"Am Ministear a' Spèileadh" le Henry Raeburn
§3 - MÙTHADH MOTAIV NA CRUITHEACHD SAN IDÈAL-SHAIDHEINS GU SMAOIN SICEÒLACH. BREITHNEACHADH HUME MU MHATAMATAIG.
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§3 - THE TRANSITION OF THE CREATION-MOTIVE IN THE SCIENCE-IDEAL TO PSYCHOLOGICAL THOUGHT. HUME'S CRITICISM OF MATHEMATICS.
Proceeding from the four invariable philosophical relations as the only possible foundation of certain knowledge, HUME began first of all with his criticism of mathematics. In the latter the adherents of the Humanistic science-ideal (including LOCKE) had till now sought their fulcrum. In HUME, however, the science-ideal has changed its basic denominator for the different modal aspects of reality. This appears nowhere clearer than here.
HUME is even willing to abandon the creative character of mathematical thought in order to be able in his epistemological inquiry to subject all the modal aspects to the absolute sovereignty of psychological thought. However, this interpretation of his criticism of mathematics has been called in question.
Contradictory interpretations of HUME'S criticism of mathematics.
In particular RIEHL and WINDELBAND believe, that HUME, together with all his predecessors since DESCARTES, shared an unwavering faith in mathematics as the prototype and foundation of all scientific thought.
WINDELBAND, however, has overlooked the distinction between natural and philosophical relations, which is extremely fundamental in HUME. Consequently, WINDELBAND completely misrepresents HUME'S conception of the certainty of mathematical knowledge (1). RIEHL, too, did not touch the real content of this conception.
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(1) Geschichte der neueren Phil. I, 340 ff. WINDELBAND thereby entirely overlooks the problem of mathesis in HUME.
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Beyond any doubt HUME displays in his Treatise a sceptical attitude with respect to the claims of mathematics to exact knowledge. RIEHL, however, tries to deprive this attitude of its sharpness by limiting it to "applied geometry", which refers the standards of "pure geometry" to "empirical reality". According to him, HUME never meant to dispute the universal validity of "pure geometry" itself. Moreover, he thinks, that even in this limited sense, HUME's criticism only affected a single point, namely the possibility, presumed by geometry, of dividing space to infinity.
RIEHL believes, that the appearance which HUME gives in his Treatise of having denied the exactness of pure geometry is only due to his unfortunate manner of expression. According to him, the inexactitude which HUME thought he had discovered in "pure geometry" is not concerned with the proofs of the latter, but only with their relation to the objects in "empirical reality" and with the concepts upon which these proofs are based (2).
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(2) RIEHL, Der phil. Kritizismus (3e Aufl.) I, 180.
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To support his view, RIEHL appeals to the distinction that HUME also made between knowledge of facts (matters of fact) and knowledge of the relations between Ideas. For in HUME mathematics indubitably belongs to the latter. Besides, RIEHL can, indeed, appeal to some statements even in the Treatise which seem to support his point of view. And, if his interpretation is adopted, the anomaly between the appreciation of mathematical knowledge in the Treatise and in the Enquiry would be overcome.
For, in HUME's Enquiry which he published after the Treatise, we encounter the statement: "That three times five is equal to the half of thirty, expresses a relation between these numbers. Proportions of this kind are discoverable by the mere operation of thought, without dependence on what is anywhere existent in the universe. Though there never were a circle or triangle in nature, the truths, demonstrated by EUCLID, would forever retain their certainty and evidence" (3).
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(3) Enquiry, Part I, Sect. IV.
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In other words, HUME here appears to have returned completely to the logicist conception of pure mathematics which lay at the foundation of the mathematical ideal of science. And, as we have seen, even LOCKE's psychologizing epistemology had capitulated in favour of the latter. Nevertheless, RIEHL's interpretation is rejected by GREEN and CASSIRER (4). In keeping with our view, they hold that at least in his Treatise, HUME's psychologism had undermined the foundations of mathematical knowledge as such.
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(4) See GREEN'S Introduction to the first part of HUME'S works; CASSIRER II, 345.
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The method of solving this controversy.
In order to take sides correctly in this controversy, we must not base our opinion upon incidental statements in HUME concerning mathematical knowledge. For it is firmly established that especially HUME's Treatise contains very contradictory statements on this point.
The problem can only be solved by answering the preliminary question as to whether or not the fundamentals of HUME's epistemology actually leave room for an exact mathematical science. Only on the basis of the answer given to this question are we able to examine critically the mutually contradictory statements concerning the value of mathematics.
In the first place we must notice that the contrast in HUME between "matters of fact" and "relations of Ideas" can no longer have the same fundamental significance as it possessed in LOCKE. From the very beginning HUME abandoned the Lockian dualism between "sensation" and "reflection", which gradually changed into a fundamental dualism between creative mathematical thought and sensory experience of reality.
In HUME reflection is no longer "original". It is only a mere image of "sensation". True "Ideas" also have become images of "impressions": the true complex "Ideas" are mental images of complex "impressions" (connected by sensory relations). And the true simple "Ideas" are such of simple "impressions".
Now, to be sure, HUME observes that not all our Ideas are derived from impressions. There are many complex Ideas for which no corresponding impressions can be indicated, while vice versa many of our complex impressions are never reflected exactly in "Ideas" (5).
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(5) Treatise I, Part I, Sect. I (p. 312/3).
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Nevertheless, when RIEHL appeals to this statement, to demonstrate the fundamental distinction between "matters of fact" and "relations of Ideas", he distorts it, and ascribes to it a meaning which is quite different from what HUME had intended. For the latter illustrates his thesis with an instance taken from the activity of our fantasy, in which, according to him, the truth and universal validity of "Ideas" are entirely excluded: "I can imagine a city like the "New Jerusalem", he writes, "whose pavements are of gold and whose walls are of rubies, although I have never seen such a city. I have seen Paris; but can I maintain, that I can form such an Idea of this city which completely represents all its streets and houses in their real and exact proportions?"
In fact all judgments, in which the "Ideas" are no longer pure copies of the original impressions, must in the light of HUME's criterion of truth, abandon their claim to certainty and exactness.
HUME drew the full consequences of his "psychologistic" nominalism with respect to mathematics.
Thus even mathematical knowledge can never go beyond the limits of possible sense impressions without losing its claim to universally valid truth.
With respect to mathematics, HUME drew the full consequences of the extreme psychological nominalism to which he adhered, and which he also ascribed to BERKELEY (6).
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(6) As we saw above, BERKELEY has later on abandoned this extreme nominalism.
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He considered it to be one of the greatest and most valuable discoveries of his time that—as BERKELEY had established—all universal ideas are nothing other than particular ones, which by universal names acquirean extended meaning, and thereby evoke other individual ideas in the imagination which exhibit a resemblance with the first.
Even abstract mathematical "Ideas" are always individual in themselves. They can represent a great number of individual Ideas by means of a general name, but they remain mere "images in the mind" of individual objects.
The word triangle, for instance, is in fact always connected with the Idea of a particular degree of quantity and quality (e.g. equal angles, equilateralness). We can never form a universal concept of a triangle that would really be separate from such individual characteristics. Our impressions are always entirely individual: `"tis a principle generally receiv'd in philosophy that everything in nature is individual, and that 'tis utterly absurd to suppose a triangle really existent, which has no precise proportion of sides and angles. If this therefore be absurd in fact and reality, it must also he absurd in Idea; since nothing of which we can form a clear and distinct Idea is absurd and impossible" (7).
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(7) Treatise I, Part I, Sect. -VII (p. 327).
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This was the radical sensationalistic nominalism LEIBNIZ combated from the very beginning. He knew that it must necessarily undermine the foundations of the mathematical ideal of science. We shall subsequently see, however, that HUME did not draw the sceptical consequences of this nominalism in respect to his psychological ideal of science.
The entire view in HUME'S Treatise concerning the Ideas of space and time and their infinite divisibility must be understood in the light of this radical sensationalism.
In HUME the certainty of mathematical knowledge remains stringently connected with the sensory impressions and their mutual sensory relations. If mathematicians seek to find a rational standard of exactness, which transcends our possible sense impressions, they are in the field of pure fictions. These fictions are as useless as they are incomprehensible and, in any case, they cannot satisfy the criterion of truth.
HUME'S psychologistic concept of space. Space as a complex of coloured points (minima sensibilia).
HUME's conception of space and time is entirely in this line. The concept of space can only be the copy of sensory impressions of "coloured points". The basic denominator, which HUME chose to compare the modal aspects of reality, does not allow any meaning to be ascribed to the concept of space other than a visual and tactual one.
If this psychical space is a complex sensory impression, it must exist in the sensory relation between simple impressions. In that case the "coloured points" — as the smallest perceptible impressions of extension or minima sensibilia — function as such simple impressions, and the concept of space is a mere copy of them. And these points must ever possess a sensory extension which itself is no longer divisible.
In this view the concept of the original mathematical point, that never can have any extension, is untenable. Even in the "order of thought" it cannot have any truth or universal validity. For, according to HUME, anything which is absurd "in fact and reality" --- that is to say, anything which cannot be given in sensory impressions — is also absurd "in Idea".
Psychologizing of the mathematical concept of equality.
The concept of mathematical equality is treated in the same way: "The only useful notion of equality or inequality is derived from the whole united appearance and the comparison of particular objects" (read: particular sensory impressions). On the other hand the so-called exact standard of equality between two magnitudes in "pure geometry" is plainly imaginary. "For as the very Idea of equality is that of such a particular appearance corrected by juxta-position or a common measure, the notion of any correction beyond that we have instruments and art to make, is a mere fiction of the mind, and useless as well as incomprehensible" (8).
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(8) Treatise I, Part II, Sect. IV (p. 353/4).
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The same holds for mathematical definitions of straight lines, curves, planes, etc.
HUME admits, that the fictions concealed in such exact definitions are very natural and usual. Mathematicians may with ever more exact measuring instruments try to correct the inexactitude of the sensory perceptions which take place without the aid of such instruments. From this the thought naturally arises that one should finally be able to reach an ideal standard of accuracy beyond the reach of the senses. But this Idea lacks all validity. The measuring instruments remain sensory instruments whose use remains bound to the standard of sensory perceptions. "The first principles" (viz. of mathesis) "are founded on the imagination and senses: The conclusion, therefore, can never go beyond, much less contradict these faculties" (9).
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(9) This entire course of thought is misunderstood by RIEHL when he thinks that HUME recognizes an exact standard for "pure geometry" independent of sense experience.
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In contradiction to RIEHL'S interpretation, it is evident from the following statement, that this thesis is not restricted to the question as to whether or not space is infinitely divisible, but is actually concerned with the entire claims of "pure geometry" to ideal exactness: "Now since these Ideas (i.e. of exact standards) are so loose and uncertain, I wou'd fain ask any mathematician, what infallible assurance he has, not only of the more intricate and obscure propositions of his science, but of the most vulgar and obvious principles? How can he prove to me, for instance, that two right lines cannot have one common segment? Or that 'tis impossible to draw more than one right line betwixt any two points?... The original standard of a right line is in reality nothing but a certain general appearance; and 'tis evident right lines may be made to concur with each other, and yet correspond to this standard, tho' corrected by all the means either practicable or imaginable" (10).
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(10) Ibid., p. 35617: The statement in question to which RIEHL appeals is certainly not clear, when HUME writes further: "At the same time we may learn the reason why geometry fails of evidence in this simple point" (i.e. the pretended infinite divisibility of space) "while all its other reasonings command our fullest assent and approbation." For it is just this very point which strikes in its entirety the claim of mathematics to exactness!
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All that HUME taught here with respect to the concept of space applies even more strongly to the concept of time. For in similar fashion, he gave only a sensationalist sense to the latter. The Idea of time is formed out of the sequence of changing sensory "impressions" as well as "Ideas". As a relation of sensory succession it can never exist apart from such successive sensory Ideas, as NEWTON thought of his "absolute mathematical time". Five notes played on a flute, give us the impression and the concept of time. Time is not a sixth impression which presents itself to our hearing or to one of our other sense organs. Nor is it a sixth impression which the mind discovers in itself by means of "reflection". Therefore, a completely static and unchangeable object can never give us the impression of "duration" or time (11).
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(11) Treatise I, Part II, Sect. III, p. 342-344.
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All false concepts in mathematics, which pretend to give us an ideal exactness beyond the testimony of the sense organs, arise through the natural associations of resemblance, contiguity, and causality. And, according to HUME, the first of these three is "the most fertile source of error."
The position of arithmetic in HUME'S sensationalism.
Now it may appear that HUME still granted the standard of ideal mathematical exactness at least to algebra and arithmetic. He writes in part III, sect. 1 of his Treatise: "There remain, therefore, algebra and arithmetic as the only sciences, in which we can carry on a chain of reasoning to any degree of intricacy, and yet preserve a perfect exactness and certainty. We are possest of a precise standard, by which we can judge of the equality and proportion of numbers; and according as they correspond or not to that standard, we determine their relations, without any possibility of error. When two numbers are so combin'd, as that the one has always an unite answering to every unite of the other, we pronounce them equal; and 'tis for want of such a standard of equality in extension, that geometry can scarce be esteem'd a perfect and infallible science" (12).
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(12) Ibid., p. 374.
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But has the meaning of number in HUME'S system in fact escaped from being rendered psychological? Not in the least. The logicistic conception of arithmetic (held to by DESCARTES and LEIBNIZ) is here only seemingly maintained.
In HUME'S thought, arithmetical unity as an abstract concept can only be the copy of a single impression. Number as unity in the quantitative relations is a fiction. The real unity, which alone has real existence, and which necessarily lies at the foundation of the abstract concept of number, "must be perfectly indivisible and incapable of being resolved into any lesser unity" (13). Number can only be composed of such indivisible unities. Twenty men exist, but only because there exist one, two, three men.
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(13) Treatise I, Part II, Sect. II, p. 338.
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What then is the true unit? In HUME'S system it can only be an impression which is perceived separately and cannot be resolved into other impressions. As LAING has correctly observed, this was the conception of unity which is to be found in SEXTUS EMPIRICUS (14).
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(14) LAING, Op. cit., p. 107.
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Let us now return to the "minima sensibilia", the coloured points of space. A sum of units can in HUME'S system only be grounded on a sensory relation between individual impressions (15).
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(15) In Part 1, Sect. vii (p. 330) HUME introduces a virtually adeaquate concept of number: "when we mention any great number, such as a thousand, the mind has generally no adeaquate Idea of it, but only a power of producing such an Idea by its adeaquate Idea of the decimals, under which the number is comprehended." But even the concept of the decimals in HUME'S system only permit themselves to be maintained as the copy of a sensory multiplicity of simple impressions.
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HUME does not see the inner antinomy in which such a reduction of the original modal meaning of number to that of sensory impression must necessarily involve itself. He does not see that sensory multiplicity pre-supposes the original multiplicity in the modal sense of the numerical aspect, and that in a sensory multiplicity as such no arithmetical meaning can hide. In his system the arithmetical laws which rule the necessary quantitative relations among all possible numbers, must be reduced to psychical laws ruling the relations of the sensory impressions. Thus, even arithmetic must abandon all claim to being an exact science. Not only the irrational, the differential and the complex functions of number, but also the simple fractions have no valid ground. Even simple addition, subtraction, and multiplication of whole numbers lack a genuine mathematical foundation in his system. It appears from the exceptional position which he ascribes to arithmetic in contradistinction to geometry, that HUME did not expressly draw this conclusion. It seems he did not dare to draw it (16). Moreover, his entire exposition with respect to number must be judged extremely summary, vague, and intrinsically contradictory.
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(16) GREEN, op. cit., p. 254, thinks that HUME saw the impossibility of reducing arithmetic to sensory relations.
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Nevertheless, the destructive conclusion here intended, lay hidden inexorably in his psychological starting-point. HUME's retrogression into the Lockian conception of mathematics remains completely inexplicable on the sensationalistic basis of his system. The position which HUME in his later work, Enquiry concerning human understanding, assumes with respect to mathematics, is actually a relapse into the Lockian standpoint; it is a capitulation in face of common opinion concerning the exactness of mathematical thought. Locke, however, could base his view upon his dualism between sensation and reflection. But in HUME'S sensationalistic nominalism, no single tenable point of contact is to be found for the traditional conception with regard to the creative character of mathematical thought.
At the utmost, the claims of mathematics to exactness and to independence of all sensory impressions can be judged valid in a pragmatic sense. For in the final analysis in both his Treatise and Enquiry, HUME did not wish to contest the practical utility of mathematics in natural science.
And, as it will subsequently appear, faith in the exactness of mathematics and in the objective universal validity of the causal judgments of physics can be explained by him from imagination and the psychical laws of association of human nature. By means of the latter he finally intended to arrest the radical Pyrrhonist scepticism. There is, however, in his system no room for the real mathematical science-ideal.
(Herman Dooyeweerd, New Critique of Theoretical Thought, Vol I/ Part 2/ Chapt 3/§3 pp 280-289)
