Chaos in Cosmic Selection by Felix Hausdorff
Walter Purkert’s abridged biography of Felix Hausdorff is now available in German and is in the process of being translated into English by David Rowe. In advance of the full volume being available, I am publishing my translations of some of the key excerpts Purkert includes from Hausdorff’s 1898 book on the nature of space and time, Chaos in Cosmic Selection. As I’ve noted in a previous Quantslut column, the book touches on questions about continuity that have continued to be important for mathematics and modern physics. In many of his works, Hausdorff spoke of the importance of interdisciplinary exchange, particularly between mathematics and philosophy. With this collection of translated quotes, I hope to begin to interest a few English-reading mathematicians and philosophers in such an exchange about his work.
At the same time, I hope to invite all readers to consider whether Hausdorff’s core conception–that there are different present moments and different paths through time for each of us– might actually be true. There are already many works of science fiction involving the ideas of time travel and alternate timelines, and I have friends who’ve shared their intuitive, inchoate feeling that in some sense, “everything happens at once.” Similar ideas are being proposed by modern physicists with the “block theory of time.” Knowing that one of our great mathematicians of the past considered similar ideas might lead some of us to take our own intuitions in this regard seriously.
As Hausdorff wrote in his introduction to Chaos in Cosmic Selection, these ideas came to him through personal, intimate moments of insight that illuminated a connected structure of ideas. He was moved to share them with others even though he didn’t have the academic credentials of a professional philosopher, and couldn’t precede them with a formal review of everything that had been said on these topics before. So I think we can feel free to read them in the same spirit, exploring how Hausdorff’s reflections on temporal experience may resonate with our own.
Hausdorff excerpts from Purkert
Page numbers in the below are to Purkert’s abridged biography of Hausdorff; quotations are from Hausdorff himself, in my translation.
The investigation of time […] can be divided into two parts from the outset. Our conception of time contains two dissimilar concepts, which I will call for short the content and the passage of time, and which are linked to each other in a way that can easily be distinguished. The first is the continuous series of states of the world, a material substrate of time, and the other a mysterious formal process through which each state of the world experiences the series of transformations future, present, past. Although these two separate elements always appear bound together in our temporal experience, we want to separate them conceptually and explore how much each of them can be varied, without distorting empirical effects.
The term I have just introduced, “state of the world,” has for me the following precise meaning. “State of the world” [Weltzustand] is a filled time period of length zero, just as “instant” [Augenblick] is an empty time period of length zero. The state of the world bears the same relation to a filled time period as the instant does to an empty one, or as a point does to a line. Filled time is simply an extended continuum of states of the world, just as empty time is simply an extended continuum of instants, and the line is simply an extended continuum of points. The question of what filled time is must be asked directly of our temporally experiencing consciousness, which receptively and productively occupies itself with nothing other than the filling of the form of time. [. . .] [T]he state of the world cannot, however, be any further described from the perspective of temporally experiencing consciousness, for which it forms an unreachable zero-limit.
One can picture the simple extended totality of all states of the world, whose constant succession makes up our temporal world of experience, as an unbounded line, the timeline; each state of the world as one of its points; and finally, the process of temporal realization as occurring through a point, the present point, which moves along this line. The time in which this movement occurs, is absolute time.
Only with the help of analytical hypotheses about what filled time truly is can one analyze the state of the world. For example, if materialism is correct, the state of the world would be a resting state of material points, the system of points in space belonging to a particular instant.
The movement of the present point along the timeline creates for us the flow of empirical time: what can be said about this movement? The naïve realist will not be able to imagine anything other than this point traversing the timeline in the same direction forever at a constant speed; this would be the direct translation into the transcendental realm of our usual opinion of the uniform flow of time. As far as I know, the only generalization that has been made in this regard is one that allows for a variable speed of the passage of time.
After this residuum is destroyed and the given “possibly” rejected, we can say: if one also removes from the idea of time the idea of temporal succession, what remains, and what has the potential to have an absolute reality, is the content of time: its inner constitution and the connection among states of the world.
The present point moves along the timeline in entirely random, constant or inconstant ways. The transcendental succession of states of the world is arbitrary and does not occur in our consciousness. (emphasis in original)
A moving point can exhibit various sorts of behavior with regard to a fixed line during a finite stretch of time. It can be outside the line; it can momentarily pass one of its points; it can rest in one of its points; it can traverse any finite bounded segment of the line in one direction or in the opposite direction. The movement of the present point on the timeline can be composed arbitrarily from these elementary actions.
A transcendental world-observer, existing in absolute time, would experience the web of simultaneous changes that makes up our empirical world in an entirely random sequence, with arbitrary succession, direction, and speed; with arbitrary interruptions, resting-points, repetitions, jumps and reversals—without the usual fluxus temporis that is familiar to us being disturbed at all.
The proof of our fundamental proposition can be seen by the form described earlier of a simple syllogism. Assume that a particular chosen succession A of states of the world produces our empirical world phenomenon. Imagining, now, that a different succession B is chosen and that this change becomes part of our consciousness, it would need to become perceptible at some point, say during the time period t. But since we ourselves, with all the contents of our consciousness, are woven into the timeline, which is the very embodiment (Inbegriff) of all the states of the world—for this reason such a switch from A to B would mean that something had come into the time period t which was not there before. Thus the switch would alter not only the succession, but compared to what was first given (die Voraussetzung), the inner structure of the states of the world. Our separation of the content and passage of time thus implies the peculiar sense that the former is entirely indifferent as to the latter, just as the form and quality of a line is indifferent to the possibility of a point movement.
In our inner structure as carriers of time, we are woven into the positively oriented timeline and excluded from the negatively oriented one—or put differently, we call the direction to which we belong the positive one.
Some readers will wish for greater clarification of the timelessness of the material substrate, which we have often stressed and will have the occasion to stress again. The persistent continuum of states of the world seems as though it could only reveal itself in a temporally successive unfolding. I can give no answer to this, and agree with Lotze that it is not our task to create such things as being, becoming, and action; they themselves must see how they manage it. We are here faced with a fact whose simple structure cannot be broken down into even simpler elements; an originary phenomenon that can be recognized and conceptually exposed, but not derived and explained. Neither a single persistent world-state nor a persistent multiplicity of world-states appears in consciousness, but only the streaming or gliding continuum of world-states whose peculiar behavior we can only understand symbolically, as a point-movement. This is a fundamental characteristic of our temporal existence, whose origins cannot be traced further back. For this reason, as soon as this fixed and unchangeable system, the timeline, is supposed to emerge from its imaginary basis and exist, it passes by as a flowing and unstoppable becoming of the knowing subject that is woven into it.
If we summarize the discussion so far, it becomes clear that even if we hold fixed the empirical picture of the world, there is such an inexhaustible fullness of metaphysical possibilities for temporal succession that regarding our main topic, there can be no question of a transcendental reality of time. Every relationship which is imposed on the absolute linkage of time elements in order to ensure the empirical is an unnecessary restriction. Temporal sequence in general, the basic form of the world as a becoming, flowing, transitory series of states, is metaphysically indeterminate and indifferent; unambiguous reality can at most be ascribed to the material substrate of events, the fixed, unchangeable nunc stans ready for realization at any time; the timeline as the indestructible, inexhaustible, always complete and simultaneously existing source of reality.
The next step in increasing generality leads us to the multiplication of the number of present moments, which trace arbitrary movements along the timeline simultaneously and independently of one another.
If the timeline is to retain its earlier, easy-to-imagine meaning as an objective reality that is valid for all the bearers of consciousness within it, then we must shift the multiplicity of individuals into the present point, and assign each individual a separate present point. These individual points move, independently of each other, arbitrarily along the timeline, and it would be the most unlikely accident, if they eternally or even temporarily coincided in a single point. […] The “simultaneity of time in different minds,” that oft-admired miracle, is probably no more real than the ubiquity of time in space.
Empirical time thus shows itself to us always filled with e, while in absolute time e can be arbitrarily exchanged with not-e, for example e can be replaced with transformed events e1, e2, e3. […] In all these cases only the case with e becomes an object of our experience, while all the cases of not-e eliminate themselves from the complex of our world. To put it differently, the case e has for us a subjective certainty, but in and of itself an arbitrary, even arbitrarily small, objective probability. I would like to call the fundamental principle expressed here, which is very important for what follows, the principle of indirect selection applied to epistemology. The word indirect, which could be more pointedly replaced with automatic, self-acting, or the like, is intended to have the meaning it does in biology. Namely, it is not that the selected case existed from the beginning as the only real one among many that are conceivable. Rather, the selected case distinguished itself through actual competition with others as unique by virtue of its inner characteristics. In the contiguity of what does and does not serve a purpose, there survives the purposeful, by virtue of its viability. Out of the mingling of chaos and cosmos there emerges into our field of vision, by virtue of its relationship to our consciousness, only the cosmic.
For us, the connection (Connex) between consciousness and its world of consciousness is defined purely conceptually; a continuum c of world-states and its appearance e in an intellect i which is woven into that continuum belong together inseparably.
A subjectively inviolable unity and lawfulness turns out to be the mere product of a selection from a massive multiplicity and randomness of a higher order, to which the unity relates as a finitude does to an infinity of infinitely high degree. I don’t know how far up in the sequence of “transfinite numbers” one would have to climb in order to count the essence (Inbegriff) of transcendental possibilities. Our usual infinity symbols of the first and second cardinality prove themselves here quite inadequate, and gaining an overview of the full scope of the randomness and arbitrariness of the transcendental world is something denied to the human brain even in abstracto.
More than one present point, more than one timeline! That is to say: the continuum of world states into which we are woven need not be the only one; in addition to this one, arbitrarily many other worlds with arbitrary content are imaginable, represented by timelines on which the present points take their course. In other words a pluralite des mondes, which it is incumbent upon us to recognize precisely because we cannot convince ourselves of it through experience, but always remain focused on our single timeline.
When we analyzed the arbitrary movement of the present point, we discussed the case in which the point is not on the line; let us ask, where else is it? Thus we place it, in our intellectual vision, in some spatial exterior; that is, we relate the timeline to a higher-order spatial multiplicity of surface-like or body-like formations. Nothing prevents us from consciously using this graphic clarification, in which the plane will suffice for now as a symbol of a higher dimensional image of time.
Our time plane, which can be immediately expanded to a time-space, a continuum of world-states extended three times and more, is naturally only a visual symbol, and only in this respect a new element of our contemplation.
Every point of the plane signifies a world-state, and every linear formation a world-course.
In this respect we remain far behind the position we reached earlier regarding generality–we content ourselves, for the sake of visualization, with seeing the ensemble of world-states as a point continuum, giving it thus the “second cardinality.” In another respect we go far beyond the earlier position, in that we fill the time plane with arbitrary content of world-states.
Only the momentary present, the world state without extension, can be grasped and fixed as absolute being, but—it escapes perception; every act of consciousness, however narrow its amplitude and creeping its tempo, requires becoming and change, an infinite succession of different world states continually following on each other.
Any arbitrarily chosen selection principle that is narrow enough and otherwise accords with our concepts of the empirical world, selects a cosmos from the chaos; that, from the totality of world states it selects a linearly extended totality of particular world states that a consciousness contained in it will represent to itself in roughly the same way that our empirical world presents itself to us.
~ Kimberly Gladman ~
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