What is a continuum? Russell knocks Hegel's logic (1903)

Bertrand Russell, in his Principles of Mathematics (1903), comments on G.W. Hegel's Logic:

271. The notion of continuity has been treated by philosophers, as a rule, as though it were incapable of analysis. They have said many things about it, including the Hegelian dictum that everything discrete is also continuous and vice versâ. This remark, as being an exemplification of Hegel’s usual habit of combining opposites, has been tamely repeated by all his followers. But as to what they meant by continuity and discreteness, they preserved a discreet and continuous silence; only one thing was evident, that whatever they did mean could not be relevant to mathematics, or to the philosophy of space and time. -- Chapter XXXV, Section 271.

Though Russell gives a page number from William Wallace's translation of Hegel's logic, I could not pin down the reference exactly. Still, the following excerpt from Wallace's version of "Being Part One of the Encyclopaedia of The Philosophical Sciences (1830)" [known as The Science of Logic] gives you an idea of what Russell meant.

From Wallace's translation:

Quantity, as we saw, has two sources: the exclusive unit, and the identification or equalisation of these units. When we look therefore at its immediate relation to self, or at the characteristic of self-sameness made explicit by attraction, quantity is Continuous magnitude; but when we look at the other characteristic, the One implied in it, it is Discrete magnitude. Still continuous quantity has also a certain discreteness, being but a continuity of the Many; and discrete quantity is no less continuous, its continuity being the One or Unit, that is, the self-same point of the many Ones.

(1) Continuous and Discrete magnitude, therefore, must not be supposed two species of magnitude, as if the characteristic of the one did not attach to the other. The only distinction between them is that the same whole (of quantity) is at one time explicitly put under the one, at another under the other of its characteristics.

(2) The Antinomy of space, of time, or of matter, which discusses the question of their being divisible for ever, or of consisting of indivisible units, just means that we maintain quantity as at one time Discrete, at another Continuous.

If we explicitly invest time, space, or matter with the attribute of Continuous quantity alone, they are divisible ad infinitum. When, on the contrary, they are invested with the attribute of Discrete quantity, they are potentially divided already, and consist of indivisible units. The one view is as inadequate as the other. Quantity, as the proximate result of Being-for-self, involves the two sides in the process of the latter, attraction and repulsion, as constitutive elements of its own idea. It is consequently Continuous as well as Discrete. Each of these two elements involves the other also, and hence there is no such thing as a merely Continuous or a merely Discrete quantity.

We may speak of the two as two particular and opposite species of magnitude; but that is merely the result of our abstracting reflection, which in viewing definite magnitudes waives now the one, now the other, of the elements contained in inseparable unity in the notion of quantity. Thus, it may be said, the space occupied by this room is a continuous magnitude and the hundred men assembled in it form a discrete magnitude. And yet the space is continuous and discrete at the same time; hence we speak of points of space, or we divide space, a certain length, into so many feet, inches, etc., which can be done only on the hypothesis that space is also potentially discrete. Similarly, on the other hand, the discrete magnitude, made up of a hundred men, is also continuous; and the circumstance on which this continuity depends is the common element, the species man, which pervades all the individuals and unites them with each other.

(b) Quantum (How Much) §101 Quantity, essentially invested with the exclusionist character which it involves, is Quantum (or How Much): i.e. limited quantity. Quantum is, as it were, the determinate Being of quantity: whereas mere quantity corresponds to abstract Being, and the Degree, which is next to be considered, corresponds to Being-for-self. As for the details of the advance from mere quantity to quantum, it is founded on this: that while in mere quantity the distinction, as a distinction of continuity and discreteness, is at first only implicit, in a quantum the distinction is actually made, so that quantity in general now appears as distinguished or limited. But in this way the quantum breaks up at the same time into an indefinite multitude of quanta or definite magnitudes. Each of these definite magnitudes, as distinguished from the others, forms a unity, while on the other hand, viewed per se, it is a many. And, when that is done, the quantum is described as Number.

§102 In Number the quantum reaches its development and perfect mode. Like the One, the medium in which it exists, Number involves two qualitative/factors or functions; Annumeration or Sum, which depends on the factor discreteness, and Unity, which depends on continuity. In arithmetic the several kinds of operation are usually presented as accidental modes of dealing with numbers. If necessary and meaning is to be found in these operations, it must be by a principle: and that must come from the characteristic element in the notion of number itself. (This principle must here be briefly exhibited.) These characteristic elements are Annumeration on the one hand, and Unity on the other, of which number is the unity. But this latter Unity, when applied to empirical numbers, is only the equality of these numbers: hence the principle of arithmetical operations must be to put numbers in the ratio of Unity and Sum (or amount), and to elicit the equality of these two modes.

I would mildly disagree with Russell to the extent that continuity and discreteness do seem to imply each other and do seem to be two sides of the same coin, rather like Change and the Absolute. If one, for example, divides a line segment into say 2ⁿ finite segments, then we assume that "at" infinity the finite segments have reached zero length, and yet each location (point) is discrete. But how can a discrete point of 0 width abut another discrete point of 0 width? The points sort of have a peculiar state of discreteness and non-discreteness that are, one might say, superposed.

Russell took a dim view of Hegel's logic in general. And, from the above passage, one can see why. Yet, we should at least grant that Hegel's Idealism meant that the mind of the observer was a candle of the flame of Spirit (or Mind), so that Russell's traditional objectivism would not do for Hegel. (Even so, Russell eventually came to the view that the cosmos must be made up of something weird, such as a mind-matter composite.)

No Turing machine can model the cosmos

Versions of this paper are found elsewhere on my blogs.

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Dave Selke, an electrical engineer with a computer background, has made a number of interesting comments concerning this page, spurring me to redo the argument in another form. The new essay is entitled "On Hilbert's sixth problem" and may be found at
http://paulpages.blogspot.com/2011/11/first-published-tuesday-june-26-2007-on.html

Draft 3 (Includes discussion of Wolfram cellular automata)
Comments and suggestions welcome  

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Note: The word "or" is usually used in the following discussion in the inclusive sense.

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Many subscribe to the view that the cosmos is essentially a big machine which can be analyzed and understood in terms of other machines.A well-known machine is the general Turing machine, which is a logic system that can be modified to obtain any discrete-input computation. Richard Feynman, the brilliant physicist, is said to have been fascinated by the question of whether the cosmos is a computer -- originally saying no but later insisting the opposite. As a quantum physicist, Feynmen would have realized that the question was difficult. If the cosmos is a computer, it certainly must be a quantum computer. But what does that certainty mean? Feynmen, one assumes, would also have concluded that the cosmos cannot be modeled as a classical computer, or Turing machine [see footnote below].
Let's entertain the idea that the cosmos can be represented as a Turing machine or Turing computation. This notion is equivalent to the idea that neo-classical science (including relativity theory) can explain the cosmos. That is, we could conceive of every "neo-classical action" in the cosmos to date -- using absolute cosmic time, if such exists -- as being represented by a huge logic circuit, which in turn can be reduced to some instance (computation) of a Turing algorithm. God wouldn't be playing dice.
A logic circuit always follows if-then rules, which we interpret as causation. But, as we know, at the quantum level, if-then rules only work (with respect to the observer) within constraints, so we might very well argue that QM rules out the cosmos being a "classical" computer.
On the other hand, some would respond by arguing that quantum fuzziness is so miniscule on a macroscopic (human) scale, that the cosmos can be quite well represented as a classical machine. That is, the fuzziness cancels out on average. They might also note that quantum fluctuations in electrons do not have any significant effect on the accuracy of computers -- though this may not be true as computer parts head toward the nanometer scale. (My personal position is that there are numerous examples of the scaling up or amplification of quantum effects. "Schrodinger's cat" is the archetypal example.)
Of course, another issue is that the cosmos should itself have a wave function that is a superposition of all possible states -- until observed by someone (who?). (I will not proceed any further on the measurement problem of quantum physics, despite its many fascinating aspects.)
Before going any further on the subject at hand, we note that a Turing machine is finite (although the set of such machines is denumerably infinite). So if one takes the position that the cosmos -- or specifically, the cosmic initial conditions (or "singularity") -- are effectively infinite, then no Turing algorithm can model the cosmos.
So let us consider a mechanical computer-robot, A, whose program is a general Turing machine. A is given a program that instructs the robotic part of A to select a specific Turing machine, and to select the finite set of initial values (perhaps the "constants of nature"), that models the cosmos.
What algorithm is used to instruct A to choose a specific cosmos-outcome algorithm and computation? This is a typical chicken-or-the-egg self-referencing question and as such is related to Turing's halting problem, Godel's incompleteness theorem and Russell's paradox.
If there is an algorithm B to select an algorithm A, what algorithm selected B? -- leading us to an infinite regression.
Well, suppose that A has the specific cosmic algorithm, with a set of discrete initial input numbers, a priori? That algorithm, call it T_c, and its instance (the finite set of initial input numbers and the computation, which we regard as still running), imply the general Turing algorithm T_g. We know this from the fact that, by assumption, a formalistic description of Alan Turing and his mathematical logic result were implied by T_c. On the other hand, we know that every computable result is programable by modifying T_g. All computable results can be cast in the form of "if-then" logic circuits, as is evident from Turing's result.
So we have

Tc <--> Tg

Though this result isn't clearly paradoxical, it is a bit disquieting in that we have no way of explaining why Turing's result didn't "cause" the universe. That is, why didn't it happen that T_g implied Turing who (which) in turn implied the Big Bang? That is, wouldn't it be just as probable that the universe kicked off as Alan Turing's result, with the Big Bang to follow? (This is not a philisophical question so much as a question of logic.)
Be that as it may, the point is that we have not succeeded in fully modeling the universe as a Turing machine.
The issue in a nutshell: how did the cosmos instruct itself to unfold? Since the universe contains everything, it must contain the instructions for its unfoldment. Hence, we have the T_c instructing its program to be formed.
Another way to say this: If the universe can be modeled as a Turing computation, can it also be modeled as a program? If it can be modeled as a program, can it then be modeled as a robot forming a program and then carrying it out?
In fact, by Godel's incompleteness theorem, we know that the issue of T_c "choosing" itself to run implies that the T_c is a model (mathematically formal theory) that is inconsistent or incomplete. This assertion follows from the fact that the T_c requires a set of axioms in order to exist (and hence "run"). That is, there must be a set of instructions that orders the layout of the logic circuit. However, by Godel's result, the Turing machine is unable to determine a truth value for some statements relating to the axioms without extending the theory ("rewiring the logic circuit") to include a new axiom.
This holds even if T_c = T_g (though such an equality implies a continuity between the program and the computation which perforce bars an accurate model using any Turing machines).
So then, any model of the cosmos as a Boolean logic circuit is inconsistent or incomplete. In other words, a Turing machine cannot fully describe the cosmos.
If by "Theory of Everything" is meant a formal logico-mathematical system built from a finite set of axioms [though, in fact, Zermelo-Frankel set theory includes an infinite subset of axioms], then that TOE is either incomplete or inconsistent. Previously, one might have argued that no one has formally established that a TOE is necessarily rich enough for Godel's incompleteness theorem to be known to apply. Or, as is common, the self-referencing issue is brushed aside as a minor technicality.
Of course, the Church thesis essentially tells us that any logico-mathematical system can be represented as a Turing machine or set of machines and that any logico-mathematical value that can be expressed from such a system can be expressed as a Turing machine output. (Again, Godel puts limits on what a Turing machine can do.)
So, if we accept the Church thesis -- as most logicians do -- then our result says that there is always "something" about the cosmos that Boolean logic -- and hence the standard "scientific method" -- cannot explain.
Even if we try representing "parallel" universes as a denumerable family of computations of one or more Turing algorithms, with the computational instance varying by input values, we face the issue of what would be used to model the master programer.
Similarly, one might imagine a larger "container" universe in which a full model of "our" universe is embedded. Then it might seem that "our" universe could be modeled in principle, even if not modeled by a machine or computation modeled in "our" universe. Of course, then we apply our argument to the container universe, reminding us of the necessity of an infinity of extensions of every sufficiently rich theory in order to incorporate the next stage of axioms and also reminding us that in order to avoid the paradox inherent in the set of all universes, we would have to resort to a Zermelo-Frankel-type axiomatic ban on such a set.
Now we arrive at another point: If the universe is modeled as a quantum computation, would not such a framework possibly resolve our difficulty?
If we use a quantum computer and computation to model the universe, we will not be able to use a formal logical system to answer all questions about it, including what we loosely call the "frame" question -- unless we come up with new methods and standards of mathematical proof that go beyond traditional Boolean analysis.
Let us examine the hope expressed in Stephen Wolfram's New Kind of Science that the cosmos can be summarized in some basic rule of the type found in his cellular automata graphs.
We have no reason to dispute Wolfram's claim that his cellular automata rules can be tweaked to mimic any Turing machine. (And it is of considerable interest that he finds specific CA/TM that can be used for a universal machine.)
So if the cosmos can be modeled as a Turing machine then it can be modeled as a cellular automaton. However, a CA always has a first row, where the algorithm starts. So the algorithm's design -- the Turing machine -- must be axiomatic. In that case, the TM has not modeled the design of the TM nor the specific initial conditions, which are both parts of a universe (with that word used in the sense of totality of material existence).
We could of course think of a CA in which the first row is attached to the last row and a cylinder formed. There would be no specific start row. Still, we would need a CA whereby the rule applied with aribitrary row n as a start yields the same total output as the rule applied at arbitrary row m. This might resolve the time problem, but it is yet to be demonstrated that such a CA -- with an extraordinarily complex output -- exists. (Forgive the qualitative term extraordinarily complex. I hope to address this matter elsewhere soon.)
However, even with time out of the way, we still have the problem of the specific rule to be used. What mechanism selects that? Obviously it cannot be something from within the universe. (Shades of Russell's paradox.)

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Footnote Informally, one can think of a general Turing machine as a set of logic gates that can compose any Boolean network. That is, we have a set of gates such as "not", "and," "or," "exclusive or," "copy," and so forth. If-then is set up as "not-P or Q," where P and Q themselves are networks constructed from such gates. A specific Turing machine will then yield the same computation as a specific logic circuit composed of the sequence of gates.
By this, we can number any computable output by its gates. Assuming we have less than 10 gates (which is more than necessary), we can assign a base-10 digit to each gate. In that case, the code number of the circuit is simply the digit string representing the sequence of gates.
Note that circuit A and circuit B may yield the same computation. Still, there is a countable infinity of such programs, though, if we use any real for an input value, we would have an uncountable infinity of outputs. But this cannot be, because an algorithm for producing a real number in a finite number of steps can only produce a rational approximation of an irrational. Hence, there is only a countable number of outputs.

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Thanks to Josh Mitteldorf, a mathematician and physicist, for his incisive and helpful comments. Based upon a previous draft, Dr. Mitteldorf said he believes I have shown that, if the universe is finite, it cannot be modeled by a subset of itself but he expressed wariness over the merit of this point.

Objection to Proposition I of 'Tractatus'

An old post that for which I no longer vouch.

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From Wittgenstein's 'Tractatus Logico-Philosophicus,' proposition 1:

1. The world is all that is the case.

1.1 The world is the totality of facts, not of things.

1.11 The world is determined by the facts, and by their being ALL the facts.

1.12 For the totality of facts determines what is the case, and also whatever is not the case.

1.13. The facts in logical space are the world.

1.2 The world divides into facts.

1.21 Each item can be the case or not the case while everything else remains the same.

We include proposition 2.0, which includes a key concept:

2.0 What is the case -- a fact -- is the existence of states of affairs [or, atomic propositions].

According to Ray Monk's astute biography, 'Ludwig Wittgenstein, the Duty of Genius' (Free Press division of Macmillan, 1990), Gottlob Frege aggravated Wittgenstein by apparently never getting beyond the first page of 'Tractatus' and quibbling over definitions.

However, it seems to me there is merit in taking exception to the initial assumption, even if perhaps definitions can be clarified (as we know, Wittgenstein later repudiated the theory of pictures that underlay the 'Tractatus'; nevertheless, a great value of 'Tractatus' is the compression of concepts that makes the book a goldmine of topics for discussion).

Before doing that, however, I recast proposition 1 as follows:

1. The world is a theorem.

1.1 The world is the set of all theorems, not of things [a thing requires definition and this definition is either a 'higher' theorem or an axiom]

1.12 The set of all theorems determines what is accepted as true and what is not.

1.13 The set of theorems is the world [redundancy acknowledged]

2. It is a theorem -- a true proposition -- that axioms exist.

This world view, founded in Wittgenstein's extensive mining of Russell's 'Principia' and fascination with Russell's paradox is reflected in the following:

Suppose we have a set of axioms (two will do here). We can build all theorems and anti-theorems from the axioms (though not necessarily solve basic philosophical issues).

With p and q as axioms (atomic propositions that can't be durther divided by connectives and other symbols except for vacuous tautologies and contradictions), we can begin:

1. p, 2. ~p

3. q, 4. ~q

and call these 4 statements Level 0 set of theorems and anti-theorems. If we say 'it is true that p is a theorem' or 'it is true that ~p is an anti-theorem' then we must use a higher order system of numbering. That is, such a statement must be numbered in such a way as to indicate that it is a statement about a statement.

We now can form set Level 1:

5. p & q [theorem]

6. ~p & ~q [anti-theorem]

7. p v q

8. ~p & ~q

9. p v ~q

10. ~p & q

11. ~p v q

12. p & ~q

Level 2 is composed of all possible combinations of p's, q's and connectives, with Level 1 statements combined with Level 2 statements, being a subset of Level 2.

By wise choice of numbering algorithms, we can associate any positive integer with a statement. Also, the truth value of any statement can be ascertained by the truth table method of analyzing such statements. And, it may be possible to find the truth value of statement n by knowing the truth value of sub-statement m, so that reduction to axioms can be avoided in the interest of efficiency.

I have no objection to trying to establish an abstract system using axioms. But the concept of a single system as having a priori existence gives pause.

If I am to agree with Prop 1.0, I must qualify it by insisting on the presence of a human mind, so that 1.0 then means that there is for each mind a corresponding arena of facts. A 'fact' here is a proposition that is assumed true until the mind decides it is false.

I also don't see how we can bypass the notion of 'culture,' which implies a collective set of beliefs and behaviors which acts as an auxiliary memory for each mind that grows within that culture. The interaction of the minds of course yields the evolution of the culture and its collective memory.

Words and word groups are a means of prompting responses from minds (including one's own mind). It seems that most cultures divide words into noun types and verb types. Verbs that cover common occurrences can be noun-ized as in gerunds.

A word may be seen as an auditory association with a specific set of stimuli. When an early man shouted to alert his group to imminent danger, he was at the doorstep of abstraction. When he discovered that use of specific sounds to denote specific threats permitted better responses by the group, he passed through the door of abstraction.

Still, we are assuming that such men had a sense of time and motion about like our own. Beings that perceive without resort to time would not develop language akin to modern speech forms.

In other words, their world would not be our world.

Even beings with a sense of time might differ in their perception of reality. The concept of 'now' is quite difficult to define. However, 'now' does appear to have different meaning in accord with metabolic rate. The smallest meaningful moment of a fly is possibly below the threshold of meaningful human perception. A fly might respond to a motion that is too short for a human to cognize as a motion.

Similarly, another lifeform might have a 'now' considerably longer than ours, with the ultimate 'now' being, theoretically, eternity. Some mystics claim such a time sense.

The word 'deer' (perhaps it is an atomic proposition) does not prove anything about the phenomenon with which it is associated. Deer exist even if a word for a deer doesn't.

Or does it? They exist for us 'because' they have importance for us. That's why we give it a name.

Consider the eskimo who has numerous words for phenomena all of which we English-speakers name 'snow.' We assume that each of these phenomena is an element of a class named 'snow.' But it cannot be assumed that the eskimo perceives these phenomena as types of a single phenomenon. They might be as different as sails and nails as far as he is concerned.

These phenomena are individually named because they are important to him in the sense that his responses to the sets of stimuli that 'signal' a particular phenomenon potentially affect his survival. (We use 'signal' reservedly because the mind knows of the phenomenon only through the sensors [which MIGHT include unconventional sensors, such as spirit detectors].

Suppose a space alien arrived on earth and was able to locomote through trees as if they were gaseous. That being might have very little idea of the concept of tree. Perhaps if it were some sort of scientist, using special detection methods, it might categorize trees by type. Otherwise, a tree would not be part of its world, a self-sevident fact.

What a human is forced to concede is important, at root, is the recurrence of a stimuli set that the memory associates with a pleasure-pain ratio. The brain can add various pleasure-pain ratios as a means of forecasting a probable result.

A stimuli set is normally, but not always, composed of elements closely associated in time. It is when these elements are themselves sets of elements that abstraction occurs.

Much more can be said on the issue of learning. perception and mind but the point I wish to make is that when we come upon logical scenarios, such as syllogisms, we are using a human abstraction or association system that reflects our way of learning and coping with pleasure and pain. The fact that, for example, some pain is not directly physical but is 'worry' does not materially affect my point.

That is, 'reality' is quite subjective, though I have not tried to utterly justify the solipsist point of view. And, if reality is deeply subjective, then the laws of form which seem to describe said reality may well be incomplete.

I suggest this issue is behind the rigid determinism of Einstein, Bohm and Deutsch (though Bohm's 'implicate order' is a subtle and useful concept).

Deutsch, for example, is correct to endorse the idea that reality might be far bigger than ordinarily presumed. Yet, it is his faith that reality must be fully deterministic that indicates that he thinks that 'objective reality' (the source of inputs into his mind) can be matched point for point with the perception system that is the reality he apprehends (subjective reality).

For example, his reality requires that if a photon can go to point A or point B, there must be a reason in some larger scheme whereby the photon MUST go to either A or B, even if we are utterly unable to predict the correct point. But this 'scientific' assumption stems from the pleasure-pain ratio for stimuli sets in furtherance of the organism's probability of survival. That is, determinism is rooted in our perceptual apparatus. Even 'unscientific' thinking is determinist. 'Causes' however are perhaps identified as gods, demons, spells and counter-spells.

Determinism rests in our sense of 'passage of time.'

In the quantum area, we can use a 'Russell's paradox' approach to perhaps justify the Copenhagen interpretation.

Let's use a symmetrical photon interferometer. If a single photon passes through and is left undetected in transit, it reliably exits only in one direction. If, detected in transit, detection results in a change in exit direction in 50 percent of trials. That is, the photon as a wave interferes with itself, exiting in a single direction. But once the wave 'collapses' because of detection, its position is irrevocably fixed and so exits in the direction established at detection point A or detection point B.

Deutsch, a disciple of Hugh Everett who proposed the 'many worlds' theory, argues that the universe splits into two nearly-identical universes when the photon seems to arbitrarily choose A or B, and in fact follows path A in Universe A and path B in Universe B.

Yet, we might use the determinism of conservation to argue for the Copenhagen interpretation. That is, we may consider a light wave to have a minimum quantum of energy, which we call a quantum amount. If two detectors intercept this wave, only one detector can respond because a detector can't be activated by half a quantum unit. Half a quantum unit is effectively nothing. Well, why are the detectors activated probablistically, you say? Shouldn't some force determine the choice?

Here is where the issue of reality enters.

From a classical standpoint, determinism requires ENERGY. Event A at time(0) is linked to event B at time(1) by an expenditure of energy. But the energy needed for 'throwing the switch on the logic gate' is not present.

We might argue that a necessary feature of a logically consistent deterministic world view founded on discrete calculations requires that determinism is also discrete (not continuous) and hence limited and hence non-deterministic at the quantum level.

[The hit counters on all of Paul Conant's pages have been behaving erratically, going up and in numbers with no apparent rhyme or reason.]

[This page first posted January 2002]

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