Heyting (5.3.1) Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Section 5.3.1, “Calcul numérique”, summary

 

by Corry Shores

 

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[The following is summary. I am not a mathematician, so please consult the original text instead of trusting my summarizations, which are possibly mistaken and probably inelegantly articulated. Also, my abilities with French are insufficient to translate reliably, so please again rely upon the quotations rather than my summarizations. Bracketed comments and subsection divisions are my own. Proofreading is incomplete, so please forgive my mistakes.]

 

 

 

 

Summary of

 

Arend Heyting

 

Les fondements des mathématiques.

Intuitionnisme.

Théorie de la démonstration.

 

Première section:
Intuitionnisme

 

5.
L'intuitionnisme brouwérien

 

5.3

Continu. Suite de choix

 

5.3.1

“Calcul numérique”

 

 

 

 

 

Brief summary:

(5.3.1.1) Coordinated choice sequences can be used to define the operations of calculation. But problems arise when inequalities are used. For Brouwer, “a is different from b”, (a ≠ b), means that a = b is impossible. But when we are dealing with the continuum, we have an additional relationship of inequality and equality that can hold between variables. Roughly: For the continuum, we additionally have the relationship “a is positively different from b” or “a is apart from [écarté de]  b” (a ⧣ b). This is fulfilled when, in the series of intervals that define a and b, two external intervals are known [to be shared by both]. a ≠ b of course results from a ⧣ b ; but the inverse cannot be affirmed. Moreover, as we can easily see, the negation of a ≠ b, and also of a ⧣ b, is equivalent to a = b.

 

 

 

 

 

Contents

 

5.3.1.1

[Inequality and Continua]

 

Bibliography

 

 

 

 

 

 

Summary

 

5.3.1.1

[Inequality and Continua]

 

[Coordinated choice sequences can be used to define the operations of calculation. But problems arise when inequalities are used. For Brouwer, “a is different from b”, (a ≠ b), means that a = b is impossible. But when we are dealing with the continuum, we have an additional relationship of inequality and equality that can hold between variables. Roughly: For the continuum, we additionally have the relationship “a is positively different from b” or “a is apart from [écarté de]  b” (a ⧣ b). This is fulfilled when, in the series of intervals that define a and b, two external intervals are known [to be shared by both]. a ≠ b of course results from a ⧣ b ; but the inverse cannot be affirmed. Moreover, as we can easily see, the negation of a ≠ b, and also of a ⧣ b, is equivalent to a = b.]

 

[In rough form: Numerical calculation.– The use of choice sequences in mathematics is based on the possibility of establishing coordination between choice sequences. The sequence β can be coordinated with the sequence α by the fact that each choice of β can be defined by using a finite initial segment of α. The definitions of the operations of calculation constitute simple examples of such a coordination. Once these have been given, we then easily have the results of a pure calculation of elementary arithmetic and algebra; however, complications can appear as soon as inequalities are used. “a is different from b”, (a ≠ b), means, in BROUWER’s terminology, that a = b is impossible. For the continuum, we additionally have the relationship “a is positively different from b” or “a is apart from [écarté de]  b” (a ⧣ b). This is fulfilled when, in the series of intervals that define a and b, two external intervals are known [to be shared by both]. a ≠ b of course results from a ⧣ b ; but the inverse cannot be affirmed. Moreover, as we can easily see, the negation of a ≠ b, and also of a ⧣ b, is equivalent to a = b. BROUWER has shown using examples [26] that there are real numbers for which we do not know if a = 0 or  if a ≠ 0. We will write a ⩺ b when we know in the series of intervals for a an interval that is external and to the right of an interval of the sequence for.b. From a ⧣ b it follows either a ⩺ b, or b ⩺ a.]

]

Calcul numérique.– L’utilisation des suites de choix en mathématiques est fondée sur la possibilité d’établir des coordinations entre des suites de choix. La suite β peut être coordonnée à la suite α par le fait que chaque choix de β peut être défini au moyen d’un segment initial fini de α. Les définitions des opérations du calcul constituent des exemples simples de telles coordinations. Une fois celles-ci données, on a alors sans difficulté les résultats de pur calcul de l’arithmétique élémentaire et de l’algèbre ; pourtant des complications peuvent apparaître dès qu’on fait emploi d’inégalités. “a est différent de b”, (a ≠ b), signifie, dans la terminologie de BROUWER, que a = b est impossible. Pour le continu, on a en outre la relation “a est positivement différent de b” ou “a est | écarté de b” (a ⧣ b). Celle-ci est remplie quand, dans les suites d’intervalles qui définissent a et b, on connaît deux intervalles extérieurs l’un à l’autre. a ≠ b résulte évidemment de a ⧣ b ; mais l’inverse ne peut pas être affirmé. De plus, on le voit facilement, la négation de a ≠ b, et aussi celle de a ⧣ b, est équivalente à a = b. BROUWER a montré par des exemples [26] qu’il y a des nombres réels pour lesquels on ne sait pas si a = 0 ou a ≠ 0. On écrira a ⩺ b quand on connaît dans la suite d’intervalles pour a un intervalle qui est extérieur et à droite d’un intervalle de la suite pour.b. De a ⧣ b il suit soit a ⩺ b, soit b ⩺ a.

(24-25)

[26] : Mathematik, Wissenschaft und Sprache. Mh, Math, Phys. 36 (1929), p.153–164 ;

(79) (boldface is mine)

[contents]

 

 

 

 

 

 

 

 

Bibliography:

 

Heyting, Arend. Les fondements des mathématiques. Intuitionnisme. Théorie de la démonstration. Paris / Louven: Gauthier-Villars / E. Nauwelaerts, 1955.

 

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Simon Duffy. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze, Chapter 1, §7 "Et determinatio negatio est"

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[The following summarizes Simon Duffy's extraordinary book, The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze, Chapter 1, §7.

Duffy's work is remarkable, so I highly recommend this book. If it costs too much, perhaps encourage your library to obtain a copy.]

Simon Duffy. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze,

Chapter 1 "Spinoza from the point of view of an idealist or a materialist dialectic",

§7 "Et determinatio negatio est"

We might say that the book lying on the table has many pages, that is, many parts, and the book itself is the whole encompassing those parts. But then we might stand further away, and take into account the whole desk with all its accompanying objects. The whole work area itself is then a whole, whose parts are the many objects contained in it and on it. We could stand even further away and consider the desk as a part of the whole office, then consider the office as a part of the whole building, then the building as a part of the whole city, and so on. [We might even venture to say that all of reality is one whole that only arbitrarily is broken into parts.] In other words, under these considerations, part and whole seem to be arbitrary divisions our mind makes for its given purposes.

Spinoza likewise does not consider parts and wholes as real things. Rather, they are useful mental fabrications, "beings of reason" he calls them. He writes in his Short Treatise:

part and whole are not true or real entities, but only things of reason, and consequently there are in Nature neither whole nor parts.

So we may consider the book as sharing a boundary with the desk. Or we may not grant such a boundary, and consider the whole work area as complete in itself, with its own arbitrarily designated boundary with the other things in the office. Each time we arbitrarily designate a boundary, we are fabricating a being of reason, rather than identifying a real being. What constitutes the boundary between the desk and the book is a negational determination: the book is not the desk and it lies outside it, and vice versa. But really there never was such an internal opposition to all of reality.

However, in our minds, we may construe such a distinction in which there is a mutual negational determination. Hence Spinoza in his 50th letter to Jelles writes determinatio negatio est. [For more on this letter, see Macherey's commentary.]

As Duffy has explained, Hegel wants to write Spinoza into his dialectical history, so to render Spinoza's substance metaphysics as a lesser developed stage in his dialectical system. This turns upon Hegel's suspicious misreading of Spinoza's determinatio negatio est. Hegel reads it as omnis determinatio est negatio, "all determination is negation." This is a misleading misreading, because it suggests that Spinoza meant that for all of being, determination is negation. But really, Spinoza just meant, for only the beings of reason, determination is negation. (18c-19)

In the 50th letter, Spinoza speaks of determination when referring to figures. But Duffy shows that even Hegel himself does not consider figures as identical with the beings bearing them [see §206 of Hegel's Science of Logic.] Here is Duffy's notable contribution to this debate. He has found compelling evidence that even Hegel himself knew he had no grounds to overgeneralize Spinoza's claim.

So for Spinoza, a determinate mode is a unique entity only in our imagination. Modes are produced by substance, and modes modify substance. So in that sense modes are determined by substance. But substance is one with its modes. For, modes are immanent expressions of substance. Modal determinations are merely fabrications of the imagination.

Hence Hegel has "no justification" for interpreting Spinoza's theory of imaginary determinations as implying that all determination is negation. For, substance determines its own modes without them also negating substance. (19a.d)

Duffy, Simon. The Logic of Expression: Quality, Quantity and Intensity in Spinoza, Hegel and Deleuze. Aldershot: Ashgate Publishing, 2006.

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Substituting Positivity

by Corry Shores
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Scott Wollschleger writes in the comments to the Substitution Substituting Creativity entry:

right, I see what you mean and see how substituting would be the wrong path if we are looking to create something new. Is there something weak in the idea of Substitution maybe? Maybe weak is not the best word. Substitutions can have a diminishing effect. Remember in high school, when a Sub would teach class, the students would torture the Substitute teacher? Maybe it had to do with the fact that a person cannot be substituted, and we had to punish the teacher, as it were, for being part of a false reality. There are times when substitution lends to flexibility, which can lead to an amplification. But in any event, thanks for the comments and we can move on to repetition, i think it offers more in the creative realm.

CS Response: I might only add that substitution is either a zero-sum, where there is no net gain, or additive, where the gains are within the same order. Deleuze's repetition as exponential is more than additive, because the repetition creates new dimensions, degrees, and levels, not just new terms. In other words, it is hyper-positive. We might contrast this with the principles of deconstruction where terms are always under erasure. For Deleuze, terms instead express new dimensions without destroying those out from which the new ones grow.

Scott Wollschleger’s Inquisitive Connection between Engineering and Positivity (From Comments to the Summary of Welchman’s “Machinic Thinking")

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Scott Wollschleger insightfully wonders (in the Welchman post):

"thanks,this is great. can you say more about engineering? also would it be ok to say that what matters is also something positive by nature?"

Scott raises the perfect questions, because they connect two of the most essential ideas in the Welchman article.

Engines are production machines: they produce force. Thermodynamics is the study of changing heat (thermos) to motion or power (dynamis). The first law of thermodynamics is the law of the conservation of energy: no more energy can leave a system than has entered (or "Energy can neither be created nor destroyed. It can only change forms"). The second principle of thermodynamics is entropy: differences in forces tend to equalize in a system. When you put hot coffee in a mug, the mug cools the coffee and the coffee warms the mug, and slowly they tend towards about the same temperature.

Deleuze is a more revolutionary thermodynamicist, because he is a “Difference Engineer”: in the first place, what matters in Deleuze’s thermodynamics is positivity, as Scott insightfully observes. We might consider a certain machine, let’s say a music-composer-machine, who designs on his drawing board intricate musical pieces that themselves are little machines, which then produce performances and recordings that produce listenings, which then produce new ideas and viewpoints in other minds, and so forth. This composer – this engineer who is himself an engine that produces more engines – this composer-engineer has a net positive of force (hence creative dynamics does not obey the law of conservation of energy) because, this music engineer does not become less productive and effective as a composer, rather, his composing powers become greater with time. His internal creative forces do not cancel each other over time, resulting in stasis, rather they build upon themselves so that he actually gains more creative power (and thus creative dynamics does not obey the principle of entropy).

The reason for these broken laws is that thermodynamics is not a matter of science for Deleuze, but instead a matter of engineering. Scientists want to isolate consistent principles that describe the way things work. Engineers want to produce things. Engineers discover principles of reality, but they do so by making things which perform a function that responds to a real and immediate problem. There are concrete forces in the world – needs, desires, wars – that fuel ingenuity. The sciences tend to be too abstract; they do not touch ground. Hence for Deleuze, we must be engineers of reality, we must produce machines and be machines, rather than think of ourselves as distinct from machines and study them impartially. Machines in Deleuze are not mechanisms that stay the way they are, but are a part of a larger machine that itself is changing according to intense forces.

So the fuel for machines are intensive forces. The reason that the world around us changes is because there are competing forces whose outcomes have to do with given conditions combined with pure chance, the dice throw. This is partly why machinization cannot be a science. It must be engineering, because engineers work on the battlefields, responding constantly to changing unpredictable conditions of war.

The positivity of course then is the absence of entropy and of the conservation of energy. The intensive forces do not decrease but instead continue raging. Although intensive forces cancel when they explicate into extensity, as when a hammer hits hot steel to produce weapons, intensive forces nonetheless are always implicated within each other, which means you can take one away, but there is forever another one left-over self-expressing from within it. So forces continue, and productions forever add reality to reality. There is no principle of destruction or dissolution, merely additive becoming.

The Positivity of Spinoza’s Semiology

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According to Deleuze, Spinoza opposes signs to expressions. Of signs there are three sorts: indicative, imperative, and revelatory. Indicative signs enable us to infer something from a modification in our body, for example, we may see a book on the table as an indication of a state of affairs. But as modes, we are finite, so we never see the whole picture of causation. Thus, ideas drawn from such indications are not adequate, because they cannot fully explain their cause. Imperative signs cause us to consider the laws of nature as moral laws: pity might naturally result from seeing suffering, and so we might consider it a moral law to have pity, when instead it is irrational and hence unethical. Revelatory signs, as inherently mysterious and obscure, do little more than cultivate an inexpressible and confused knowledge of God, rather than a rational adequate sort.

In these cases, signs are representational, which implies a negativity or an inherent absence: the referent is not the sign itself, but is missing from it. However, expression is immanent and univocal. What is implicit in an explication is not absent, it is fully there, fully expressed, only it is expressed complicatively: it is expressed both implicitly and explicitly. This is the “positive content” of expression that Spinoza brings-out by contrast with signs.

(Expressionism in Philosophy: Spinoza 181-182. Spinoza et le problème de l’expression 164-165)

Deleuze, Gilles. Spinoza et le problème de l'expression. Paris: Les Éditions de Minuit, 1968.

Deleuze, Gilles. Expressionism in Philosophy: Spinoza. Trans. Martin Joughin. New York: Zone Books, 1990.

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