Freudenthal M's Phil. Program Copy

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Internationales Jahrbuchdes Deutschen IdealismusInternational Yearbookof German Idealism

8 · 2010Philosophie und Wissenschaft Philosophy and Science

Herausgegeben von/edited byFred Rush (Notre Dame) und/ and Jürgen Stolzenberg (Halle/S.)Redaktion/ Associate editors

Paul Franks (Toronto) und/ and Lars-Thade Ulrichs (Halle/S.)Wissenschaftlicher Beirat/Editorial Board Karl Ameriks (Notre Dame), Andreas Arndt (Berlin), Manfred Baum (Wuppertal), Frede-rick C. Beiser (Syracuse), Robert Brandom (Pittsburgh), Daniel Breazeale (Lexington),Rüdiger Bubner† (Heidelberg), Claudio Cesa (Pisa), Konrad Cramer (Göttingen), KlausDüsing (Köln), Michael N. Forster (Chicago), Eckart Förster (Baltimore), Manfred Frank(Tübingen), Hans Friedrich Fulda (Heidelberg), Karen Gloy (Luzern), Henry S. Harris(Toronto), Vittorio Hösle (Notre Dame), Rolf-Peter Horstmann (Berlin), Michael In-wood (Oxford), Wilhelm G. Jacobs (München), Jörg Jantzen (München), Walter Jaeschke

(Bochum), Salvi Turró (Barcelona), Charles Larmore (Chicago), Béatrice Longuenesse(New York), Frederick Neuhouser (New York), Robert B. Pippin (Chicago), ClaudePiché (Montreal), Terry Pinkard (Georgetown), Alain Renaut (Paris), Michael Rosen(Oxford), Birgit Sandkaulen (Jena), Hans-Jörg Sandkühler (Bremen), Dieter Schönecker(Siegen), Ludwig Siep (Münster), Pirmin Stekeler-Weithofer (Leipzig), Dieter Sturma(Essen), Charles Taylor (Montreal), Violetta L. Waibel (Wien), Michael Wolff (Bielefeld),Allen W. Wood (Stanford), Günter Zöller (München)

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Gideon Freudenthal

Maimon’s Philosophical Program.Understanding versus Intuition

Maimon entwickelt seine Philosophie anhand einer Analyse der Mathematik, die,seiner Ansicht nach, das Beste an menschlicher Erkenntnis darstellt. Aber nichteinmal die Mathematik genügt seinem Kriterium der Rationalität, das in der Ein-sehbarkeit durch den Verstand und nicht in der Evidenz der Intuition oder Einbil-dungskraft besteht. Maimon versucht die Mathematik auf analytische Aussagen

zurückzuführen, und, wo sich dies als unmöglich erweist, sie zumindest so zu for-mulieren, dass sie seinem synthetischen Kriterium genügen. Dieser „Satz der Be-stimmbarkeit“ unterscheidet reelle Synthesen von Kategorienfehlern. Beide Pro- gramme konnten ihr Ziel nicht erreichen, aber Maimon gab sein Kriterium der Verstandeseinsicht nicht auf, sondern stellte einen unendlichen Fortschritt auf die-ses Ziel hin in Aussicht.

1. The Argument

Maimon’s philosophy is imbued with mathematics. To him, only mathematics isknowledge proper, and he develops his theses in discussion of mathematicalexamples. However, even mathematics does not satisfy his criteria of ration-ality.1

Maimon’s main concern is intelligibility. His criterion of rationality is in-sight of the understanding as opposed to intuition and imagination. Intelligibi-lity is specified to analytic and synthetic criteria, which form the bases for twocomprehensive philosophical programs, one of which is more demanding thanthe other. The first criterion is logical truth, and the program consists in the

reduction of all synthetic propositions to analytical ones, concepts of substanceto concepts of function. The less demanding criterion was Maimon’s Principle(or Law) of Determinability. The principle formulates the conditions of a “realsynthesis”. In “real synthesis” a new object is produced, from which new con-sequences follow, that follow neither from the original subject nor from thepredicate concepts alone, but only from their synthesis. Thus a triangle has cer-tain “consequences” (e.g., that the sum of its internal angles equals two rightangles), whereas the Pythagorean Theorem is a consequence of the synthesis of “triangle” and “right angle”. This criterion does not dispense with intuition,

1 There have been a number of cursory discussions of Maimon’s philosophy of mathe-matics, but the only serious analysis is Lachterman, 1992.


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nor does it substitute analytic for synthetic judgments, but it accepts the (tem-porary) reality of synthetic judgments (a priori). Both criteria presuppose onesupreme concept from which they proceed either analytically or synthetically.

The central motif of Maimon’s philosophy is hence that proper knowledgemust be based on the understanding. Intuition is not only opaque to reason butmay also deceive us. Maimon learned this lesson from The Guide of the Perplex-ed of Maimonides, his early source of philosophical education. Maimonides dis-cusses asymptotes. The imagination (or intuition) shows that these two linesmust intersect; the understanding proves that this is false. Which source of knowledge do we trust, the imagination (or intuition) or the understanding? Inhis commentary, Maimon emphasizes the prerogative of reason over imagina-tion as that which alone establishes the preeminence of man over beasts, andsupplies three pages of discussion with a simplified version of Apollonius’proof accompanied by a diagram. He was evidently very proud of this proof,mentioning it also in his Lebensgeschichte.2

Maimon’s philosophical program was not successful in either version.Mathematics depends on axioms, postulates, and natural numbers, which arenot the product of the understanding but are imposed on us in intuition. Hetherefore concludes that even mathematics is only subjectively necessary andnot objectively necessary and apodeictic. We thus have before us the followinghierarchy: pure logic is objective and apodeictic; arithmetic and, even more so,geometry are subjectively necessary; mathematical physics is contingent; andpropositions dependent on perception (“the surface is red”) are not yet know-ledge in this form.

The uniqueness of Maimon’s philosophy consists in upholding these criteriaof rationality, on the one hand, and claiming that they have not been met evenby the best of human knowledge, on the other. The gap between actual and idealknowledge is a permanent challenge and – because it diminishes through theprogress of knowledge – it is also a motivation to further efforts. The completefulfillment of the program is the prerogative of the “infinite intellect”. The pre-

sent state of mathematics – and its gradual transcendence towards the ideal of the infinite intellect! – are the share of the finite intellect. Insisting that ourknowledge is not based on firm ultimate foundations and does not conform tothe criteria of proper knowledge, that it rather begins and stops in the “middle”in a mixture of logic and intuition, and also that philosophy – his own includ-ed! – is merely hypothetical, is the core of Maimon’s anti-Kantian philosophy of human finitude. The optimistic counterpart is the claim that we proceedtowards ever more objective knowledge. If indeed an isomorphism obtains be-

84 Gideon Freudenthal

2 Maimonides’ example is the leg of a hyperbola and its asymptote, which is the outlineof the cone itself. See The Guide of the Perplexed I, p. 73. See Apollonius Conica II,pp. 1, 2, 14. See Maimon’s commentary in GM, pp. 142–149, esp. 146–148;Lebens- geschichteI, p. 381; see also GW III, p. 232.


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tween the knowledge of the finite and the infinite intellect, then we may hopethat we progress in the right direction. But because we cannot know this, ourprogress may be an aberration. This is Maimon’s radical skepticism. The hopethat from the “middle” we progress towards “proper knowledge”, and theskeptical fear that this might be an illusion, designate the opposite poles of Maimon’s “Rational Dogmatism” and skepticism.

2. Construction in Geometry

Kant famously stated that mathematics is so successful because it constructs itsconcepts in intuition (CpR, A 714/ B742). 3 Does it really?

In an optimistic presentation of his project, Maimon compares constructionin mathematics to creation, man to God:

God, as an infinite power of representation [ Vorstellungsvermögen], from alleternity, thinks himself as all possible essences [Wesen], that is, he thinks himself aslimited in every possible way. He does not think as we do, discursively; rather, histhoughts are at one and the same time presentations/complete exhibitions [ Dar-stellungen]. If someone objects that we have no concept of such a way of thinking,my answer is: We do in fact have a concept of it, since we possess it in part. Wethink and at the same time exhibit all mathematical concepts as real objects [reelle

Objekte ] by construction a priori. In this, we are similar to God ( Progressen,p. 20).

Our cognition stands in the same relation to the objects of mathematics as theinfinite intellect to all objects of nature. This intellect thinks human beings aswe think triangles. In fact, it resolves all objects to “real definitions”, to therules of their construction ( Bruno, p. 54; Tr, p. 377). To the infinite intellect allmathematical truth is analytic (KU, p. 76; Tr, pp. 61, 181).

Now, if we could replace the object (in intuition) with rules of construction(of the understanding) and forsake intuition, at least know that our operations

in intuition correspond to logical operations, then, at least in this respect, wewould be “similar to God” for whom thinking and creating are one and thesame. The attempts to fulfill this promise taught Maimon that in geometry con-struction depends on “givens” in intuition, not on the concept alone ( Logik,p. 125; Kat, pp. 163–164; Tr , pp. 105–106).

Maimon’s Philosophical Program. Understanding versus Intuition 85

3 See alsoProlegomena § 4; AA IV, p. 272 and AA VIII, pp. 191n–192n.


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2.1. The Construction of the Straight Line

The inquiry as to whether geometry can construct its objects and dispense withintuition reasonably begins with the straight line and the circle. This is so be-cause Euclid introduces these two objects with postulates and then constructsall other objects from these. If their construction requires intuition, then all of geometry depends on intuition. In his Critique of Pure Reason , Kant claims that“[w]e cannot think a line without drawing it in thought, or a circle withoutdescribing it” (CpR, B 154). However, Kant does not explain by what rule wecan “think” or draw the straight line. In order to construct an object we need aconstruction rule and a definition to which the constructed object should con-form. How, then, do we distinguish “drawing a straight line” from drawing acurve if we have neither an adequate definition nor a construction rule for theformer? Kant never answers the question. As if making a pun on Kant’s asser-tion “[w]e cannot think a line without drawing it in thought”, in which under-standing and intuition seem confused, Maimon presents this rationalist alter-native:

For the understanding to think a line, it must draw it in thought, but to present aline in intuition, it must be imagined as already drawn. For the intuition of a line,only consciousness of the apprehension (of the taking together of mutually extern-al parts) is required, whereas in order to comprehend [ begreifen] a line, a real de-finition [Sacherklärung] is required, i.e. the explanation of the way it arises [dieErklärung der Entstehungsart ] (Tr, pp. 35–36).

“Drawing in thought” hence means formulating a rule of construction of theline (an equation), neither synthesizing the successive perceptions of the parts,nor imagining the motion of a point. Like Kant, Maimon has no rule of con-struction for the straight line and cannot construct it.

2.2. The Proof that the Straight Line is shortest between two Points

To substantiate his claim that geometry is based on synthetic judgments a priori,Kant argued:

[T]he straight line between two points is the shortest is a synthetic proposition, formy concept of the straight contains nothing of quantity, but only a quality. Theconcept of the shortest is therefore entirely additional to it, and cannot be extract-ed out of the concept of the straight line by any analysis. Help must here be gottenfrom intuition, by means of which alone the synthesis is possible (CpR, B 16).

Maimon, who wishes to demonstrate that intuition is but “confused” know-ledge, attempts to prove that this proposition is in fact analytic, i.e. that theproperty “shortest between two points” can be inferred from the definition of astraight line.



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The proof proceeds from Maimon’s interpretation of Christian Wolff’s defi-nition of a “straight line”. The definition states that the straight line is the line,“the parts of which are similar to the whole”. Maimon interprets this as statingthat all parts have the same direction. If we abstract from the magnitude, theparts of a line can be distinguished from each other only by their “direction”(Richtung) or their “position” ( Lage). But if this is so, then a straight line(abstracted from its magnitude!) has no parts or is one line only, since it is de-fined by its singular direction. A not-straight line is in fact “several” lines in-dividuated by the change of direction. This reduction of a perceptual quality(“straight”) to quantity (“one”, “several”) contradicts, of course, Kant’s viewthat the “concept of the straight contains nothing of quantity, but only a quali-ty” (CpR, B 16). Now, Kant leveled the criticism that we cannot define “direc-tion” without using “straight” (AA XI, pp. 53–54) and that the definition is,therefore, circular. Maimon answered that no better definition than Wolff’s hasbeen proposed (Tr, p. 68), and he ventured to prove that the predicate “shortestbetween two points” is implied by the subject term “straight line”. If successful,this proves that the allegedly synthetic proposition, “a straight line is the short-est between two points”, is in fact analytic.

Suppose that between points A and C there is one (i.e. straight) line segmentand also the line segments AB––– and BC –––. The broken line ABC –––– forms a trianglewith the sides: AB–––, BC –––, and AC –––. Euclid’s Elements I, prop. 20 proves that twosides of a triangle are longer than the third, and hence AB–––+ BC –––> AC –––. Now,since any multilateral figure can be analyzed into triangles to which Euclid’s I,prop. 20 applies, it follows that “several” line segments, i. e. all other line seg-ments between points A and C are longer than the unique [straight] line AC –––

(Tr, pp. 65–67).However, curved lines are not considered by Euclid. Maimon attempts to

apply Euclid’s proposition also to curved lines. The proof proceeds, as Maimonsays, per substitutionem: the curve between two points is substituted by a bro-ken straight line, which can be resolved into triangles to which Elements I,

prop. 20 apply (Tr, pp. 65–66, 68). The equivalence between a curved and a bro-ken line was widely accepted; in fact, it was part of the definition of the curvedline. Christian Wolff, whose definition of a straight line Maimon adopted (a line“the parts of which are similar to the whole”), defined in a complementary waythe curved line as the line “the parts of which are not similar to the whole line orcan be well distinguished from it”, and which is “compounded of infinitelysmall straight lines” or “a many-sided polygon of infinitely many and infinitelysmall sides” (Wolff, 1978, pp. 749, 460–461).

Maimon’s proof depends on the substitution of the polygon for the curved

line salva veritate . However, in his discussion of the circle’s construction Mai-mon discovered that these are not equivalent.



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2.3. The Circle

Kant considers the definition and the construction of a circle in the context of his discussion of definition in general. He maintains that only in mathematicsdo we encounter appropriate definitions. The example is the circle (CpR, A234/B 287).

Kant refers to the construction of the circle by turning a segment aroundone of its ends.4 This is the “real definition” of the circle and it proves not onlythat the concept is free of contradiction, but also that the object defined is pos-sible, in contradistinction to Leibniz’s decahedron or Kant’s biangle. The latterobjects can be flawlessly defined and yet prove impossible: they cannot be con-structed. Here, in contrast, the “possibility [of a circle] is […] given in the de-finition of the circle, in that it is actually constructed by means of the definitionitself”.5

However, this “definition” is not the Euclidean definition of the circle. InEuclid, the definition reads: “A circle is a plane figure contained by one linesuch that all the straight lines falling upon it from one point among those lyingwithin the figure are equal to one another” ( Elements I, def. 15). The definitiondemands that the circumference of the circle be “one line”, i. e. continuous. It ishence not enough to show that “every point” or all points considered on thecircumference are equidistant from the center, but it is required that “all points”on the continuous line are equidistant, or all radii be equal.

Kant’s definition does not mention this differentia specificaof the circle; itrather seems implied because the same radius constructs all points on the cir-cumference. This construction certainly appeals to the imagination, but it alsoinvolves paradoxes that were known since antiquity and discussed again byGalileo and others (including Abraham Gotthelf Kästner, a mathematical au-thority recognized by Kant), e.g. the rota Aristotelis.6 If we do not presupposethat motion constructs continuous lines (straight or curved), then we have toprove that the line so constructed is continuous. This Kant explicitly denies:

In the concept of a circular line, nothing more is thought than that all straight linesdrawn from it to a single point (the centre) are equal: this is a merely logical func-tion of the universality of the judgement in which the concept of a line constitutesthe subject and refers merely to each [eine jede] of the lines, not to the totality [ das All ] of the lines that can be described on a plane to a given point; if it did not, then


4 Letter to Marcus Herz, May 26, 1789: Correspondence, p. 315; AA XI, p. 53. See alsothe letter to Reinhold of May 19, 1789: Correspondence, p. 306; AA XI, p. 43.

5

Letter to Marcus Herz, May 26, 1789: Correspondence, p. 315; AA XI, p. 53. Cf. Let-ter to Reinhold, May 19, 1789: Correspondence, p. 306; AA XI, p. 43.6 For Maimon’s discussion of the rota Aristotelis, see Tr, pp. 230–238 and my “Defini-

tion and construction” in Freudenthal, 2006. A revised version of this extended essaywill appear in my forthcoming book on Maimon’s philosophy.


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every line would with equal right be an idea of the understanding [this is Maimon’sterm – GF], because they all contain lines (as parts) that can be thought betweenany two arbitrary [ nur denkbaren ] points in them, whose number equally goes to

infinity (Letter to Hertz, May 26, 1789: AA XI, pp. 52–53).Discussing the construction of a circle, Maimon proceeds from Euclid’s and notfrom the alleged “real definition” of a circle by motion. We assign a point andmark around it equidistant points. For simplicity’s sake, let these points be alsoequidistant from each other. All these point do not form the “one line” requir-ed, they must be connected. Every two points can be uniquely connected onlyby a straight line (applying postulate 1 and 2 of the Elements). The apexes of thefigure are equidistant from the center, but the figure is not a circle but a regularpolygon. Maimon therefore says: “A regular polygon is in respect to a circle[…] [a] concept” (PhWb, pp. 162, 170, 172).

The attempt to construct the elementary objects of geometry and to provethat “the straight line is shortest between two points” showed that the dualityof understanding and intuition appears here also in this form; in intuition, spa-tial objects appear continuous but we have no concept of the understanding forthem. The rationalist alternative to Kant’s treatment of geometry failed in threepoints: it failed to replace the property “straight” given in intuition with a con-cept of the understanding. It also failed to prove that the straight line is theshortest between two points; this, too, is obvious in intuition but cannot beproven by the understanding. Finally, it failed to construct a continuous circlefrom its concept. Whereas Kant accepted geometrical practice and, therefore,also intuition as authoritative, Maimon upheld criteria of rationality that hecould not satisfy. Because axioms and postulates are not transparent to reasonbut are imposed on intuition, geometry cannot be accepted as objectively ne-cessary. It is merely subjectively so.

3. Number

Maimon devotes incomparably less space to arithmetic than to geometry. This istrue with regard to both explicit discussions of the foundations of arithmetic incomparison to those of geometry and references to arithmetic (respectively geo-metry) as exemplifying philosophical issues. Geometry is ubiquitous in Maimon’swritings, arithmetic rare. Maimon (and not only Maimon, but also others of hiscontemporaries) often writes “mathematics” when he means, in fact, geometry.

Maimon does not explicitly define the concept of number. We find only aninformal elucidation of the concept and a few casual references. The elucidation

follows Maimon’s philosophical objectives to dispense with intuition. Numbershould not be understood as a “collection of units” (“given” in intuition), butrather as “ratios of magnitudes”. Concepts of ratio and relations in general arepure concepts of the understanding and independent of intuition.



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The brief discussion of number in Transcendentalphilosophie reads thus:I view the understanding as merely a capacity for thought, that is, for producing

pure concepts by means of judging. No real objects are given to it as material for itto work on. Its objects are merely logical and they only become real objects in thefirst place by means of thought. It is an error to believe that things (real objects)must be prior to their relations. The concepts of the numbers are merely relationsand do not presuppose real objects because these relations are the objects them-selves. For example, the number 2 expresses a ratio [Verhältnis ] of 2:1 at the sametime as it expresses the object of this relation […]. These pure concepts of theunderstanding and relations (that always come in pairs) define one another re-ciprocally, i.e., in a circle (Tr, p. 191; cf.Logik, pp. 115–116).

The origin of the idea that this relation, in fact: a correlation (2:1; 1 :2) is analy-

tic is in Aristotle’s Categories 10. Aristotle discusses there “correlation” as onekind of “opposition”. The idea is that, if one of the members and the specificcorrelation are known, then the other member is determined. If x is the doubleof y, then y is the half of x. In his first book, Maimon claimed that propositionsbased on correlatives are identical, in later years that they are not strictly iden-tical, but nevertheless analytical (Tr, p. 37; cf. Kat, p. 66).

It is clear that in the above Maimon does not define the number 2, for thiswould be an obvious petitio principii: “2” in the expression “2:1” would be usedto define “2” as a cardinal number. Moreover, Maimon also refers without fur-

ther ado to intuition in order to introduce the numbers 1 to 10, and also foraddition of any two numbers between 1 and 9. He concedes that “the proposi-tion 3 + 2 = 5 is given to me in intuition” (Tr, p. 323), as Kant claimed. The pro-positions of arithmetic that do not exceed the base of the number system (10 inthe case of the decimal system) are synthetic, cannot be proven, and are to becounted among the principles ( Grundsätze ) “since they are known by no analy-sis but immediately synthetically in intuition” (KU, pp. 176–177; cf. also Baconpp. 182–183;Logik, pp. 273–274).7

What remains of the claim that “number” is a ratio is that it should be

understood as the ratio of specific number to the “absolute unit”, not as the“multitude of units” as in Euclid (Elements VI, def. 2) even if it is so construct-ed. Like his predecessors in the seventeenth century, Maimon insists that the“unit” (in Maimon’s terms: the “absolute unit”) is 1, an abstract number (theancient unit is not a number because it is not a multitude), that can be thoughtand does not refer to whatever may be called one in intuition. 8 It is not “that byvirtue of which each of the things that exist is called one” ( Elements VI, def. 1)but a symbol of the “absolute unit” in thought. Whereas natural numbers can be


7 In German geometry books of the eighteenth century Euclid’s axioms and postulateswere put together under the heading “Grundsätze”.

8 Here and subsequently I follow Klein, 1968. On the distinction between the “one” asthe unit of arithmetic and “one” counted object, see especially pp. 39–40, 49, 54.


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understood as counting units of some kind, the ratio of two numbers or magni-tudes is given by the quotient, which is indifferent to the nature of the magnitu-des compared. The “absolute unit” in thought cannot be divided in infinity as aunit (object) in intuition. Number is not affected by the nature of what it isapplied to (Tr, pp. 350–51, 353–54; KU, p. 29). Because of the failure to con-struct numbers from ratios on the one hand and the interpretation of number asa ratio and not as a “multitude of units” on the other, we find in Maimon dis-cussions in which number seems to conform both to the ancient and to themodern concept of number. Maimon’s conception is very similar to the views of John Wallis (see Klein, 1968, pp. 211–224; esp. 220–222).

Consider the analogous case of “cause” and “effect” which Maimon men-tions but does not elaborate. These, too, are correlative terms, and therefore, asHume and Maimon observed: analytic. 9 The proposition “x is the cause of y”implies logically the proposition “y is the effect of x”. However, the correlativeterms “cause and effect” do not replace physics; specific causal relations cannotbe derived from them. They rather subsume single regularities under a universalcategory. Apart from their application to concrete cases “cause and effect” haveno meaning (Tr, p. 368).

Thus, Maimon does not construct the individual numbers from the notionof number as a ratio any more than he constructs specific physical connectionsfrom the pure concept of cause. Rather, he begins with the natural numbersgiven in intuition – we count fingers and add small numbers by means of group-ing single representations in intuition – and observes that they fall under thecategory of correlatives. The common property of 1, 2, 3 … is that they standfor ratios to 1, not for a quantity. The numbers and the objects numbered arethus strictly severed:

It is not the number but that which is subsumed under the concept of a numberthat can be greater or lesser [ größer oder kleiner ], i.e. be subsumed under a greateror lesser number. Hence the concept of magnitude has no meaning whatever inpure arithmetic, abstracted from its usage. It is not a number, but that which can be

counted which is a magnitude, and is determined by a specific number. Likewise, itis not an arbitrarily assumed unit that is at the basis of a number, but the absoluteunit, which as such cannot be constructed but merely thought. The doctrine of fractions, which is based on an arbitrarily assumed unit, has no place in arithmetic.We thus see that the concepts of unity and number on which arithmetic is basedpresuppose the concept of a continuous magnitude (which is the object of geome-try), without which they have no meaning. (KU, p. 26; cf. Tr, pp. 353–354; Logik,pp. 115–116).

The exclusion of fractions from arithmetic and the reference to geometry soundstrange, but here too, Maimon reflects the transition to the modern concept of number, apparent also in Wallis and Wolff. Wallis hesitated to include fractions


9 Hume, Treatise on Human Nature , Bk. I, Pt. iii, § 3.


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among the numbers and referred to geometry to justify the divisibility of theunit, and he, too, conceived number as a ratio. Whether Maimon was acquaint-ed with Wallis or not, he was certainly acquainted with Christian Wolff’s text-book of mathematics; in fact, Maimon wrote a Hebrew textbook on the basis of Wolff’s Latin textbook. 10 And in Wolff we find similar views to Wallis. On thebasis of Euclid’s definition, “it cannot even be proven that one itself is a num-ber”, observed Christian Wolff (1734, “Zahl”, p. 1345). Wolff, too, defines in away similar to Wallis “number” as “everything that refers to unity as a straightline to another” (1732, def. VIII, p. 18). In Scholion II to this definition Wolff explains what the advantage of this definition is that “[n]umber is to be definedin general so that it be valid for integers as well as fractions, for rational as wellas irrational numbers” (1732, def. VIII sch. II, p. 18). And the advantage of ageneral definition is that we can apply arithmetical operations to lines: “Theutility of this shows when we apply algebra to geometry” (Wolff, 1734, “Zahl”,pp. 1345–1346).

Maimon differs from Wolff in some respects. For him number is the rationot of continuous magnitudes, but of a number to the “absolute unit”, dis-tinguished from a unit given in intuition and that can be infinitely divided(Tr, pp. 350–351, 353–354; KU, p. 29). Moreover, Maimon does not recognizeirrational numbers as numbers. The reason is that an irrational number cannotbe constructed; it cannot be definitely given although it can be approached evermore. It is, therefore, not an object, but an “idea”. And yet, we can operate withthis concept by moving from the object to its “real definition”, to the rule of itsproduction. When we have the rule of a series we can produce the series. Wecannot present an infinite series in intuition, but we can by all means think therule of its production and substitute it for the series itself, and thus also dispensewith the intuition of time. Moreover, convergent series and objects (numbers)may be substituted for each other (Tr, pp. 227–228). If we substitute “real defi-nition” for objects, we can accept also irrational numbers although their con-struction cannot be completed. √ 2 is, therefore, not a possible object, but it is

a possible concept (i. e. it does not contain a contradiction). 11 Thinking of √ 2 asa number, i. e. as an object, means thinking of it both as a convergent andnon-convergent series, as rational and irrational, and this thought “contains acontradiction, since it is both an object and not an object at the same time”(Tr , p. 164). This impossibility applies to the number, not to the magnitude. Themagnitude itself can be constructed as a geometrical magnitude (Tr , pp. 374–375). Moreover, although it is contradictorily determined as an object of arith-


10

The textbook was not printed; the manuscript seems lost. LebensgeschichteII, p. 236.11 In this it is different from √ -a. The latter is a contradictory concept because the pro-duct of two identical numbers cannot be negative; see Tr, pp. 58–60, 361–362. And yet,Maimon does not reject even the use of √ -a, because it enables the consistent applica-tion of mathematical rules without restrictions.


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metic, we can know its ratio to other magnitudes ( LebensgeschichteII, p. 43).Finally, and most important: the value of √ 2 can be ever more approximated.Although √ 2 is therefore not a number, i.e. not an object, it is not nothing, it isan idea (Tr , p. 229).

We have seen that numbers can be interpreted as ratios, but they cannot beconstructed from ratios. Intuition cannot be dispensed with entirely. Purelyrelational thought is apparently the privilege of the infinite intellect. A yearafter the publication of the Transcendentalphilosophie, Maimon conceded thatonly God can construct number from ratio, matter from its form alone:

The infinite intellect, may He be exalted, actualizes by means of the forms of theunderstanding their subjects which are the objects of knowledge ( mv>klvt). Butthis possibility will become clear by the example of the objects of arithmetic,because the numbers are nothing but known ratios, I wish to say the forms of knowledge and their subjects. But the finite intellect must necessarily distinguishin its knowledge the form of apprehension from the object apprehended itself, notan essential but at least a formal distinction, namely that with it [the finite intellect]the form of apprehension is an apprehended ratio; and the apprehended object,although it is in itself also an apprehended ratio, nevertheless it is for it [the finiteintellect] the subject of the ratio [ nv>X hir? ], since it [the finite intellect] does notapprehend this ratio clearly [ deutlich] (GM, pp. 107–108).

There is, hence, an important difference between what numbers are and how

they are known to the finite intellect. Whatever is given as an object is not clear-ly apprehended. If it were, it would be a pure relational concept – not an intui-tion. A finite intellect cannot apprehend without unclear rests, i. e. without in-tuition, i. e. without “given” objects, which resist analysis, and it cannot con-struct numbers from mere ratios. A few years later Maimon finally answeredthe question concerning “form” and “matter” of number first raised in hisTranscendentalphilosophie: the ratios between the objects of mathematics arethe “form of their knowledge”, but their “matter” determines them as specificobjects (Logik, p. 115).12 Differently put, to human beings number (matter)

must be given in intuition, the ratios of numbers (including n:1) are the form bywhich they are thought by the understanding. The infinite intellect constructsthe object from pure relations with nothing given to it. This gap between thefinite and the infinite intellect can be reduced but not closed.


12

In a note to his Transcendentalphilosophie, Maimon asked: “What is a determinatenumber? It is not an a posteriori object (something given), because it is merely a deter-minate way of thinking an object. It is not an a priori form because it is not a conditionof an object. It is not an a posteriori form because this has no meaning at all, as eachform can be nothing other than an a priori condition. What is it then?” (Tr, p. 424).


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4. Differentials

Maimon valued the infinitesimal calculus even more than arithmetic. In fact,Maimon ascribed to the calculus a quasi-religious meaning. The differential cal-culus is a “sparkle of divinity” and a “patent of nobility” testifying to the originof the human spirit in the “pure intelligences” ( Logik, p. 266). The concepts of the infinite in mathematics teach man to transcend his finitude, to assume theperspective of an infinite intellect. The interest behind the engagement with themathematical infinite is the same that also directs man to contemplate “God,universe, immortal soul”, etc. (KU, pp. 163–165). The ideas of the infinitelylarge in mathematics, Maimon says elsewhere, are “sublime” (PhWb, “Erha-ben”, pp. 30–31).

But above all, the infinitesimal calculus was essential to Maimon’s reformedmonadology, or – as he expressed himself – to his “improved Leibnizianism”(Baco and Kant, p. 121). He attributed the close connection between monadolo-gy and the calculus to Leibniz himself: in the Transcendentalphilosophie he sug-gests “that the great Leibniz came upon the discovery of the differential calcu-lus through his system of the Monadology”; in his Progressen he reverses theassertion: Leibniz is here said to have come upon his monadology through hiscalculus.13

4.1. “Improved Leibnizianism”

Before turning to the calculus, we should consider what philosophical problemsthe calculus should help solve. Maimon’s concept of “differentials” appearsaccording to context in two versions. In the epistemological context (in hiscritique of Kant), Maimon speaks of “differentials of sensations”, in the ontolo-gical context (in his critique of Maimonides and the philosophy of Kalam) of “elements of bodies”. In Maimon’s monadology, the world consists of individu-al substances, which are forces of representation differing from each other bythe degree of this force;14 these two versions are thus not alternatives but rathercomplementary perspectives. 15 If this is accepted, then the dualism of mind andmatter disappears and is replaced by the degrees of the forces of representationwithin consciousness. That the same individual substance represents different“external bodies” means that the same force of representation determines itself


13 Tr, pp. 27n–28n; Streifereien, p. 31.14

See GM, pp. 8, 126–127, 136–137.15 Achim Engstler argues that the function of the “differentials” is the solution of the“quid juris?” question and not, as Kuntze suggested, to explain different representa-tions of the same cognitive faculty; see Engstler, 1990, pp. 143 and 47 ff., 167. I suggestthat these are two aspects of the same philosophy.


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in different ways. The dualism between the continuum of the forms of intui-tion – space and time – and the discrete – the individual substances – assumesthe form that the interaction of the individual substances appears in (human)consciousness as an unclear continuous representation, as matter or space.

Kant’s quid juris question is here part of a much more comprehensive prob-lem: in Aristotelian language the question is how “form” relates to “matter”,and the question is posed within consciousness itself. Specifically, it is no longera matter of how concepts are applied to the totally heterogeneous intuitions,but how clear representations (pure concepts of discrete substances) are appliedto confused representations (continuous intuitions, both a priori and a poste-riori):

If we want to consider the matter more carefully, we will find that the question“quid juris?” is one and the same as the important question that has occupied allprevious philosophy, namely the explanation of the community [ Gemeinschaft]between soul and body, or again, as the explanation of the world’s arising (withrespect to its matter) from an intelligence. For, we ourselves as well as the thingsoutside of us (insofar as we are conscious of them) can be nothing other than ourrepresentations themselves, representations that are rightly divided into two prin-cipal classes. 1) The forms, i.e. the representation of the universal modes of ouroperations [ Arten unserer Operationen ], which must be in us a priori. 2) The mat-ter, or the representation of particular objects that is given to us a posteriori andthat, in connection with the first, yields consciousness of particular objects […].How can the understanding subject something (the given object) to its power (toits rules) that is not in its power? In the Kantian system, namely where sensibilityand understanding are two totally different sources of our cognition, this questionis insoluble as I have shown; on the other hand in the Leibnizian-Wolffian system,both flow from one and the same cognitive source [ Erkenntnisquelle ] (the differ-ence lies only in the degree of completeness of this cognition) and so the questionis easily resolved (Tr, pp. 62–63, 191–192; cf. p. 362).16

4.2. Indivisibilia

The concepts of the understanding, the categories, apply to the “differentials”,which are the principles of the sensual world but not sensual themselves. 17 Mai-mon himself seems to have been well-informed about the calculus. He mentionsand quotes Newton and Leibniz, evidently studied Euler, 18 and he also com-


16 Cassirer calls this more comprehensive question an “überraschende and kühne Wen-

dung” (Cassirer, 1920, pp. 128–129). Kant summarizes Maimon’s conception in hisletter to Hertz of May 26, 1789; see Correspondence, pp. 311–316.17 GM, p. 18; Tr, pp. 355–356;Bacon, p. 193.18 See LebensgeschichteII, p. 246: “I taught Euler’s Algebra to a young man”; cf. Wolff,

1813, p. 89.


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pares the style of presentation in Kästner’s and Clairault’s textbooks. 19 And yet,when discussing it in connection with his metaphysics, he uses indiscriminately“infinitely small [magnitudes]”, “differentials”, “fluxions”, and “indivisibilia”(Tr, p. 274; KU, pp. 159–160, 209–210). When discussing the circle’s construc-tion Maimon referred to the antique method of exhaustion, and when discuss-ing the conceptual problems of the infinite he repeatedly refers to Euclid X.1,not to contemporary discussions. 20 Euclid X.1 (with the porism) reads:

Two unequal magnitudes being set out, if from the greater there is subtracted amagnitude greater than its half, and from that which is left a magnitude greaterthan its half, and if this process be repeated continually, there will be left somemagnitude which will be less than the magnitude set out. And the theorem cansimilarly be proved even if the parts subtracted are halves (Elements III, pp. 14, 15).

Maimon distinguishes between a magnitude “smaller than any given magni-tude” – as here in Euclid – and a magnitude “smaller than any magnitude thatcan be thought”. The first involves no contradiction, the second does; the for-mer suffices for the mathematician, the latter is of concern to the philosopher(KU, p. 164). Infinitesimals are “limit-concepts” ( Grenzbegriffe), or “ideas”,i.e. “representations, which cannot be fully presented as an object, the completepresentations of which we can nevertheless approach ever more in infinity”(KU, p. 155).21 We see that the same considerations apply to the infinitesimal asto √ 2. The question whether the limit belongs to the class or not is frequentlydiscussed by Maimon, and his answers vary. Is the infinitesimally small cosineof a right angle still a cosine? “The only reason we are nevertheless able to de-signate these states (that quanta can never reach) is because they are limitconcepts, i.e. a merely symbolic infinitely small […]. The symbolic infinite ismerely an invention of mathematicians that lends generality to their claims”(Tr, p. 353; cf. Tr, pp. 165, 286–288). No object corresponds to the concept(Tr, p. 412). The dilemma is this: if the infinitesimal is an object, a magnitude (inintuition), it must be infinitely divisible. The concept of an infinitely small lineis contradictory (Tr, p. 288). If it is not a magnitude in intuition, then the inte-gral cannot be a finite magnitude and the concept loses its function. At times,Maimon suggests that the differential is a magnitude, at times that it is not adetermined magnitude (the area is not composed of lines) but a ratio of magni-tudes (Tr, pp. 352–355), at times that it designates a changing numerical ratio(Tr, pp. 373–374), at times that it is merely a fiction, a concept useful to under-stand phenomena but with no ontological import. Maimon himself remarkedthat elucidating philosophical notions by means of the calculus may appear to


19 PhWb, p. 154: “Vortrag und Stil”.20 See KU , pp. 164, 209, 213; Tr, p. 353.21 See Tr, pp. 9, 82, 118, 192, 373, 377. Compare the discussion of five kinds of ideas and

the different meanings of “fluxions” in KU, pp. 160–161.


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be clarifying the obscure by what is even more obscure (Tr, pp. 27–28). In orderto maintain that the infinitesimal is a magnitude, Maimon tacitly changes thetraditional definition of magnitude. Instead of “that which can be augmentedand diminished”, Maimon substitutes “either […] or”. Magnitude is that “of which something either larger or smaller can be thought; consequently what isomni dabili majus as well as what is omni dabili minus, i.e. the infinitely largeand the infinitely small, is a magnitude” (Tr, p. 353). In this, the differential issimilar to the “absolute unit” in arithmetic which is “treated as an object of purearithmetic itself because it can be augmented even if it cannot be diminished”,and both should not be thought of as actual (in intuition) but as symbols(Tr, pp. 353–534). Conceiving an infinitesimal magnitude as actual (in intuition)originates neither in reason nor in sensibility but in imagination ( Einbildungs-kraft ). The understanding thinks a rule, imagination presents it as an object(Logik, pp. 204–206), the understanding thinks discrete entities, imaginationpresents a continuum ( Logik, pp. 218–219). This may be at times helpful inmathematical practice (Tr, pp. 274–275; Bacon, p. 282), but it is not true. Philo-sophical analysis exposes this mistake. Finally (probably following one of Leib-niz’s suggestions), Maimon also considers infinitesimals as “fictions”:22

The method of indivisibles, the infinite series, the differential calculus and such likenecessarily lead to contradiction if they are considered to be more than meremethods. Imagination palters with us and presents its fictions as real objects. Butreason does not mind this and declares them to be what they really are: mere fic-tions (Logik, pp. 205–206).

“Fictions” are not opposed to “reality” but to “actuality”, to existence in intui-tion.

4.3. Fictions in Mathematics and Philosophy

We have seen that Maimon suggested conceiving of some rules or ratios as con-cepts of objects, with the proviso that they be understood as “fictions”. Thesame should also apply to philosophy. What does “fiction” mean? “Fiction[Erdichtung ] in its most general meaning is an operation of the imagination[Einbildungskraft ] by which a not objectively necessary unity of the manifoldof an object is produced” (PhWb, p. 36).

In our context “fiction” refers, more specifically, to conceiving something asan object which cannot be given as an object: “[a]n object which in some respectis variable according to rule may be considered as if it arrives at the highestgrade of its variation, i.e. as if it were the same and not the same object at the


22 Bendavid suggested that infinitesimals are “qualities,” not quantities. Maimon firstaccepted this suggestion (Tr, pp. 291–292); later he rejected it (Tr, pp. 355–356).


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same time” (Progressen, p. 17). √ 2, so Maimon maintained, is thus conceivedboth a convergent and non-convergent series, a rational and irrational number,and “both an object and not an object at the same time” (Tr , p. 164). And yet,the method of fictions is productive (and deceptive) not only in mathematics,but also in physics: the parallelogram of forces, the backbone of mechanics of point masses, is a good example (Pemberton, pp. 198–201).23

These considerations should also apply in philosophy. Leibniz’s mona-dology appears inconsistent since it seems to claim that an infinitely divisiblebody is compounded of indivisible monads. But what if his philosophy is to beunderstood as a methodus invidisibilium, i.e. as a method and not a dogmaticontology? On this view, monads are understood as concepts defined by thetheory of which they form part and applied to determine relations of bodies andyet as “fictions” in an ontological perspective ( Progressen, pp. 29–30)? Moro-ever, what if Leibniz didn’t conceive of monads as “things-in-themselves” butas fictions and merely retained the traditional ontological modus loquendi (Pro- gressen, p. 56), if they “are not real objects but merely what differential magni-tudes are in mathematics, limits of ratios” ( Streifereien, p. 271)?24 We learn twolessons. The first is that à la rigueur metaphysique ontological language can beunderstood as “fictional”. The second lesson is more important. Philosophy canlay claim to truth in the same way that the sciences do; its concepts and theoriesare developed to account for phenomena, and they are ascribed validity accord-ing to their success. They are not supposed to offer absolute knowledge of thefirst and last elements of reality. This is the prerogative of the infinite intellect. 25

4.4. Taking Stock

The central motif of Maimon’s philosophy is his insistence that proper know-ledge must be based on the understanding. Intuition is not only opaque to rea-son but may also deceive us. If knowledge dependent on this medium is valid,


23 Compound motion is constructed by conceiving the action of simultaneous forces assuccessive accelerations and assuming the time each such force acts as “infinitelysmall”; “But then the conception of a compound motion is a mere idea or fictionwhich our presentation may approach ever closer but never fully reach” ( Pemberton ,pp. 200–201).

24 On Maimon’s method of fictions and its critique by Reinhold, see Breazeale, 2003.25 Engstler maintains that this interpretation undermines Maimon’s claim to have solved

the quid juris question: “[w]enn Maimon die Differentiale nämlich tatsächlich alsBegriffe fiktiver Elemente auffaßte, wäre seine Theorie nicht in der Lage, die Möglich-

keit objektiv gültiger empirischer Urteile zu erklären, und sie böte mithin auch keineLösung des Problems quid juris” (Engstler, 1990, p. 140). This is so if philosophyclaims absolute truth, or truth more valid than ascribed to the differential equations of classical mechanics. I argue that Maimon’s unique stance is that he ascribes them thesame validity, neither more nor less.


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then it is the task of philosophy to analyze it and show that it can be reformulat-ed in terms of the understanding. Maimon’s attempts to reduce intuition to theunderstanding and objects to ratios and construction rules were not successful.The conclusion he draws from his failure to prove that the straight line is shor-test is rather surprising:

By contrast [to Kant – GF], I pose the question in the following way. Since all apriori cognition must be analytic, and can be derived from the principle of contra-diction, how can we make those propositions that are synthetic due to a lack in ourcognition into analytic ones? […] I do not want to take on the task of developingall such propositions in this way in order to make them satisfy my requirement; itis enough that I hold it not to be impossible (Tr, pp. 178–79, my emphasis; cf. Tr,p. 323).26

The failure to reform mathematics is no reason to change our criteria of ratio-nality.

5. Philosophizing in the Middle

Absolute knowledge of the infinite intellect assumes two forms, producingobjects from relations and producing concepts of objects (respectively objects)by synthesis of more general concepts and specific difference. The former pro-gram failed. The latter, synthetic program is repeatedly alluded to in the con-struction of geometrical objects. Maimon maintains that if there are synthetic judgments a priori, there must also be a principle of reason such judgments.This is his Principle of Determinability. 27 The principle examines the construc-tion of concepts “top-down”, i.e. by further determination of given generalobjects. The procedure conforms to Aristotle’s notion of definition by genus proximum and differentia specifica. In geometry, this means that objects shouldnot be constructed from more elementary objects but, to the contrary, frommore complex ones, not by composition, but by specification, in fact by “deter-mination” according to the Law of Determinability. The “necessity and univer-sality required for science” could be attained, so Maimon believed, “if we couldsubordinate all objects of human knowledge to one and the same concept”.(Progressen, pp. 42–43). With this suggestion, Maimon presents an entirelydifferent and new notion not only of construction, but also of a conceptual“system”, philosophy included.

Take, for example, the construction by conic sections. From a general al-gebraic definition we can infer the properties not only of the circle but also of


26 The universality of the mathematical truths “certainly must have an objective ground,i. e., to an infinite understanding the proposition must be analytic; but we cannot haveany insight into this ground” (Tr, p. 181).

27 On Maimon’s principle of determinability, see Schechter, 2003.


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all other objects that fall under this general definition, as well as the relationsamong these (Progressen, p. 42). Moreover, we also obtain rules of constructionfor these objects from this definition:

Thus in ordinary geometry the circle e. g. is defined as a line all parts of which areequally distant from a certain point (the center). The consequences to be drawnfrom this concept are only valid for the circle, not also for other curved lines. Inhigher geometry, the circle is determined as a curved line of the second order by ageneral equation. The consequences to be drawn from this equation are thereforevalid not only for the circle, but for all lines of this order, etc. ( Ueber die Schwär-merei, p. 44).28

Consider a simple case: a circular cone cut by a plane. The boundary curve of the intersection is a conic section. According to the angle of intersection, thisconic section is an ellipse, a circle, a parabola or a hyperbola, yes even a point; astraight line and intersecting straight lines can be thus produced. If the plane

intersects the apex of the cone parallel to its napes it produces a straight line(or intersecting lines), if it intersects the vertex of the cone parallel to its base, itproduces a point.

We see here “genetic definitions” or construction rules by genus proximumand differentia specifica. In this way the relations between these different objects are transparent (especially in the algebraic representation). But in the case


28 Kant, too, once considered the construction of a circle as a conic section. Moreover, he

also noted that then a property that Euclid proved for the circle ( Elements III, p. 35)can be proved for all conic sections. However, he does not draw from this exampleconsequences concerning mathematics but rather that physics has to conform to thegeometrical properties of space as proven in geometry. See Prolegomena, § 38; AA IV,pp. 320–321.

parabola

straight line

ellipse

point hyperbola

circle

Fig. 1


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of the finite mind, the petitio principiiis obvious: In order to construct the circleand the straight line as conic sections, we need a cone (and a plane). In order toconstruct a cone, we need a circle and a straight line. 29 The attempt to constructgeometry ab ovo fails either way.

5.1. Hypothetical Metaphysics

Conceiving the entire fabric of human knowledge as constructed “top-down”requires that we begin with the most general concept of a determined object,know all its possible determinations, use them as specific differences, and thusproduce the more specific concepts, and so on to the most specific concepts. If this were possible, the principle of determinability would serve as the “rule of construction” of all further objects. A project of this kind fails for two reasons.First, the most general concept, the “I” or “consciousness in general” ( Bewußt-sein überhaupt ),30 analogous to “space” in geometry, remains for the finiteintellect an unattainable idea. 31 Second, the principle does not produce all pos-sible determinations, but is only a criterion by which we judge whether a givendetermination is proper or produces a category mistake. The principle showsthat “right-angle triangle” is a possible synthesis (but, “sweet triangle” is not),but it cannot suggest “right angle” as a specific predication of an “angle”. Thereis nothing in the concept “angle” that suggests that “right angle” is a propertydistinct from all other inclinations of two straight lines.

How, then, can philosophy proceed? We should note, says Maimon:

that both the primitive consciousness of a constituent part of a synthesis […] aswell as the consciousness of the complete synthesis are mere ideas, i. e. they are thetwo limit-concepts of a synthesis, in that without synthesis no consciousness ispossible, but the consciousness of the completed synthesis grasps the infinite in it-self, and is consequently impossible for a limited cognitive faculty […]. So webegin in the middle with our cognition of things and stop in the middle again. It is

the same as, for example, in calculating with our number system, where weproceed according to the very same rules both forwards and backwards in relationto an extended magnitude (through decimal fractions) […]. The absolutely first inthe consciousness of a thing is a mere idea that we reach by infinitely decreasing it,i.e. that we never reach in intuition (Tr, pp. 350–351).


29 In fact, this is the definition of a cone in Elements XI, def. 18: “When a right trianglewith one side of those about the right angle remains fixed is carried round and restor-ed again to the same position from which it began to be moved, the figure so compre-

hended is a cone”. Appollonius’ definition of the cone involves the rotation of astraight line around the circumference of a given circle.30 See the references in Bergmann, 1967, pp. 164–166.31 Tr, p. 193; Logik, 243–245. “Object as such” (Gegenstand überhaupt ) is the subject

matter of logic.


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The analogy Maimon draws here between philosophy and arithmetic is not atall accidental. He conceives philosophy as a science, a science a priori of the“form of knowledge”, distinguished from mathematics by its greater generality:

Philosophy is the idea of a science of the possibility of a system [das Ganze ] of knowledge, i.e. its object is merely the form of a science or of a whole of knowled-ge. Here the difference between philosophy and mathematics (which are both puresciences a priori) becomes obvious. The object of (pure) philosophy which abstrac-ts from determined subjects is only the form or the way in which a system of kno-wledge in general is possible. Also [the object of] mathematics, as a science, whichrefers to determined objects (albeit a priori), is the possibility of a system, but nota system of knowledge in general [überhaupt ], but a system of knowledge of thedetermined objects of mathematics (Kat, pp. 120–121; cf. p. 118).

Two important consequences follow from this beginning and stopping “in themiddle”: the hypothetical and uncertain nature of metaphysics, and the infinitequest for truth. As is often the case, Maimon attributes his own views to thephilosophers he discusses. In Maimon’s interpretation, Kant assumes the apo-deictic truth of Newtonian science and mathematics as an answer to the ques-tion quid facti, and then develops his transcendental philosophy as a “hypothe-sis” to account for the possibility of such knowledge:

Mr. Kant takes experience as an indubitable fact as the basis of his critical systemand demonstrates hypothetically from there the reality of the principal conceptsand propositions a priori ( Obereits Widerruf , p. 108: GW III, p. 420).32

This sounds like an ironical commentary on Kant’s announcement in the pre-face to the second edition of Critique of Pure Reason that he will first presenthis “Copernican turn” as a hypothesis “in order to draw attention to the cha-racter of these first attempts at such a change”, but then prove it “apodeicticallynot hypothetically” (CpR, B xxiii). Maimon himself accepts only logical infer-ence as “apodeictically not hypothetically”; all the rest is as hypothetical asscience is (in his interpretation). However, the ever better approximation of intuition by mathematics and physics suggests that we may proceed in the rightdirection.

If philosophy never reaches a secure beginning or end but begins and stopsin the hypothetical middle, however, then it follows that the progress of inquirycontinues forever and also retains its hypothetical character. This is the deepmeaning Maimon ascribes to a Talmudic quote which he repeatedly used andwith which he also closed his Transcendentalphilosophie:

Our Talmudists (who, from time to time, have certainly expressed thoughts wor-thy of a Plato) say, ‘the students of wisdom find no rest, neither in this life nor yet


32 See also Antwort auf Obereit , pp. 102–103: GW III, pp. 458–459; Streifereien,pp. 203–204, 207.


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in the life to come’, to which they relate, in their own way, the words of the Psal-mists: ‘They go from strength to strength, to appear before the Almighty in Zion’(Tr, p. 444).

Or not. 33

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Bergmann, Shmuel Hugo (1967): The Philosophy of Solomon Maimon, trans. Noah J. Jacobs, Jerusalem.

Breazeale, Daniel (2003): “Reinhold gegen Maimon über den Gebrauch der Fiktionenin der Philosophie”, in: Bondeli, Martin (ed.): Die Philosophie Karl Leonhard Reinholds, Fichte-Studien-Supplementa, Bd. 16, New York, pp. 123–151.

Bruno, Giordano (1793): Auszug aus Von der Ursache, dem Prinzip und dem Einen in:Gnothi Seauton oder Magazin zur Erfahrungsseelenkunde als ein Lesebuch für Gelehrte und Ungelehrte , Bd. X/2, pp. 49–84 (in: GW IV, pp. 617–652).

Cassirer, Ernst (1920): Das Erkenntnisproblem in der Philosophie und Wissenschaft der neueren Zeit, Bd. 3: Die nachkantischen Systeme, Berlin.

Engstler, Achim (1990): Untersuchungen zum Idealismus Salomon Maimons, Stutt-gart-Bad Cannstatt.

Euclid (1926):The Thirteen Books of Euclid’s Elements, trans. Thomas Heath, 2nd ed.,New York. (Cited as “Elements”.)Freudenthal, Gideon (2006): Definition and Construction. Salomon Maimon’s

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Hume, David (2000): Treatise on Human Nature , ed. Norton, David Fate, Oxford.Kant, Immanuel (1903 ff.): Kant’s gesammelte Schriften, ed. Preußische Akademie der

Wissenschaften etc., Berlin. (Cited as “AA”.)Kant, Immanuel (1999): Correspondence, ed. Zweig, Arnulf, Cambridge (cited as Cor-

respondence).Klein, Jacob (1968): Greek Mathematical thought and the Origin of Algebra , Cam-

bridge (Mass.) and London, (originally in German, 1934, 1936).Kuntze, Friedrich (1912): Die Philosophie Salomon Maimons, Heidelberg.Lachterman, David R. (1992): “Mathematical Construction, Symbolic Cognition and

the Infinite Intellect: Reflections on Maimon and Maimonides”, in: Journal of theHistory of Philosophy, vol. 30, no. 4, pp. 497–522.

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33 This essay has been written during my stay at the Max-Planck-Institute for theHistory of Science, Berlin. I am grateful to the director of Department I, Jürgen Renn,for the invitation to work in this inspiring setting.


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Maimon, Salomon (1792): “Antwort auf das Schreiben des Herrn Obereit an HerrnS. Maimon”, in: Gnothi Seauton oder Magazin zur Erfahrungsseelenkunde als einLesebuch für Gelehrte und Ungelehrte , Bd. IX/3, pp. 100–105 (in: GW III,pp. 456–461). (Cited as “Antwort auf Obereit”.)

Maimon, Salomon (1792): “Obereits Widerruf für Kant. Ein psychologischer Kreis-lauf”, in: Gnothi Seauton oder Magazin zur Erfahrungsseelenkunde als ein Lese-buch für Gelehrte und Ungelehrte , Bd. IX/2, pp. 106–143 [author: J.H. Obereit;notes: S. Maimon] (in: GW III, pp. 418–455). (Cited as “Obereits Widerruf”.)

Maimon, Salomon (1792/93): Salomon Maimon’s Lebensgeschichte. Von ihm selbst geschrieben und herausgegeben von K.P. Moritz, in zwei Theilen, Berlin (in: GW I,pp. 293–588). (Cited as Lebensgeschichte.)

Maimon, Salomon (1793):Bacons von Verulam Neues Organon. Aus dem Lateinischenübersetzt von George Wilhelm Bartoldy. Mit Anmerkungen von Salomon Maimon.Mit Kupfern, Berlin (in: GW IV, pp. 295–530). (Cited as Bacon.)

Maimon, Salomon (1793): Anfangsgründe der Newtonischen Philosophie vonDr. Pemberton. Aus dem Englischen mit Anmerkungen und einer Vorrede vonSalomon Maimon. Erster Theil mit vier Kupfertafeln, Berlin (in: GW IV, pp.531–580, cited asPemberton ).

Maimon, Salomon (1793):Ueber die Progressen der Philosophie in Salomon Maimon’sStreifereien im Gebiete der Philosophie. Erster Theil , Berlin (in: GW IV,pp. 23–80). (Cited as “Progressen”.)

Maimon, Salomon (1793):Salomon Maimon’s Streifereien im Gebiete der Philosophie.Erster Theil , Berlin (in: GW IV, pp. 1–294). (Cited as “Streifereien”.)

Maimon, Salomon (1793): “Ueber die Schwärmerei”, in: Gnothi Seauton oder Maga-zin zur Erfahrungsseelenkunde als ein Lesebuch für Gelehrte und Unge-lehrte, Bd. X/2, pp. 43–48 (in: GW IV, pp. 611–616). (Cited as “Ueber die Schwär-merei”.)

Maimon, Salomon (1794): Die Kathegorien des Aristoteles. Mit Anmerkungen erläu-tert und als Propädeutik zu einer neuen Theorie des Denkens dargestellt , Berlin (in:GW VI, pp. 1–271). (Cited as “Kat”.)

Maimon, Salomon (1794):Versuch einer neuen Logik oder Theorie des Denkens. Nebstangehängten Briefen des Philaletes an Aenesidemus, Berlin (in: GW V). (Cited as“Logik”).

Maimon, Salomon (1797): Kritische Untersuchungen über den menschlichen Geist oder das höhere Erkenntnis- und Willensvermögen , Leipzig (in: GW VII, pp. I–373).(Cited as “KU”.)

Maimon, Salomon (1965, 2000):Giv’at Hammore (commentary on Maimonides’ TheGuide of the Perplexed , Part I), ed. Bergmann, S.H./Rotenstreich, N., Jerusalem.(Cited as “GM”.)



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Maimon, Salomon (2010):Essay on Transcendental Philosophy, trans. Nick Midgley,Henry Somers-Hall, Alistair Welchman and Merten Reglitz, London / New York.

Maimonides, Moses (1969):The Guide of the Perplexed , trans. Shlomo Pines, Chicago.Schechter, Oded (2003): “The Logic of Speculative Philosophy and Skepticism in Mai-

mon’s Philosophy: Satz der Bestimmbarkeit and the Role of Synthesis”, in: Freu-denthal, Gideon (ed.): Salomon Maimon: Rational Dogmatist, Empirical Skeptic;Critical Assessments,Dordrecht, pp. 18–53.

Wolff, Christian (1734): Vollständiges mathematisches Lexicon, Leipzig.Wolff, Christian (1732): Elementa matheseos universae, in Qui Commentationem de

Methodo Mathematica, Arithmeticam, Geometriam, Trigonometriam Planam, & Analysim, tam Finitorum quam Infinitorum complectitur , Tom. I, Geneva.

Wolff, Christian (1978): Mathematisches Lexicon, ed. Hofmann, J.E., Hildesheim /New York.

Wolff, Sabattia Joseph (1813): Maimoniana oder Rhapsodie zur Charakteristik Salo-mon Maimons, Berlin.


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