Saturday, January 31, 2009

Counting to infinity by infinity…

[Read this post for a later attempt to make this same point.]

MAJOR PREMISE: It is not possible to measure an infinite magnitude.

POSTULATE: The very act of measuring, or quantitatively delimiting, puts an incoherent 'cap' on infinity.

MINOR PREMISE: It is possible to measure the universe. It is possible at the very least to measure matter, and material objects, as the constitution of the universe.

CAVEAT: The inability to measure the universe to an infinitely precise degree does not negate the fact of its measurability per se. Indeed, the very ability to challenge or refine one measurement, is itself based on a competing standard of measurement (viz., a measurement obtained in and of the actual spacetime manifold can only be refined, or, indeed, rejected by measuring it against other quantifiable objects). This is significant, because potentially infinite divisibility (i.e., by increasingly precise measuring devices), does not equate to actual infinity. If everything were actually infinite, then anything we measured, at any scale, would be infinite, not measurably finite, as our measurements report.

CONCLUSION: The universe is not of an infinite magnitude, and is not itself a material 'entity' of infinite magnitude. Nor can it, therefore, be eternal.

DENOUEMENT: I would even go so far as to say the idea of an infinitely large material substance, as well as an eternally 'old' temporal object, is incoherent, since in either case, the categories of materiality and temporality presuppose finite divisibility, i.e., quantitative divisibility and measurability as distinct objects in spatiotemporal relation to others. To be an object of empirical scrutiny is to be quantitatively delimited by others, and to delimit others objects in the same way. This holds for objects' fourth dimension as well. Unless one is prepared to deny science can measure anywhere in the cosmos––i.e., sectors or 'levels' of the universe are metaphysically simple––then one admits the universe, from top to bottom, is subsumed by the finite categories of quantitative spatiotemporal extension. To reject the universe's 'subsumption' under finitude is to posit an inifnite (and eternal) universe. Again, though, an infinitely extended quantity is incoherent on the grounds that an infinite magnitude cannot be a "quantum" (i.e., a discrete amount). One infinity cannot be more "magnus" than another, and therefore neither can be of any magnitude.

[I would like to add comments from the discussion about this post at PhilPer. They help get at what I was shooting for in the original post, above this memo. Vero, "semper homo bonus tiro est."]

Now, perhaps someone will object that there are provably larger and smaller classes of infinity (Cantor, Dedekind, et al.), and therefore infinity does admit of a sort of quantification. So I'm wrong. To which I reply:

Leaving aside the controversy among number theorists about how viable Cantorian infinities are, I would reiterate the fact that infinity is not a quantity, and therefore cannot be ‘manipulated’ in actual reality like quantities and magnitudes can. Much of the basis for Cantorian innovations in set theory was to get an arithmetical grip on unwieldy infinity problems (e.g., ∞ + ∞ ≠ 2∞, = ∞; ∞ – ∞ ≠ 0, = ∞; etc.). They are purely theoretical entities, not physical quantities. Therefore to say one infinity is “larger” than another is simply a way of arithmetizing them in order to tighten up the meaning of zero and limits. It’s a formal convenience, not a physically coherent possibility.

Further, to head off the objection that we can measure the number line even though it is infinite, I would stress two points. First, the infinitude of the number line is based on its existence as a conceptual reality. There is no number line fully existent in actual reality. There is no limit (n) on a number line, since we can always posit n + 1, but physical reality doesn’t work that way: as soon as we posit "+ 1" to a measured reality, we cease to be talking about that measured reality itself in its specific dimensions.

Second, there are plenty of finite number sets (e.g., the natural numbers {1…10}, all prime numbers less than 1 x 10^6, etc.), so while there may be an infinitely vast universe “somewhere out there”, I deny it is measurable like ours is. Ours happens to be, as it were, a finite “set” cosmos. (This, incidentally, is why, following the work of Stanley Jaki, I think Gödel's incompleteness theorems are the death knell for any necessarily true and absolute "theory of everything." Contingency will simply never be vanquished.) Such sets we can measure easily (say, on our fingers, or on graphing paper, or in bits, etc.), because they are concrete, actual entities with specifiable magnitudes. I’m willing to grant that the “entire number line” is an analogy for prime matter––i.e., the infinite scope of raw possibility in nature––but that doesn’t help us much scientifically, since prime matter is technically nonexistent and utterly non-measurable (because non-distinct). When physicists find a singularity, in which an otherwise sound physical theory breaks down under infinite energies, densities, durations, and the like (such as in a black hole), they must renormalize those infinities into meaningful physical terms, knowing that such an "infinite failure" only means the theory is not complete yet, is not yet adequately describing the singularity as a part of physical reality.

Lastly, even if one were to grant an infinite-eternal universe, as St. Thomas admits, this would not solve its contingency in various ways (i.e., its presumed measurability, its unique causal ‘history’, etc.).

A further worry might be that I appear to equate “physically realizable” with “measurable”—which isn’t really sound in light of the Heisenberg uncertainty principle. In point of fact, in this discussion, Heisenbergian uncertainty vis-à-vis measurability needs to be qualified in at least two ways. First, we must avoid the fallacy of equivocation which suggests that because an event cannot be determined (i.e., measured) exactly, it cannot be determined (i.e., caused) exactly. This qualification is important because the indeterminacy of a measured quantum state does not entail a diremption of physical causality. Quantum indeterminacy is a physical reality.

Second, the quantum flux is not a discrete (measurable) physical phenomenon, but a field of immeasurable potentiality. Wolfgang Smith has very penetrating comments about this topic in The Quantum Enigma. Insofar as a quantum state CAN be measured in physical reality, it is no longer that theoretically infinitely potential field of superpositions. Uncollapsed, the quantum state is purely deterministic according to the Schrödinger equation, but is only determined once it is collapsed (i.e., observed) BY SOMETHING OUTSIDE THE STATE ITSELF, whereupon the non-discrete potentialities becomes a discrete physical reality. We don’t measure (and therefore cannot predict) the infinite possibilities of an uncollapsed quantum state, but only the uniquely determined state itself in the very act of measurement-collapse. In this way I think quantum mechanics is implying exactly what I mean about the unmeasurability of an infinite field or magnitude: we simply cannot get there from here. Once the state is collapsed, however, it is measurable––because measured––AND THUS not infinite. The obverse holds. Insofar as it is an infinite potentiality, a quantum state is neither measured nor measurable. At the very root of known physical reality, therefore, we find that theoretical infinitude is inimical to physical measurement. All of this is to say that it does no good to confuse the methodological inexactitude of our measurements with the empirical fact of measurement per se. The uncertainty of a quantum measurement, like the inability to reach “n + 1” is an empirical problem. It is not a refutation of the fact that a measured quantum state exists only AS MEASURED. Infinity is exactly what gave quantum mechanics the push it needed (viz., Planck)– cf. the ultraviolet paradox in black body radiation–so I can’t imagine physicists want to tangle with it again. Indeed, the very idea of a quantum is that it is a FINITE AMOUNT (quantum finitum) of energy. Thus, as I’ve said, I think an infinite quantum is literally a contradiction in terms, like asking how long an eternal hour is, or how tall an infinite meter is.

"Hucusque auxiliatus est nobis Dominus" (1 Sam 7:12)––but unfortunately another, theological, objection lurks: “If we have bodies in heaven and the bodies move, there is time in heaven, and heaven is eternal.” Therefore, the fact that time is infinite in heaven does not seem to "fit" my theory about the immeasurability of infinity. In light of this objection, I should perhaps rephrase my complaint thus: If we can measure at all, we can only meaningfully obtain finite measurements. Since, obviously, we can measure, then we cannot be obtaining finite measurements of infinite quantities. We can, however, obtain finite measurements of finite segments which are theoretically (or formally) contiguous with infinite quantities.

We cannot measure TIME and SPACE as such, since the very position and moment from which, and in which, we take that measurement would itself be spatiotemporal, and thus lie outside the supposed “fullness” of space and time we are measuring. The very notions of measuring, measurers, and measurees, as it were, require all three be not only notionally distinct but also spatiotemporally discrete. If we found an infinite mass, would it include us and our scale? If so, then what instruments or observer takes the measurement? If not, then how can it be considered infinite insofar as it is delimited (“pressed up against”, “stopped”, etc.) by us and out instruments?

All we can do is measure discrete objects, velocities, etc. IN space and time. The measured duration and size of the universe are not themselves measurements of the whole of space and time, since they apply only to the PAST career of spacetime. If we had a measurement of time in its fullness (i.e., infinitude), then all ‘future’ moments would be measurably present. They are, however, not present. Thus, while the fullness of empirical time is formally contiguous with the infinitude of ‘ultimate’ time (in heaven or otherwise), ‘our’ time is not itself infinite. It makes no sense to ask “When is time?” or “Where is space?” This is because they are not properly called magnitudes, and thus, as bizarre as it sounds, not actually measurable. We cannot ask how much time is in a minute, but can ask how many minutes are there in time. Nor can we ask how much space is in a cubic liter, but how many cubic liters there are in space. The former are measurable; the latter, being the very criteria of measurement, are not.

I’m being a bit pedantic because I believe my intuitions emerge from the very face of the things we are discussing. A measurement is by definition FINISHED, counted, ended (otherwise we would still be counting up to it, not stating it). An object, in its actual measurable dimensions, can only begin and end where others end and begin. It is FINIShed, and therefore not INFINIte. Likewise, space and time as formal IMMENSIties are literally without (im-) measure (-mensura). Infinity is literally incalculable, while exact measurement is the lifeblood of calculus. God, being infinite and eternal and entirely immaterial, is not measurable, although His actions are measurable (e.g., how many loaves and fish, the corporeal dimensions of Christ on the Cross, the velocity of prophetic words in the air, etc.). Intellection, likewise, because it is immaterial, has no measurable properties, though its operations do (musing, asking, wondering, neural activity, etc.). In an analogous way, while space and time as such are formally immaterial (since they cannot be wholly materialized “all at once”––when?!––or “all in one place”––where!?), they ‘contain’ finite proper parts which do admit of quantification.

All of this should remind us that creation, on multiple levels, partakes of eternity, without actually ‘enclosing’ it. Additionally, the participation of the higher in the lower is wholly concrete: we know Time only in the discrete moments that comprise our life, Space only in the discrete areas that comprise our range of action, Wisdom only in discrete acts of “mentation”, and God only in the discrete sacramental disclosures He gives of Himself. I add this last point to reiterate my “Aristhomism”, as opposed to the creeping Platonism some of my comments suggest. Not that it is so bad to be a Platonist; I am just more of a “Thomistotelian”.

Finally, it may be wondered why I think it’s important to argue that the universe not be infinite-eternal. What's the hullabaloo? The main reason why I find an infinite but empirically coherent universe (or any substance) repugnant is because I believe the “infinite” (and related “homogeneous”) impulse in cosmology has always been trouble for exact physical science. It’s driven, then, by a sort pragmatic urge to preserve the intelligibility of the universe AS we MEASURE it in reality. But a related motive is that I do believe the very idea is incoherent. And I think it befits a philosopher in fieri (moi!) to address that.

I close with Sapientia 11:21–27 [Wisdom 20–26]:

…uno spiritu poterant occidi,
persecutionem passi ab ipsis factis suis,
et dispersi per spiritum virtutis tuæ :
sed omnia in mensura, et numero et pondere disposuisti.
22 Multum enim valere, tibi soli supererat semper :
et virtuti brachii tui quis resistet ?
23 Quoniam tamquam momentum stateræ,
sic est ante te orbis terrarum,
et tamquam gutta roris antelucani quæ descendit in terram.
24 Sed misereris omnium, quia omnia potes ;
et dissimulas peccata hominum, propter pœnitentiam.
25 Diligis enim omnia quæ sunt,
et nihil odisti eorum quæ fecisti ;
nec enim odiens aliquid constituisti aut fecisti.
26 Quomodo autem posset aliquid permanere, nisi tu voluisses ?
aut quod a te vocatum non esset conservaretur ?
27 Parcis autem omnibus, quoniam tua sunt, Domine,
qui amas animas.

…men could fall at a single breath
when pursued by justice
and scattered by the breath of thy power.
But thou hast arranged all things by measure
and number and weight.
[21] For it is always in thy power to show great strength,
and who can withstand the might of thy arm?
[22] Because the whole world before thee is like
a speck that tips the scales,
and like a drop of morning dew that falls upon the ground.
[23] But thou art merciful to all, for thou canst do all things,
and thou dost overlook men's sins, that they may repent.
[24] For thou lovest all things that exist,
and hast loathing for none of the things which thou hast made,
for thou wouldst not have made anything if thou hadst hated it.
[25] How would anything have endured if thou hadst not willed it?
Or how would anything not called forth by thee
have been preserved?
[26] Thou sparest all things, for they are thine,
O Lord who lovest the living.

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