how the dialogue would have went if Bohr, like many of those who followed him, advanced his position as resting on the reality of the subject and requiring an indeterminism that allows for freedom.

Immateriality as a negation and not a property

All Thomists are argue from the universality  of thought to the immateriality of the intellect. Why not make the more immediate inference from the universality of thought to the universality of the intellect? This is in some ways very silly, but it raises interesting questions.

The silliness is that universality is properly a way things are known, not a way in which things exist. We shouldn’t expect anything proper  to the mode of knowing to be a feature of being. Still, the Thomist proof doesn’t work unless one takes it as given that universals exist. But what does it mean to say this? We slide over the problem when we simply argue that the intellect is non-material. This is a pure negation, not a property. It’s only true that the intellect is non-material in the same way that it’s a non-banana, or a non-sjkldhfv or a non-squarecircle. As Aristotle points out, non-X (like “immaterial”) is not a word – it has no signification because it cannot place our thought on one side or another of the principle of contradiction. Both what exists and does not exist are equally non-X. So what is it about intellect, or things like it, that makes us use such terms to describe it? And how do we resolve the tension of saying that the intellect both exists, and yet is primarily described by terms that are indifferent to both being and non-being?

gods with borders (1)

If it does not please you to serve the LORD,
decide today whom you will serve,
the gods your fathers served beyond the River
or the gods of the Amorites in whose country you are now dwelling.

Joshua’s line immediately after this is the famous one ” as for me and my family,  we will serve the Lord.” But these verses make clear what the other option is. Who do we serve if we do not serve the Lord?  The gods your fathers served beyond the River or the gods of the Amorites in whose country you are now dwelling. First, notice what is left out: there is no mention of idols or demons or even that these gods are false. They are simply gods with borders. The true God is opposed to the sort of god that begins or ends at the river.

Platonic notes

-Aristotle: the reason many things are alike are because they have the same form. Very well, but you also say there are many like forms: John’s soul, Mary’s soul…

-God and the universe are distinct, for God is not material. God and the universe are not distinct, for he exists in no other place. Since each soul has its own body, the same reasoning applies. So why not form and matter generally?

-Dekoninck: for all we know, the non-living world (and thus even the living so far as it is conditioned by it or studied by physics and chemistry) might be ontologically one thing. But this opens the possibility that Aristotle’s Categories cannot serve as a guide in either physics or metaphysics, which is exactly how Aristotle himself takes it. He’s quite convinced that what is not said of nor present in is a description of a fundamental ontological reality. If this is not the case, and ontological monism is possible, then Aristotle’s distinct substances are accidental divisions of a fundamental substance. This seems to be what Plato thought they were (Timaeus makes all physical reality various warpings of some underlying screen) and it seems to be how we want to see them. What scientist is content with irreducible dualities in the physical realm? Are any of us?

-St. Thomas, in explaining the knowledge of Christ, gives an account of what a complete human science would be: a complete science for every sensible species. This would have to be every ontologically distinct sensible species. So how many is that, and how do we answer the question?

-All agree science is about the universal as opposed to this X existing now.  But then science is about X throughout history. History – with all of its messiness, particularity, irrationality and facticity-as-opposed-to-truth – must enter into the scientific universal. Into essence.

- The non-living world follows a logic of unification. There is an element of this in the study of the living too (so far as the living is conditioned by the merely physical or descending from a single ancestor) but it also goes in the reverse. Advances in biology must also multiply the Kingdoms.

-Dialectic tends to unity even while it requires division – a dialectical opinion is essentially one side of a contradiction. Dialectic thus tends to an essentially unattainable goal. If you like, it tends to the impossible and therefore to nothing at all. It is not clear what we should do about this in light of Aristotle’s vary accurate accounts of agents only being able to tend to defined ends (is it even coherent to tend-to-nothing? You can’t remove something to tend to and then proceed to tend to it.). On the one hand, inquiry and science do not advance by physical motion and immanent act are complete at every moment. But there is some sort of perfection involved in the advance of knowledge.

- If all our knowledge is scientific, then there is necessarily an epistemic gap, even if all portions of it can be illuminated. Call it the epistemic Hilbert hotel. If a gap is necessary, what do we make of God-in-the-gaps?

-The gap is finite, so far as we tend to something; it is unbounded to far as we cannot attain it by the nature of our own powers. We are tending to a nature above ourselves.

The bad infinite (1)

One can make any number of horror-movie monsters by removing the limits from some natural desire or process: Alien is reproduction without regard for anything else; The Blob is pure and unlimited growth (with unlimited consumption as a corollary); The Thing is a sacculina-esque parasite that places no limits on its own desire to survive etc.. There are two sides to removing a limit: on the one hand we get a monster and on the other hand we get godlike power. These two aspects can be reflected in the plot when protagonists want to kill the unlimited thing as a monster and antagonists (government officials) marvels and desire the unlimited thing as a god. The marvel of the antagonists is tempered by the fact that they do not simply marvel at the godlike thing but also desire to possess and control it. Such possession itself is a claim to unlimited power, and so is a redoubling of the monster.



A government of laws and not of men. That is, the organizing structure of human relations should not be capricious, partial to some, and make-it-up-as-you-go; and it also ought to be knowable and communicable to all, unlike the private motivations of individuals. But it should be pretty clear that the sheer multiplication of laws, or even widespread respect for the law does not suffice to attain what the slogan advocates, and can even be the cause of it. A billion laws are as unknowable and incommunicable as a man’s private intentions – certainly in the practical sense, which is the only relevant sense here. So the reverse is also true, sc.

A government of men and not of laws. That is, the law should only mediate between a rational person and the persons in power. Law should be a structure by which rational persons feel and know that their actions are supported (usually tacitly) by all the force of those who have been given the use of force, and reason is always broader and more fluid than what can be written in law. Law therefore imperfectly mediates rational persons: its purpose is to ensure that a rational person can assume the support of the coercive power of force which he has handed over to his rulers. When a rational person makes a decision and then has to go off and figure out whether the law supports it or not, it might be more reasonable and humane to build our own forts, arm ourselves to the teeth, and fight out our politics without civilization.

(For more of this, done better, watch Philip K. Howard, and read his “The Death of Common Sense“)

What is a metaphysical man?

Just for fun, let’s make things as needlessly controversial as we can:

The theory of evolution cannot explain metaphysical living things, which can only arise by the intention of mind.

The claim is either obvious or vacuous and demands that we flesh out the first adjective: what is a metaphysical thing? Is it an abstract thing? A sort of idealization or ghost of the reality or “brute facts” in front of us? Is it a mystical or mysterious account of things invoking occult powers and gnostic incantations?

It might be any of these things, but it needn’t be. Explaining Athanasius’s cosmology below provides one very down-to-earth way of understanding metaphysical things. Athanasius’s claim is that we cannot explain anything as a whole having distinct parts considered precisely as a whole having distinct parts, without tracing its origin to mind as opposed to chance. That this is not the only way of considering things goes without saying: we can consider them as collections of traits, as members of a population, as phenotypes related to genotypes, as having masses we can measure on a scale, as being picturesque. etc. ad infinitum. In fact, we might consider one and the same reality in all these different ways. This is not a matter of dividing up things – a genotype or a mass is might well be a whole, but to consider something as a mass is not to consider it as a whole, and it is the way of considering things that divides up the sciences. You might consider the number of chromosomes in a human being (46) as an even number, but this does not make the study of chromosomes (biology) is the study of even numbers (mathematics). Likewise, you might consider an animal or living thing as a whole, but when you do this is not the study of animals (one of the natural sciences) but the study of wholes as wholes.  I say this is a metaphysical account. Why?

Aristotle’s simplest account is that every science leaves certain things out: physics does not study the Pythagorean theorem (even if it uses it); mathematics does not deal with the genes (even if it is relevant to their study), etc. But there is a certain group of things that are common to all things, and whole and part are perfect examples. Both triangles and turnip plants are wholes, both the hypotenuse and the foot are parts. It is this class of things that are not peculiar to some distinct science that Aristotle first defines as metaphysical. This is what is first meant by “the science of being as being”, i.e. it is the study of what is common as opposed to what is proper to any particular science. If you want to make a proper study of wholes, you can’t reach for any particular science that considers a part of being. But these common ideas are emphatically not “more abstract” or “less concrete” or “more like ideas”. John Smith is a real, concrete whole just as much as he is a real, concrete animal. The study of being as being is not the study of some vast, universal thing any more than biology is the study of some vast universal animal.

In considering wholes as wholes in relation to their separate parts, Athanasius gives a proof that all things in the cosmos arise by purpose and design as opposed to chance. This is not the only mode of consideration of things, and it leaves open the possibility of infinite sciences that find no evidence of design or purpose.

St. Athanasius contra Epicurian Cosmology

From the day I’ve read it 15 years ago till now, I’ve been fascinated by Athanasius’s argument against the Epicureans:

For if, as they say, everything has had its beginning of itself, and independently of purpose, it would follow that everything had come into mere being, so as to be alike and not distinct. For it would follow in virtue of the unity of body that everything must be sun or moon, and in the case of men it would follow that the whole must be hand, or eye, or foot. But as it is this is not so. On the contrary, we see a distinction of sun, moon, and earth; and again, in the case of human bodies, of foot, hand, and head. Now, such separate arrangement as this tells us not of their having come into being of themselves, but shows that a cause preceded them; from which cause it is possible to apprehend God also as the Maker and Orderer of all.

De Incarnatione c. 1

Athanasius is thus arguing from to the existence of God merely from the separation or division of things, saying that the consequence of making everything independent of purpose would make everything uniform, undifferentiated, and homogeneous. Though he speaks of this meaning “everything would be sun…[or] the whole body must be a hand”, this is not to be taken absolutely – Athanasius clearly understood that you could clearly dissolve the separation and division of the parts of the hand in the same way, if you considered them as somehow wholes. What is so fascinating about the proof is not only its novelty but its absolute minimalism: there is no reference to the beauty or complexity of the separated things, merely and appeal to separation of the parts of a whole as opposed to absolute homogeneity. How does this work?

Two answers:

1.) The basic argument is this:

The universe is a combination of distinct, differentiated parts.

When parts are combined by chance, they tend toward homogeneity and absence of differentiation.

Automatic mixers, for example, use chance to blend parts together by just heaping them together and churning them around. But the only reason we use such a method to combine things is because we want the parts mixed together evenly, so that they are homogeneously distributed throughout.

Chance does often lead to distinct results, like the probability of results of throwing dice or various poker hands. But this does not seem to touch Athanasius’s point: the results of dice throws are not combinations of parts, as the universe is; and while poker hands are combinations of distinct parts, these hands are essentially parts of a larger whole, and cannot be taken as wholes themselves. You’re not playing poker (that is, gambling or playing a game of chance) if the only cards in the deck are the ones in your hand. They have to be parts of a shuffled whole: that is, you perform an action ordered towards giving all the players an equal chance.

2.) The proof takes as its datum the separation of the parts of the whole, saying that this is impossible without mind. Now the separation of parts in a whole requires two things: the unity of the whole and the multiplicity of parts. The unity of the whole is either purely artificial and stipulated, or intrinsic to the things. If it is stipulated, artificial and extrinsic, then it clearly is the product of a mind stipulating, making, or drawing together things that are in themselves separate. But if it is natural or intrinsic to the things, then it cannot be by chance, which is axiomatic to all of us through the theory of evolution. If A causes B by chance (say, the coal plant caused moths to be selected for black wings) then there is no intrinsic connection between A and B.

Reasoning and Experience

Robert Hutchinson on proofs for the existence of God:

If I were to debate myself, I would never use mystical experience as an argument for God’s existence because it is non-falsifiable, it is an unfair trump card that avoids logical reasoning. But just as my love for my wife is not the result of a logical demonstration, so, too, my faith in God is not the result of a chain of deduction. Reason can perhaps confirm what we know already by faith, but faith is rarely the result of reason. What’s more, I have this sense that the God of Abraham, Isaac and Jacob is not the Prime Mover of Aristotelian logic… and that to argue for his existence, using the paltry weapons of the human mind, seems almost presumptuous.

The opposition he is drawing here is not between reasoning about God and experiencing him, but between reason as such and experience, where “experience” means immediate knowledge. Briefly, the whole force of this argument comes not from the difference between mediate knowledge of God (theistic proof) and experience of God (mystical or religious experience), but is a general feature of the way in which immediate knowledge is always preferable to mediate knowledge, whether in theology, physics, history, biology, whatever.

It’s very often the case that the things we reason to are more worth knowing than the things we know immediately, but immediate knowledge is a better way of knowing. Ceteris paribus, if  we had a choice between a direct knowledge of some reality by an immediate manifestation and a mediate, indirect awareness of it, we’d all choose the first. It would be better, for example, if we could just see some historical event happen rather than have to argue about it through monument inscriptions, testimonies, and probability arguments; it would be better if we could just look up in the sky and see black holes as opposed to inferring them from microwave readings, etc. We only reason because a.) we can’t know something any other way or b.) we can’t unify the things we know any other way c.) we can’t communicate some things we know any other way.

Reasoning is always a substitute for experience, or something that we have to settle for given the impossibility of an immediate knowledge. Since reasoning by definition has no direct access to what it seeks to know and finds itself in a situation where the reality does not manifest itself, reasoning presupposes world that substitutes for the manifestation itself in which it can reason (cf. Chapters 1 and 2 of Sophistical Refutations). This is the world of signs i.e. things which being themselves immediately known make other things known. Instrumental signs thus both manifest something and are a veil in front of it, or perhaps a response to the veil that is, for us, over reality. It is the world of signs, considered as substitutes for things that do not manifest themselves,  which opens the possibility of deception, sophistry, lying, and that nagging sense we all have that there is something unreal and even inhuman about reasoning. Just as the continual repetition of a word makes it lose all of its sense in become an irrational clanging sound, so too arguments which go on too long for someone tend to sound unreal to him. Who hasn’t heard someone call an argument “mere wordplay” or a bunch of “empty logic”?

Though the opposition between “logic” and “mystical/religious experience” is a feature of the general opposition between mediate and immediate knowledge, the personhood of God makes this opposition more acute, painful, and troubling. Personal relations give an added force to the desire for immediacy. One could never exchange letters with a friend without wanting him to be present, but we might do history for quite some time without ever wishing to just see the things we study.

« Older entries


Get every new post delivered to your Inbox.

Join 144 other followers