This essay will explore what can be said, from Objectivist premises, about existence as a whole -- the totality of that which exists, the universe.
A classical Objectivist would expect such an essay to be rather brief. Once one has acknowledged that 'existence exists' is axiomatic, no further analysis or explanation is possible or necessary. What exists exists and has the properties, the identity, it has -- and there the matter rests. No deeper explanation is to be found. If there could be one, existence would no longer be axiomatic and we would be trying to explain existence by non-existence.
Causality is a non-issue. Causal analysis reduces to the properties of existents. New states, actions, or changes are manifestations of the identity, the properties and attributes, of existents. Causality of all that exists is a non-concept; it too implies an explanation of existence starting from non-existence.
There ends an Objectivist discussion of the totality of that which exists. Or does it?
At this point, a warning: I intend to adhere rigorously to fundamental objectivist concepts of reality, identity, and reason, but will make arguments and reach conclusions not to my knowledge part of classical Objectivist thought. I offer these arguments as a tribute to the potency of objectivist principles consistently applied.
A crucial tool, already used in previous essays, will be the non-conceptual character of 'existent infinity'. The 'concept' violates the necessity for that which exists to have an identity -- to be something specific. To define conceptually, to identify, is to limit. We will explore this issue in further depth. Even Objectivists, as we shall see (in part II), do not always apply this insight consistently.
Next we will further emphasize, in context, what it means to make 'existence exists' an axiom -- the fundamental axiom -- of reality. Remember, or review from my introductory essays, in what manner meaning precedes axiomatics, and why even laws of logic may be misapplied if meaning is not retained every step of the way.
In this context I will show how Objectivists may be misled into making erroneous assertions about reality. Some are often too quick to criticize various physical formulations on the grounds that rules of logic are supposedly being violated. They are also too quick to assert limits to scientific discovery on supposed axiomatic grounds.
Third, we will explore some of what can be said meaningfully about time and space, their divisibility and extensibility, and how physical time relates to the more primitive concept of sequential states.
Finally, applying these principles, we will explore what paradigms for the origin and nature of the universe are consistent with Objectivist principles. You can then make your own judgment as to just what Objectivist principles can contribute to a meaningful philosophy of science and cosmology. You can also decide whether 'A is A' has more than tautologic potency.
We turn now to infinity. If you are an Objectivist, you already reject 'existent infinity' in whatever guise it appears, in whatever form it is asserted -- time, space, matter, causality, divisibility. Infinity as process is acceptable -- some and more -- but however far you proceed, wherever you are, whenever you stop, however much you have already done, you have not left the finite.
Mathematical infinities have been discussed in previous essays and individual communications exchanged in relation to such matters as rigorous foundations for limit processes, Cantorian infinities, and the foundations of mathematics itself.
I won't discuss the mathematical aspects in full depth at this time except to emphasize that it is the placement of meaning prior to axiomatics and formalism that is crucial.
The logicist school places axiomatics first, and then runs head on into logical contradictions in its deepest set-theoretic core. The formalist school self-admittedly plays meaningless games with meaningless symbols, and runs into the strictures of Goedel's Theorem which renders impossible of achievement their goal to formalize all of mathematics. Goedel's Theorem is itself a meta-mathematical argument deriving conclusions about formalisms from more fundamental considerations.
The intuitionist school places meaning first. No contradictions have ever appeared or can ever appear in intuitionist mathematics -- for the very same reason that there are no contradictions in reality. Meaning, appeal to reality, is retained every step of the way.
This though is not to imply that intuitionists are Objectivists; they usually believe that primitive mathematical truths are true independent of reality and appeal directly to the intellect. The Objectivist position is that they are facts of reality and are directly observed and perceived in reality.
Mathematics and meaning, the non-tautologic character of mathematics, the experiential underpinnings of mathematics, deserves a whole essay. Objectivism is right on target when it starts with sensation, the 'that it exists' (existence), progresses to perception, the 'what that exists' (identity), and finally to conception (consciousness), the 'unit' which is the fundamental mathematical intuition. [see Rand's Introduction to Objectivist Epistemology or the abstract quoted in the discussion of 'existent' in The Ayn Rand Lexicon]
As to infinity, intuitionist mathematics accepts only denumerable mathematical infinities, constructible sequential processes, or entities defined in terms of such processes as valid, meaningful, and understandable.
The term 'set' is used validly in two different ways. One use of the word 'set' is as a one-to-one correspondence between the integers and some objects (or the notion that no further objects exist). For example, suppose I have ten coats hanging on a wall and they are numbered. By saying, for each integer, if it is between one and ten inclusive refer to the coat that far in from the right, otherwise refer to the notion that there are no more elements, I have defined a set. This rule can only define denumerable sets.
The other use is a rule of inclusion and exclusion. For example, if I state that for any entity, count its legs. If it is two or three include it otherwise exclude it, I have defined a set. This rule can define non-denumerable sets such as the set of all reals.
Consider the ''set of all reals''. This set is non-denumerable, that is its members cannot be placed into one-to-one correspondence with the set of all integers. An intuitionist mathematician would interpret the phrase ''set of all reals'' to mean a rule of inclusion or exclusion to be applied to entities, classifying some of them as members, some of them as non-members, and possibly some of them as indeterminate; however, an intuitionist mathematician would not accept that the ''set of all reals'' is itself a defined entity. There exists no process for constructively specifying 'all' the reals.
Turning now from the mathematical to the physical, one candidate for an `existent infinity' is temporal infinity. Note however that, if we talk of an infinite number of future events, future states, that is not an existent infinity -- it is infinity as process.
Leaving aside for the moment the mathematics and physics of continuity -- the infinitely small, the infinitely brief -- however many discrete `instants' elapse from this moment, they comprise only a finite number. The number of instants `traversed' will eventually exceed any integer you care to name, but like the enumeration of the integers themselves, this is what infinity-as-process means.
What about 'going' backwards in time? That is another matter entirely and one that we will address in part II. The fact that the present moment has actually been reached changes everything.
What about spatial infinities? An `infinitely expanding' universe, one where no subsequent collapse or crunch occurs, likewise is not an existent infinity. However far space 'expands' it is still 'some' and capable of 'more'.
But how can space be finite? The ancient Greeks wondered what would become of an arrow released 'outward' at the very edge of finite space. Where could it go? What sort of 'thing' is a boundary between something and nothing? For that matter, what sort of 'thing' is space itself?
We now talk of physical space curving in on itself, and of arriving back at one's starting point by proceeding 'straight ahead'. These are aspects of existents that we observe in reality, but cannot easily visualize. Still, they can be conceptualized mathematically and applied physically. They seem to conflict with our 'common sense' understanding of physics, but we must appreciate that we are now trying to extend our understanding into new realms. Tommorow's common sense may render today's 'eternal mysteries' laughable. (The Earth supported on the back of an elephant, itself supported on the back of a turtle...)
The concept of a finite yet unbounded space of three dimensions has its analog in the finite unbounded two-dimensional surface of a sphere. In neither case, however, does ''unbounded'' mean infinite.
Note also that physical time and physical space are ways we talk about matter. They are ways of describing the configurations and properties of material existents -- using ''material'' in its broadest sense to encompass matter, energy, and all their transformations. This is why science talks of the non-existence of time and space at the 'naked singularity' from which the known universe apparently evolved after the big bang. We do not 'then' have matter to define space or sequential states to define time.
There is no such thing as time in a universe without sequential states of 'matter', and no such thing as space in a universe condensed into a zero-dimensional naked singularity, whatever that may mean.
What about matter as infinite? An existent infinity of extended matter is not meaningful in precisely the same manner as infinite space is not. But what about continuity of space or infinite divisibility of matter? Can we have an infinity in the realm of the very small? Physical science has already gone from sensible matter to molecules to atoms to several levels of sub-atomic particles.
How much further can the process extend? Must there come a smallest size, a final level? Quantum Mechanical principles deal with such `graininess' in matter as well as in energy, momentum, position, and other aspects of reality. Quantum Mechanics seems to rescue us from a conceptually meaningless existent infinite divisibility.
Even mathematically, infinite divisibility is redefined meaningfully only in terms of finite processes. Infinitesimals, entities somehow neither zero nor non-zero, are expunged from the foundations of calculus. This matter is discussed elsewhere in the presentation and analysis of Zeno's Paradox.
As an addendum to that discussion, consider the arrow version of Zeno's paradox. However finely we subdivide the time and space in which a flying arrow is embedded, at any instant it must always be at a specific location, it can only be where it is and not somewhere else. Thus there are no available instants in which it can actually move from here to there.
Again the resolution, as a problem in reality, demands the rejection of infinite divisibility. Time does not consist of mathematical instants nor space of mathematical points nor matter of infinitesimal components. A graininess, whether that of current Quantum Mechanical theory or some possible later formulation is necessitated.
The paradox is resolved by denying infinite divisibility to matter or momentum or energy or position or space or time. The arrow is here, then it is there, at the `next' quantum position. At some level a particle does not `move' from here to there qua particle, by traversing intermediate states. At some level there are no intermediate states. Do not confuse a mathematical model with reality.
Now, let us look again at 'existence exists' as axiom. To most Objectivists, it means that at some stage we point to 'that' and assert that it is not to be explained further. It exists and we can say no more. This recognition, that at some point there comes an end to explanation and there is left only assertion, pointing, is too frequently misapplied by Objectivists.
For example, George Smith in Atheism: The Case Against God, arguing from Objectivist fundamentals, is critical of those who ask why the numerical constants of nature have the values they do -- mass of a proton, charge or spin of an electron, velocity of light, the gravitational constant, and so on. The entities display these constants because they have the intrinsic properties they have; they exist with that identity and that's that.
Yet, within the past several months, there has appeared a new formulation of basic physical principles which predicts accurately, that is, derives from more fundamental properties, several of the fundamental constants of physics. Similarly, there are theories to relate the origin of the big bang to 'prior' quantum vacuum states. The big bang was previously regarded as an unexplainable starting point that had to be accepted as the way nature is -- the way existence exists.
The moral here is simply to avoid surrendering too soon to the unexplainable. We do not know how to know when deeper explanation must cease and we must point to an axiomatic existence. Science does well to always look further.
This lesson is very applicable, for example, to the rejection by some Objectivists of some of the newer concepts of physical reality because they supposedly contradict macroscopic notions of naive reality.
We discussed previously Peikoff's rejection of Quantum Mechanical explanations of physical reality because they 'contradict' the law of the excluded middle. An object may be here or there, but surely not in both places at once. The previous Zeno's Arrow analysis even uses this very dichotomy in the analysis of particle motion qua particle.
But what happens when particles at the sub-atomic level also display wave characteristics? What happens when suddenly, between observations, not only can we not say where the particle is, but its very existence as particle, with specific position and momentum, become physically unsupportable?
We are in the presence of a new physical paradigm at the micro level having no counterpart in previously observed macroscopic reality. What would Peikoff have -- mathematical continuity in motion with absolute position at every instant? Then we would indeed have an existent infinity irreconcilable with Objectivist premises, irreconcilable with reality!
Peikoff cannot arbitrarily extract 'position' or 'particle' from their macro-physical conceptual net and expect them to apply unchanged in new realms.
Now let us make a few observations about physical time and the concept of sequential states. We note that the two are not interchangeable. To say state-of-the-universe B succeeds state-of-the-universe A does not per se place A and B in physical time.
An example. We talk of the big bang initiating not just matter and energy, but space and time as well. As we saw, space and time are undefined when the state of the universe is a naked singularity. Time remains undefined until we have sequential states. After the big bang, we have sequential states and physical time.
Yet, physical theory talks of the possibility of big bang followed by big crunch followed by another big bang, and so on -- depending on whether enough undetected matter exists to enable gravitational attraction to overcome the inertia of expansion and initiate a contraction phase.
We may then have successive singularities occurring in successive Bang-Crunch cycles. Each singularity is itself outside physical time, yet the cycles and the components of the cycles exhibit sequentiality.
We do not know if such cycles correspond to physical reality, but the possibility is not inherently self-contradictory. Thus physical time does not coincide conceptually with sequential states. ``Sequential states-of-the-universe'' is a broader concept which includes, but is not limited to, physical time.
This concept of sequential states is very solidly grounded. It is a fusion of the very same concepts, derived from reality, that are at the foundation of intuitionist mathematics -- oneness, twoness, nextness, betweenness, moreness. These cardinal/ordinal concepts are immediately given -- immediately intuited from reality -- and carry genuine meaning at the most fundamental level.
These concepts can only be `pointed to' and understood, not derived from or proven from axioms with lesser clarity. Any such attempted axioms would themselves already subsume the very concepts to be derived. Any further mathematical or scientific formalism cannot help but use them.
The original intuited concepts, as we have seen, ground denumerability and constructibility. Other less lucid `axioms' such as those of set theory or the axiom of (infinite) choice yield unclear existent infinities. Russell's Theory of Types, which partially finitizes set theory by establishing a sequential hierarchy of constructible `types' of sets was specifically formulated to avert the Russell Paradox type contradictions of unrestricted set theory.
It is the foundation of mathematics in reality that is the source of what physicists themselves describe as ``the unreasonable effectiveness of mathematics'' in grounding our understanding of physical reality. It is this foundation I count on in later utilization of the concept of ``sequential states'' (in part II).
Nor do Einsteinian considerations of the relativity of simultaneity affect the nature of sequential states. If event A at location L precedes event B at location L in some frame of reference, it does so in every frame of reference.
The role of physical time is to tell us something about the nature, the identity, of matter. All statements about physical time are translatable into assertions about successive configurations of matter -- successive manifestations of the identity of matter.
The broader concept of sequential states not only underlies physical time but grants access to states of the universe with no matter as we know it, states open to future attribution of content, future elucidation of identity, by ongoing scientific endeavor.
This is precisely where we now stand with our current concept of naked singularity. It is a `state' with no real conceptual content, yet physics talks of the `cause' or predecessor states to a naked singularity in terms of quantum vacuum fluctuations.
We await the insights of future science. The key point is not to close off explanatory possibilities too soon by an appeal to the axiom of existence.
Now, using the concepts we have developed let us attempt an Objectivist exploration of existence and cosmology. What can we say meaningfully about the origins and the nature of the universe as a whole?
... To be continued.
Entire contents Copyright (C) 1993-94 by Joel Katz, All rights reserved, except as below:
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