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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Maybe you're God

What if the kooky idea of solipsism is a divine hint?

A solipsist is a person who thinks his is the only mind in the universe and that all the "reality" he perceives is an invention of his mind. Of course, we know that one's mind does have a unique and rather strong influence on what is taken for reality, but a solipsist takes this view to the ultimate extreme.

It is unlikely that anyone who is not schizophrenic really believes in such a philosophy.

And yet, what if there is a single mind of God who is talking to himself in a multiplicity of human voices? That is, what if your mind is, at root, actually God's? Doesn't Genesis say that we are made in his image? Doesn't Jesus, speaking to men, quote scripture: "You are gods"? (He also warns that some people around us are tares, the work of the devil, who will vanish in flame.)

The Fall represents the spiritual death of a man, whereby he is unable to commune with God. God has become alienated from Himself (you).

The reconciliation provided by Jesus permits these alienated sons (us) to commune with God as Jesus does. One can compare the Trinity -- three aspects of God in the individuals of Father, Son and Spirit -- to the light from a spectrum, which decomposes into three colors, but then recomposes into a single white light godhead when a reverse prism is set in place. But, even further, another prism can decompose white light into many colors, representing Jesus and his brothers (us), who all share in God's single mind.

The value of Mill's canons of logic (reply to Bradley)

Yes, of course J.S. Mill oversold his famous "canons of induction" in his book A System of Logic (1843). No doubt Francis Herbert Bradley in his Principles of Logic (1883 version) had some right to play the curmudgeon in scorning the canons as worthless insofar as logical proofs go.

A useful Wikipedia exposition of the canons:
https://en.wikipedia.org/wiki/Mill%27s_Methods

Here are the canons (from Bradley):

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1. If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is, is the cause (or effect) of the given phenomenon.

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2. If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every instance in common but one, that one occurring only in the former: the circumstance in which alone the two instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.

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3. If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common but the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.

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4. Subduct from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents.

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5. Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation.

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That Mill did not see these canons as hard and fast rules is even noted by Bradley who pointed out that Mill had observed that Method 1, in particular, along with other methods, had on occasion given rise to false results.

Still, Bradley claims that scientific men such as Rudolph Hermann Lotze, Christoph von Sigwart, William Whewell and William Stanley Jevons "had taken a view of the process of scientific discovery that was not favorable" to the canons.

No doubt not only Mill, but Bradley, would have benefited from the advances of modern statistics -- in particular the methods of quantitative correlation. Further, a solid course in the logic of set theory would have been enlightening to these gentlemen. But these developments were yet some years off.

On the other hand, though Mill may have been imprudent in his zeal for his canons, I think it is quite obvious that any empirical approach to discovery is subject to the criticism that an airtight logical proof is impossible. I expect that Bradley would have been familiar with Hume.

The "inductive logic" of Mill and his followers was fundamentally flawed, Bradley says. Because Mill's canons "presuppose universal truths, therefore they are not the only way of proving them. But if they are the only way of proving them, then every universal truth is unproved."

Again, any reader of Hume already is aware of this issue. (Further, one might question Bradley's reliance on universals [which came to replace Plato's Forms]. Bertrand Russell was one philosopher who questioned whether universals can be said to exist.)

Bradley, while generally praising Jevons, devoted a chapter to errors he had uncovered in Jevons' Principles of Science (1874), which outlined Jevons' equational logic, a precursor of modern symbolic logic that supplanted Booles' pioneering work. Bradley objects to Jevons' use of the sign "=" to indicate equality. Rather, says Bradley, the sign indicates identity. But in that case, if two propositions are identical, then to say that "A = B" is to say that "A = A" is to say nothing. (Russell later addressed this point; see posts on this blog.)

Bradley goes on to caution that, though Jevons' "logical machine" (a "logic piano" he invented) could calculate and do a form of reasoning, it fell short of what the human mind can do.

There may have been some mathematical insecurity showing here.

Readers, notes Bradley, may wonder why his critique of Jevons didn't include a mathematical analysis. Bradley is "compelled to throw the burden of the answer on those who had charge of my education, and who failed to give me the requisite instruction." Even so, the "mathematical logician" failed to impress Bradley, considering that "so long as he fails to treat (for example) such simple arguments as 'A before B and B with C, therefore A before C,' he has no strict right to demand a hearing."

In any case, it seems likely that when Mill used the word logic he had in mind processes of thought that were somewhat larger than those entertained by the scholastic philosophers and their modern formalist descendants.

On ground that Mill's methods are not truly inductive, Bradley thumps Mill's logic as a fiasco. "And if I am told that these flaws, or most of them, are already admitted by Inductive Logicians, I will not retract the word I have used. But to satisfy the objector I will gave way so far as to write for fiasco, confessed fiasco."

Admittedly, Mill's writings in general show a not brilliant -- though very intelligent -- mind at work. Yet, I can only admire the brilliance of his analysis of "induction." Do we not have here a precise portrayal of the scientific method of investigation? He gives us rules for weeding out probable causes and for reduction of error. Though his rules are useless for absolute proof, as in a mathematical theorem, they would certainly be helpful if studied by potential jurors, judges and officers of the court.

A case adjudicated by Mill's standards would result in a verdict in which jurors usually were convinced beyond a reasonable doubt. In that sense, Mill's canons do indeed undergird what is accepted as proof.

Scientists and engineers similarly use Millsian standards -- perhaps bolstered by modern statistical methods which in fact stand upon Millsian assumptions -- and accept "proofs" that they realize are subject to revocation and revision.

So, though I am quite enjoying Bradley's quirky and bombastic book on logic (a subject he eventually dropped in favor of Hegelian metaphysics), I cannot quite accept his pedantical bruising of Mill. Allowances, I think, should be given for Mill's rather loquacious and often impressionistic style, even when he is honing in on fine points.

Bradley's Logic, for all its irascibility, is a fascinating critique of the various assumptions of 19th century logicians. He had grounds for questioning the law of the excluded middle^[1] long before Ludwig Brouwer and the intuitionists. And his off-beat analyses demonstrate that there is more to the subject of logic than most of us have been taught.

Take this tidbit, for example:

... if we refuse to isolate a relation within [a conceptual] whole, if we prefer to treat the entire compound synthesis as the conclusion we want, are we logically wrong? Is there any law which orders us to eliminate, and, where we cannot eliminate, forbids us to argue?...

... for the conclusion is not always a new relation of the extremes; it may be merely the relation of the whole which does not permit the ideal separation of a new relation. And, having gone so far, we are led to go farther. If, the synthesis being made, we do not always go on to get from that the fresh relation, if we sometimes rest in the whole we have constructed, why not sometimes again do something else? Why not try a new exit? There are other things in the world besides relations; we all know there are qualities, and a whole put together may surely, if not always at least sometimes, develop new qualities. If then by construction we can get to a quality, and not to a relation, once more we shall have passed from the limit of our formula.

Earlier in his book, Bradley had taken a dim view of the whole notion of relation as it applies to logic.

Another interesting thought: Does comparison count as a form of inference? Bradley asks. The suggestion that whenever we compare, we are reasoning runs counter to our established ideas. But how can we repulse it? he adds. "We start from data, we subject these data to an ideal process, and we get a new truth about these data."

We are liable to compare ABC and DBF and observe that they are alike in B, notes Bradley. "No doubt we may question the validity of this inference, but I do not see how we can deny its existence."

In modern parlance, we would say X = {A,B,C} and Y = {D,B,F}. Hence X ∩ Y = {B}. One might even contend that the set theoretic mode helps us pin down better what we mean by likeness.

I suppose that when logic is restricted to meaning-free formulas, many of Bradley's insights and concerns are bypassed. That is to say, for example, if P is held true and Q is held true, the formula "P implies Q" is deemed to be true.

But, one is uncomfortable with the proposition, "If Socrates is a man, then Mars is a planet." We demand more. The proposition "If Socrates is a man and all men are mortal, then Socrates is mortal" is usually accepted. The Socrates syllogism of course dovetails nicely with naive set theory, as in

(all x ∈ M)(x has property of mortality) & x_o ∈ M. ∴ x_o has the property of mortality

(where x_o stands for Socrates).

By accepting parts of naive set theory as axiomatic, we avoid Russell's troubled efforts to define sets in terms of propositions.

In this regard, some logicians make a distinction between adjunctive operations and connective operations. If adjunctive reasoning is used, a false proposition implies every proposition and a true proposition is implied by every proposition. Connective reasoning is more common in routine discourse. The distinction works out to mean that a truth table may be read in both directions in the adjunctive case but in only one direction in the connective case.

The adjunctive camp has been warring against the connective camp since antiquity, according to Hans Reichenbach, ^U1 who is comfortable with both approaches. In any case, those who believe that "implication" should mean formal derivability, as in the Socrates set theoretic formulation, would favor the connective interpretation. (Reichenbach notes that he prefers the term "adjunctive implication" to Russell's "material implication.")

Returning to the issue of syllogisms, Bradley insists that the concept of major and minor premise is delusional. Though he may have a point as to how these terms were generally used, it seems to me that in a set theoretic context, we have the set which contains the members. A particular member then might be seen as equivalent to a minor premise. Of course, something I have not shown is my consideration that the various syllogisms (and their symbolic equivalents) yield Euclidean-style proofs when we can establish set membership (even though we don't generally bother).

Bradley, I assume, wrote his Logic in order to pave the way for his coming foray into metaphysics, Appearance and Reality (1893). His entire Logic is an attack on the assumptions of "popular" logic and reasoning. I will agree that he raises many interesting points, though I would say that at least some have been effectively answered by recent developments, such as information theory.

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1. Sometime in the 1930s (I'll have to track down the precise reference at some point), Bertrand Russell pointed out that the assumption that a proposition was either true or false was based on a metaphysical notion that the cosmos is completely describable with facts (propositions with a true or false value). But this notion is assuredly unprovable. In the thirties there was no way to know whether there were any mountains on the dark side of the moon. It was presumed that the assertion or its negation must be true. Yet, one could as well have said the proposition was undecidable and hence neither true nor false.
It is noteworthy that experiments in quantum mechanics demonstrate that propositions such as "Schroedinger's poor cat has expired" are undecidable until a quantum event has been recorded. And, it is not altogether absurd to surmise that the lunar dark side's mountainous terrain was in superposition with a flat surface prior to the dark side's observation via spaceborne cameras three decades after Russell's comment.

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U1. Elements of Symbolic Logic by Hans Reichenbach (Macmillan 1947). Chapter II, Section 7.