From: Paul Healey <Paul.Healey- at>

Subject: The Mathematical Definition of a Circle

In reply to Stephen W. Taylor's message of 05/12/98

Your notion of the circle interests me, but perhaps it is not reaching the right audience ?, or maybe they cannot see the value of the dialectics ?; I think a carefully written example relating to the praxis of Circle Maths might help.

My concern is that, I think the formal thinker fails to realise, the notion as the absolute - the mind has an identity without difference, and the body has an identity within difference. This is why, I think Hegel would say, that it is not enough to settle for mathematical proofs, like L.E.J. Brouwer's: the principle of excluded middle is false ( see his Cambridge Lectures ).

Consider the 3 moments in relation to the absolute, when the mathematicians try to envisage the world that their procedures depend on:

  1. Its measurement; What values can be assigned to its algorithms
  2. Its quality; the cardinal value of its essence
  3. Its quantity; the temporal displacement of its dimensions

For I see a grounding of the dialectics, as the only way to reconcile science with religion with ideology with ... etc. For how should we reconcile the desire that everybody should live according to the same principles, with the one that everybody should live according to their own. Is this not relevant to our discussion group ?; I think it would be regrettable, and even fatal if too many people became so absorbed with developing their own home pages, that they lost sight of what Hegel's project was all about i.e. I did not forward Annette's last post, as it consisted mainly of an apology for other commitments and ill health.

It is of course equally important to me that other people should not be excluded from the group, even if they haven't read enough. There contributions are always valued by someone. I of course, like I assume Andy would, suggest that they read Hegel, and those who have written about him e.g. Marx, Lukacs, Llyenkov, E. Harris, and Mac Gregor etc. And if they don't understand a particular passage, ask someone else what their thoughts are on it are. My reading is of course directed mainly at Hegel's method of reasoning, as it relates to Chance; in particular the absolute notion of a dialectical truth schema.

Law of Identity

Any comments, or references on the following post, which I sent to the Hegel list at Bucknell would be much appreciated:

In response to the 02Dec98 post by Paul Healey:

You are asking for a doctoral thesis here, Mr. Healey. In brief I will say that the empiricist and the transcendentalist both presume that the so-called law of identity is inviolable. The dialectician does not.

How can this, be reconciled with the proposition, 'Everything is a syllogism', if in fact it a proposition at all ?

I accept, that its meaning has a realisation, as a notion. But, is it a syllogism ?, for where is its middle term ? No procedure is evident in the assertion itself. It's identity, can be constued as being violated - for a syllogism is also a paradox - its procedure cannot determine the validity of it's own principles. When a proposition declares something, that it is true for what it is saying, I see no problem with it i.e., everything is either a syllogism or it is not a syllogism. This I think, is not a syllogism in itself. Hence, this is why Hegel agrees with Descartes when he makes an issue, of the man, that calls Descartes maxim: COGITO, ERGO SUM, a syllogism ( see, Section 64. TSL ). In fact, he goes further:

Such a feeble exemplification of the syllogism may be seen in the magnet. ( Section 24. TSL )

A stronger exemplification can be seen, in the proposition that, the truth is a syllogism, where the truth is taken to be synonymous with a trancsendental schema - there is not only one right way of constructing it. For when he remarks about The Maxim of Excluded Middle, belonging to the definite understanding ( see section 119. TSL ), it is this understanding when applied to reason, that makes it one sided. How reason is encounted, determines whether we see it as an immanently grounded system of shapes. For when our conception of the middle term gives rise to a consciousness, that denies that which is for itself, the whole cannot be present in it. This is how, I think Hegel sees the structure of a syllogism; one which creates the illusion that the middle terms can be a part of its conclusion, by a rearrangement of its terms. It also appears to be a causal relation, when you consider that its argument can be reversed, whereas the actual fact of the matter is the arrangement of its terms has no effect on its structure. However, if its terms were not variable, contradictory categories could not be deduced ( See section 295 in Hegel's Phenomenology of Spirit ). From this it follows, that unity thus conceived, as a Transcendental system of ideas, by and through the use of a syllogism is fatally flawed e.g. consider the doxology of the trinity in relation to the three moments of the spirit's movement ( see sections 530-534. PS ). And this is, I think why Hegel is fully justified in saying,

It follows that the categories are no fit terms to express the Absolute - the Absolute not given in perception - and understanding, or knowledge by means of the categories, is consequently incapable of knowing the Things-in-themselves. (Section 44. TSL )

What else could he be referring to , if not that which is the resultant of a transcendental system ? In terms of Chance, the notion of a syllogism appears, as if its structure includes a third term in an attempt to absorb the middle one. In this case what is actually absorbed, is its spirit, as it is set against itself. An absolute without essence. By the use of Either-or, reason can also be set against itself, as there is no unity. Being without consciousness. But since reason itself, is not absorbed by such a notion, it is only those that hold onto such a ground, who take out their own spirit or soul.

What the resultant of my research revealed to me, is that the process, which is evident in a dialectical schema is not absorbed - its procedure is much better when it comes to accounting for chance. What I think does require a thesis, is to present a coherent argument, that Hegel's dialectical schema also accounts for consciousness.

Again -- the principle of identity is the principle of Analysis and it is quite useful and necessary in the consideration of finite things. It is not surprising that the finite Sciences lay so much stress upon it. However, the principle of identity-in-difference is the principle of Synthesis, and it is only with this new logical construct that Hegel is able to rescue Metaphysics from the state of inanity in which modern logicians have left it after Kant.

Now, since everything Kant says about the use of reason, as observed in the syllogistic process, belongs to understanding, it follows that the transcendentalist is a subjectivist:

Hegel thought so, too. Kant denied the term, but Hegel reaffirmed it as it applies to Kant. It's a controversy.

This is why he wants to rule out the dialectic, which he attempts to do ( See CPR. III. Of the Division of General Logic into Analytic and Dialectic ). So, like the Empiricism of Hume, he throws away any chance of demonstrating the categories from the outset...

Kant was a strange bird in logic. He held that the law of identity was sacrosanct, but he wisely saw its limitations. It is limited. It *knows* it is limited. It celebrates its own limitations. All it can do is to break things apart. That's all. It cannot build things up. It depends on things being *given* to it from the outside so that it can break them apart. Hegel agrees with this. Where Hegel disagrees is the prejudice that the human mind is therefore limited just because Analysis is limited. So Kant built up Analysis with one hand and slapped it down with the other, especially with his doctrine of the Unknowable Thing-in-itself. Hegel provides the cure for this fallacy.

Hegel says,
From what has been said the principles of logic are to be sought in a system of thought-types or fundamental categories, in which the opposition between subjective and objective, in its usual sense vanishes. (Hegel, SCIENCE OF LOGIC, Section 24 )
which can only be done in a dialectical sense, by showing how they can be conjoined, as a multiplicity in their immediacy ( with nothing coming between them i.e. no third term ): we can know what is meant by the truth.

Hegel is absolutely correct. The syllogism, and any full logical form which contains a beginning, a middle and an end, i.e. a movement, i.e. a dynamic process, i.e. a cycle of change, does indeed represent the full truth of the Absolute -- that it is Change.

So no wonder Hegel says that the arguments [Kant] puts over for the antinomies are mere shams of a demonstration. The contradiction of the categories into reason was done so by a transcendental understanding not a dialectical one. For it is by his syllogistic method of reasoning that Kant dismisses the dialectical argument ( CPR. Section VII. Critical Solution of the Cosmological Problem ). Reason is not connected to understanding ( see section 80. TSL ). Such a connection, is in fact illusory.

Yes, those are Kant's key errors, Mr. Healey.

Understanding, is therefore conceived of as something quite different by both philosophers.

Hegel subscribes to Kant's notion that Reason is superior to the Understanding -- Reason is infinite and metaphysical while the purest Understanding remains finite and empirical.

And this is why Hegel later points out the fundamental defect of every dualistic system, and especially that of Kant's, is one that belongs to their realisation of unity...

Right, Mr. Healey. Well put.

In general, the transcendental system includes a third term, so the middle one can be excluded - it does this by appealing to a conjoining of terms to make it appear as if there is a synthesis, in its multiplicity and immediacy. It then looks as if there is nothing between the two categories, but in actual fact their true procedure has not been discerned - for there is no way of knowing which category its identity belongs to.

That's a pretty good way of stating Hegel's criticism, Mr. Healey. You have indeed been reading Hegel's words. For this you deserve more credit. To read Hegel's SCIENCE OF LOGIC at any level is a rare event, and you are breaking new ground, even on the Hegel List, by raising these technical issues in your own way.

Both the empiricist and the transcendentalist are forced to recognise the principle of excluded middle, not to mention a decision procedure that has no causality attached to it...

This section of your paragraph is correct in all respects, Mr. Healey. You are getting closer to Hegel's grasp of the Synthetic method.

Where the dialectical method differs, as my reading of Hegel's work sees it, is that Being along with its Essence play an equal part in discerning what principles belong to reason and what do not.

Your wording is a trifle clumsy here, but I see where you are going with this. Hegel does not ask us to choose: 'Either Being or Essence and No Middle Term!' The Truth is a changing, dynamic movement of Being and its Essence in a precise series of moments which are related to each other in clear patterns. This is Hegel's idea of Phenomenology.

For in the PHENOMENOLOGY OF SPIRIT, Hegel is quite clear, about where probability stands in relation to truth ( see Phenomenology of Spirit, p152. Trans. Miller, A. V. 1977. OUP ) - in its objective sense it can obtain an insight into its notion which Kant's Transcendental Logic would deny. ( see CPR. TRANSCENDENTAL LOGIC, SECOND DIVISION, TRANSCENDENTAL DIALECTIC INTRODUCTION, I. Of Transcendental Illusory Appearance )

Kant also knew where probability stands in relation to truth -- it is one of the moments of the Mode of Understanding. What is possible, what is probable and what is necessary form a Triad. Kant knew that too, Mr. Healey.

There is also undoubtedly a difference between what is being measured, and our notion of the principle and its procedure, that is being used to measure it. What is being measured, is relative, while the truth of the procedure being used to measure it is absolute.

We grow farther apart with this, Mr. Healey. Your words are not metaphysical in Hegel's new sense, his two-sided sense. Your words are one-sided and so they miss Hegel's metaphysic. There is no dualism between the Procedure and the Being in Hegel's new science. That would be transcendental. That would suggest a Spirit hovering over Being. That is absurd. It is not what Hegel means at all.

Yes, you are right. I was trying to account for the proposition that I gave in the previous post; 'the truth is absolute and relative' - which I now think could more easily be explained in a dialectical sense, by saying truth is equivalent to a dialectical schema - there is only one right way of constructing it. That is, its value is in accordance with what it is in itself, but not necessarily our apprehension ( understanding its perception ) of it as an event ( how I can catch hold of it, or gain by it ). Reason according to Hegel, is an immanently ( the power of the universe to include the whole ) grounded system of thoughts.

A syllogism is not absolute, simply because its notion has no reason for being - it can't answer the question, what is the truth? It cannot say, whether there is an accordance of the cognition with its object. Hence, Kant's use of an analytic/synthetic dichotomy. It is if you like, a schema that has been presupposed without demonstration.

While it is true that a Proposition cannot answer the question, What is Truth, nor can a Syllogism taken as a mere series of Propositions answer that question in a conclusion, this is not what Hegel is getting at. When Hegel said that 'Everything is a Syllogism,' he meant that the principle of the Syllogism, the thesis, antithesis and synthesis, is indeed the pattern of the entire Cosmos. It is the pattern of the human Soul. It is the pattern of Everything. It is the Absolute itself.

This is why, I would hold it is necessary to understand, how a dialectical truth schema works: it has nothing in common with a formal one, which cannot help but confounding what Hegel's message is really about. If you do not think so, are you not the slightest bit excited by the prospect of the empiricist and the transcendentalist losing a simple bet? Is not your faith tested, by your willingness to make a bet on it? -- Paul Healey

There is no point in making a wager, Mr. Healey, unless there is some outside judge who can decide the winner. But Hegel's revival has no such judge today. Reason is the only judge, and folly often believes that it is Reason itself. So a wager is absurd.

But I will agree with you up to this point: A dialectical-speculative Truth schema has nothing in common with an Analytic one. To think otherwise is to confound Hegel's message.

Best regards, -- Paul Trejo

Paul Healey

Paul Healey