Cosmology: Constructive and Non-Constructive
27 November 2013
Immanuel Kant, in an often-quoted passage, spoke of, “…the starry heavens above me and the moral law within me.” Kant might have with equal justification spoken of the formal law within and the starry heavens above. There is a sense in which the formal laws of thought are the moral laws of the mind — in logic, a good thought is a rigorous thought — so that given sufficient latitude of translation, we can interpret Kant in this way — except that we know (as Nietzsche put it) that Kant was a moral fanatic à la Rousseau.
However we choose to interpret Kant, I would like to quote more fully from the passage in the Critique of Practical Reason where Kant invokes the starry heavens above and the moral law within:
“Two things fill the mind with ever new and increasing admiration and awe, the oftener and the more steadily we reflect on them: the starry heavens above and the moral law within. I have not to search for them and conjecture them as though they were veiled in darkness or were in the transcendent region beyond my horizon; I see them before me and connect them directly with the consciousness of my existence. The former begins from the place I occupy in the external world of sense, and enlarges my connection therein to an unbounded extent with worlds upon worlds and systems of systems, and moreover into limitless times of their periodic motion, its beginning and continuance. The second begins from my invisible self, my personality, and exhibits me in a world which has true infinity, but which is traceable only by the understanding, and with which I discern that I am not in a merely contingent but in a universal and necessary connection, as I am also thereby with all those visible worlds. The former view of a countless multitude of worlds annihilates as it were my importance as an animal creature, which after it has been for a short time provided with vital power, one knows not how, must again give back the matter of which it was formed to the planet it inhabits (a mere speck in the universe). The second, on the contrary, infinitely elevates my worth as an intelligence by my personality, in which the moral law reveals to me a life independent of animality and even of the whole sensible world, at least so far as may be inferred from the destination assigned to my existence by this law, a destination not restricted to conditions and limits of this life, but reaching into the infinite.”
Immanuel Kant, Critique of Practical Reason, 1788, translated by Thomas Kingsmill Abbott, Part 2, Conclusion
This passage is striking for many reasons, not least among them them degree to which Kant has assimilated the Copernican revolution, acknowledging Earth as a mere speck in the universe. Also particularly interesting is Kant’s implicit appeal to objectivity and realism, notwithstanding the fact that Kant himself established the tradition of transcendental idealism. Kant in this passage invokes the starry heavens above and the moral law within because they are independent of the individual …
Moreover, Kant identifies both the starry heavens above and the moral law within not only as objective and independent realities, but also as infinitistic. Just as Kant the idealist looks to the stars and the moral law in a realistic spirit, so Kant the proto-constructivist invokes the “…unbounded extent with worlds upon worlds” of the starry heavens and the moral law as, “…reaching into the infinite.” I have earlier and elsewhere observed how Kant’s proto-constructivism nevertheless involves spectacularly non-constructive arguments. In the passage quoted above both Kant’s proto-constructivism and his non-constructive moments are retained in lines such as, “exhibits me in a world which has true infinity,” which by invoking exhibition in intuition toes the constructivist line, while invoking true infinity allows a legitimate role for the non-constructive.
When it comes to constructivism, we can see that Kant is conflicted. He’s not the only one. One might call Aristotle the first constructivist (or, at least, the first proto-constructivist) as the originator of the idea of the potential infinite, and here (i.e., in the context of the above discussion of Kant’s use of the infinite) Aristotelian permissive finitism is particularly relevant. (Aristotle would likely not have had much sympathy for intuitionistic constructivism, which its rejection of tertium non datur.)
The Greek intellectual attitude to the infinite was complex and conflicted. I have written about this previously in Reason in Moderation and Salto Mortale. The Greek quest for harmony, order, and proportion rejected the infinite as something that transgresses the boundaries of good taste and propriety (dismissing the infinite as apeiron, in contradistinction to peras). Nevertheless, Greek philosophers routinely argued from the infinity and eternity of the world.
Here is a famous passage from Democritus, who was perhaps best known among the Greek philosophers in arguing for the infinity of the world, making the doctrine a virtual tenet among ancient atomists:
“Worlds are unlimited and of different sizes. In some worlds there is no Sun and Moon, in others, they are larger than in our world, and in others more numerous. … Intervals between worlds are unequal. In some parts there are more worlds, in others fewer; some are increasing, some at their height, some decreasing; in some parts they are arising, in others failing… There are some worlds devoid of living creatures or plants or any moisture.”
…and Epicurus on the same theme of the infinity of the world…
“…there is an infinite number of worlds, some like this world, others unlike it. For the atoms being infinite in number, as has just been proved, are borne ever further in their course. For the atoms out of which a world might arise, or by which a world might be formed, have not all been expended on one world or a finite number of worlds, whether like or unlike this one. Hence there will be nothing to hinder an infinity of worlds.”
Epicurus, Letter to Herodotus
There were also poetic invocations of the idea of the infinity of the world, which demonstrates the extent to which the idea had penetrated popular consciousness in classical antiquity:
“When Alexander heard from Anaxarchus of the infinite number of worlds, he wept, and when his friends asked him what was the matter, he replied, ‘Is it not a matter for tears that, when the number of worlds is infinite, I have not conquered one?'”
Plutarch, PLUTARCH’S MORALS, ETHICAL ESSAYS TRANSLATED WITH NOTES AND INDEX BY ARTHUR RICHARD SHILLETO, M.A., Sometime Scholar of Trinity College, Cambridge, Translator of Pausanias, LONDON: GEORGE BELL AND SONS, 1898, “On Contentedness of Mind,” section IV
Like poetry, history had particular prestige in the ancient world, and here the theme of the infinity of the world also occurs:
“…Constantius, elated by this extravagant passion for flattery, and confidently believing that from now on he would be free from every mortal ill, swerved swiftly aside from just conduct so immoderately that sometimes in dictation he signed himself ‘My Eternity,’ and in writing with his own hand called himself lord of the whole world — an expression which, if used by others, ought to have been received with just indignation by one who, as he often asserted, laboured with extreme care to model his life and character in rivalry with those of the constitutional emperors. For even if he ruled the infinity of worlds postulated by Democritus, of which Alexander the Great dreamed under the stimulus of Anaxarchus, yet from reading or hearsay he should have considered that (as the astronomers unanimously teach) the circuit of whole earth, which to us seems endless, compared with the greatness of the universe has the likeness of a mere tiny point.
Ammianus Marcellinus, Roman Antiquities, Book XV, section 1
Like the quote from Kant quoted above, this passage is remarkable for its Copernican outlook, which shows that the ancients were not only capable of thinking in infinitistic terms, but also in more-or-less Copernican terms.
Lucretius was a follower of Epicurus, and gave one of the more detailed arguments for the infinity of the world to be found in ancient philosophy:
It matters nothing where thou post thyself,
In whatsoever regions of the same;
Even any place a man has set him down
Still leaves about him the unbounded all
Outward in all directions; or, supposing
moment the all of space finite to be,
If some one farthest traveller runs forth
Unto the extreme coasts and throws ahead
A flying spear, is’t then thy wish to think
It goes, hurled off amain, to where ’twas sent
And shoots afar, or that some object there
Can thwart and stop it? For the one or other
Thou must admit; and take. Either of which
Shuts off escape for thee, and does compel
That thou concede the all spreads everywhere,
Owning no confines. Since whether there be
Aught that may block and check it so it comes
Not where ’twas sent, nor lodges in its goal,
Or whether borne along, in either view
‘Thas started not from any end. And so
I’ll follow on, and whereso’er thou set
The extreme coasts, I’ll query, “what becomes
Thereafter of thy spear?” ‘Twill come to pass
That nowhere can a world’s-end be, and that
The chance for further flight prolongs forever
The flight itself. Besides, were all the space
Of the totality and sum shut in
With fixed coasts, and bounded everywhere,
Then would the abundance of world’s matter flow
Together by solid weight from everywhere
Still downward to the bottom of the world,
Nor aught could happen under cope of sky,
Nor could there be a sky at all or sun-
Indeed, where matter all one heap would lie,
By having settled during infinite time.
Lucretius, De rerum natura
The above argument is one that is still likely to be heard today, in various forms. If you go to the edge of the universe and throw a spear, either it is stopped by the boundary of the universe, or it continues on, and, as Lucretius says, For the one or other, Thou must admit. If the spear is stopped, what stopped it? And if it continues on, into what does it continue?
The contemporary relativistic cosmology has a novel answer to this ancient idea: the universe is finite and unbounded, so that space is wrapped back around on itself. What this means for the spear-thrower at the edge of the universe is that if he throws the spear with enough force, it may travel around the cosmos and return to pierce him in the back. There is nothing to stop the spear, because the universe is unbounded, but since the universe is also finite the spear will eventually cross its own path if it continues to travel. I do not myself think that the universe is finite and unbounded in precisely the way the many modern cosmologists argue, but I am not going to go into this interesting problem at the present time.
Other than the response to Lucretius in terms of relativistic cosmology, with its curved spacetime — a material response to the Lucretian argument for the infinity of the world — there is another response, that of intuitionistic constructivism, which denies the law of the excluded middle (tertium non datur) — i.e, a formal response to Lucretius. Lucretius asserted that, For the one or other, Thou must admit, and this is exactly what the intuitionist does not admit. As with the relativistic response to Lucretius, I do not myself agree with the intuitionist response to Lucretius. Consequently, I believe that Lucretius argument is still valid in spirit, though it must be reformulated in order to be applicable to the world as revealed to us by contemporary science. Consequently, I take it as demonstrable that the universe is infinite, taking the view of ancient natural philosophers.
Within the overall context of Greek thought, within its contending finitist and infinitistic strains, Greek cosmology was non-constructive, and the Greeks asserted (and argued for) the infinity of the world on the basis of non-constructive argument. Perhaps it would even be fair to say that the Greeks assumed the universe to be infinite in extent, and they at times sought to justify this assumption by philosophical argument, while at other times they confined themselves to the sphere of the peras.
Much of contemporary science is constructivist in spirit, though this constructivism is rarely made explicit, except among logicians and mathematicians. By this I mean that the general drift of science ever since the scientific revolution has been toward bottom-up constructions on the basis of quantifiable evidence and away from top-down argument. I made this point previously in Advanced Thinking and A Non-Constructive World, as well as other posts, though I haven’t yet given a detailed formulation of this idea. Yet the emergence of a “quantum logic” in quantum theory that does away with the principle of the excluded middle is a clear expression of the increasing constructivism of science.
In A Non-Constructive World I also made the point that the world appears to have both constructive and non-constructive features. In several posts about constructivism (e.g., P or not-P) I have argued that constructivism and non-constructivism are complementary perspectives on formal thought, and that each needs the other for an adequate account of the world.
In so far as contemporary science is essentially constructive, it lacks a non-constructive perspective on the phenomena it investigates. This is, I believe, intrinsic to science, and to the kind of civilization that emerges from the application of science to the economy (viz. industrial-technological civilization). By the constructive methods of science we can attain ever larger and ever more comprehensive conceptions of the universe — such as I described in my previous post, The Size of the World — but these constructive methods will never reach the infinite universe contemplated by the ancient Greeks.
How could the logical framework employed by a scientist have any effect over what they see in the heavens? Well, constructive science is logically incapable of formulating the idea of an infinite universe in any sense other than an Aristotelian potential infinite. No one can observe the infinite (in the philosophy of mathematics we say that the infinite is “unsurveyable”). And if you cannot produce observational evidence of the infinite, then you cannot formulate a falsifiable theory of an infinite universe. Thus the infinity of the world is, in effect, ruled out by our methods.
No one should be surprised at the direct impact the ethos of formal thought has a upon the natural sciences; one of the fundamental trends of the scientific revolution has been the mathematization of natural science, and one of the fundamental trends of mathematical rigor since the late nineteenth century has been the arithmetization of analysis, which has been taken as far as the logicization of mathematics. Logic and mathematics have been “finitized” and these finite formal methods have been employed in the rational reconstruction of the sciences.
I look forward to the day when the precision and rigor of formal methods employed in the natural sciences again includes infinitistic methods, and it once again becomes possible to formulate the thesis of the infinity of the world in science — and possible once again to see the world as infinite.
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