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Truth of Geometry

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Truth of Geometry

Part 1 – Aristotle

“Without an image thinking is impossible. For there is in such activity an affection identical with one in geometrical demonstrations. For in the latter case, though we do not make any use of the fact that the quantity in the triangle is determinate, we nevertheless draw it determinate in quantity” (On Memory 450a1–4).

According to Aristotle, the subject matter of geometry consists of things contained in or bounding natural bodies, things such as surfaces, volumes, lines, and points. A light-ray or a line scored in a stone are studied by the geometer only as lines. It is likewise for surfaces, volumes, and points. In thought the geometer separates these things from bodies. She does not treat them as the limits of a natural body, and no falsity results from this. Neither does she consider these attributes as the attributes of bodies. “Geometry investigates natural lines but not qua natural” (Physics 194a10).

Geometric objects are not prior to sensibles in being, though sensible bodies presuppose geometric objects. In claiming geometric objects are in or bound bodies and that they can be separated in thought from bodies, Aristotle is not saying they exist in bodies or apart from bodies in the way we say body parts exist (Metaphysics 1077a15–b17).

“If we suppose things separated from their attributes and make any inquiry concerning them as such, we shall not for this reason be in error, any more than when one draws a line on the ground and calls it a foot long when it is not; for the error is not included in the propositions” (Metaph. 1078a16–21). Suppose we are trying to see if some furniture at the store can be fitted into a nice arrangement for your living room. We take some graph paper and say “let the squares this size be each a square foot.” We draw the outline of your living room floor to scale on the paper. To the same scale, we cut another sheet of graph paper into the horizontal cross-sectional areas of the furniture pieces. We then test their arrangement on the paper with the floor outline, knowing our results apply to the room, even though what we took as a square foot on paper was not actually a square foot.

The truths of geometry are of the real much as Aristotle’s foot-representing line is of a line one foot long. In Euclid’s geometry, we reason about perfectly exact planes, points, lines, and figures. As the objects of Euclidean geometry, planes are perfectly flat, straight lines are perfectly straight, circles are perfect circles, and so forth. We draw on paper icons of those perfect elements. The icons can deviate somewhat from the perfect items we conceive and about which we reason using those icons. This does not mean that the perfect objects of geometry, thence their relations, are not instantiated in the material world. In Aristotle’s view, they are (Lear 1988, 240–43; Lennox 1986, 33–38). Descartes and Newton concurred with Aristotle in that view. That something can only be accessed by abstraction does not entail that the something does not exist concretely, beyond thought. It is no great mystery in Aristotle’s view that geometry can supply the reasons the rainbow has some of the characteristics it has (Meteorology 271b26–29, 375b17–76b21, 376b28–77a11; R; Lennox 1986, 44–49).

In the diagrams and reasoning of Greek geometry, unlike the thinking with the furniture floor-plan, we are not making an (indirect) empirical test. The natures of geometric objects as geometric are uncovered by proofs from assumed starting propositions, including definitions, and from permitted elementary acts for constructing diagrams.

It is of the nature of a triangle in Euclidean geometry that its interior angles sum to two right angles. One does not need to accept that on authority; one can follow Euclid’s proof and thereby understand that this proposition on the nature of the triangle is true and why (Elements I.32; 34–35*). Assuming only what a triangle most simply is, one shows that it necessarily has angles summing to two right angles (Phys. 200a16–17).

Euclid lived after Aristotle, but geometric proof by the time of Aristotle was as we have it in Euclid (Netz 1999, 275). Aristotle discovered logic, in particular the theory of the syllogism. Geometers were proving geometric propositions before then. Aristotle thought the kind of syllogisms he counted as demonstrations, or proofs, were what geometric proofs came down to. That was incorrect (Prior Analytics 40b18–41a20, 48a29–39; see Lear 1980, 10–14, 39, 48, 51–53, 65; also Friedman 1992, 57–66; 2000, 187, 202). But geometric proofs are like his syllogistic demonstrations in carrying necessary truth of starting points by strict rules of development into necessarily true conclusions explained by the premises. The geometers and Aristotle have also been in accord in thinking that a geometric proof is not only necessary for establishing that the sum of interior angles of a triangle is two right angles (180°), but that the proof is sufficient to establish that truth.

The construction of lines and figures is usually part—an essential justificative part—of a proof in Euclid and in later Greek geometers, such as Apollonius and Archimedes (Norman 2006, 20, 79–86; Netz 1999, 26–43, 187–88, 264–65). In the technology of geometric diagrams, there are two elements: (i) straight-edge and variable compass, both without scale marks, and (ii) letters naming points of line intersection in the diagram.

Each element redefines the infinite, continuous mass of geometrical figures into a manmade, finite, discrete perception. Of course, this does not mean that the object of Greek mathematics is finite and discrete. The perceived diagram does not exhaust the geometrical object. This object is partly defined by the text, e.g. metric properties are textually defined. But the properties of the perceived diagram form a true subset of the real properties of the mathematical object. This is why diagrams are good to think with. (Netz 1999, 35)

The diagram sets up a world of reference, which delimits the text. . . . This is a result of the role of the diagram for the process of fixation of reference. (31)

References to [the diagram] are references to a construction, which, by definition, is under our control. . . . The diagram which one constructed oneself, however, is also known to oneself, because it is verbalized. Note the combination: the visual presence allows a synoptic view, an easy access to the contents; the verbalization limits the contents. . . . The unit composed of the two is the subject of Greek mathematics. (181)

David Hilbert famously recast Euclidean geometry into a logical order that took propositions implicit in Euclid’s diagrams and explicitly stated them. In calling Hilbert’s topic geometry in this accomplishment Euclidean geometry, I mean for example that Hilbert was in this work doing plane geometry on the same plane that Euclid was studying, as distinct from, say, the hyperbolic plane of another geometry that had been developed in the nineteenth century. Hilbert’s axiomatization of Euclidean geometry fortifies the truth of Euclidean geometry beyond Euclid, that is, the truth of that geometry concerning its objects, which are objects of our understanding and which have instances in the world (see also Norman 2006, 19, 73–86; further, Mumma 2011*) I should mention that the non-Euclidean geometry that is hyperbolic geometry has instances in the world, such as the plane geometry of a half-sphere, which is of course a surface and geometry embedded in Euclidean space. An important potential instance of hyperbolic geometry is the large-scale structure of four-dimensional spacetime, in which local three-dimensional space is Euclidean; whether this potential is actual will be decided by physics, which is to say by empirical test (Martin 1975, chap. 26; Friedman 2000, 206–8; Bolte and Steiner 2012).

Aristotle had it that “the objects of mathematics exist, and with the character ascribed to them by mathematicians” (Metaph. 1077b33). Instances of geometry are in the sensible, but in geometry one does not treat its occurrence in the sensible as sensible, and geometry does not on account of its sensible occasion become a science of the sensible (1078a2–4). “There are attributes of things which belong to things merely as lengths or as planes” (1078a7–8).

Furthermore, “in proportion as we are dealing with things which are prior in formula [even though not prior in being] and simpler, our knowledge will have more accuracy, i.e. simplicity” (1078a9–10). In his 1992 lecture series The Art of Thinking, Leonard Peikoff remarked that the proviso “in the present context of knowledge” is not applicable to truth of mathematical axioms, because of the very delimited subject matter in mathematics (cf. Peikoff [Prof. E] in ITOE Appendix 203).

The following oral exchange took place between Allan Gotthelf and Ayn Rand in her epistemology seminar (c. 1970).

Gotthelf – You have said two things about the mathematical field. One was that once the base has been established, we can proceed without direct reference to perceptual reality. The second point was that the mathematical field was more precise than the conceptual. . . .

Rand – [Mathematics] is a science that defines the entities it deals with very simply. For instance [in arithmetic], all you have as the basis of your operation is the arithmetical series. You don’t need any further definitions as a base. From there on you work with that base Whereas in other conceptual knowledge you deal with such a complexity of phenomena that your definitions can change as your knowledge expands . . . . That’s one of the differences.

Gotthelf – In other words, you have all of the material before you from the beginning in mathematics. There’s no new information which you are going to integrate into your concepts. Rather you are going to build up abstractions from abstractions, such as “function,” “limit,” and so on.

Rand – That’s right. This is not to imply that non-mathematical concepts necessarily have to be in some way less exact than mathematical concepts. No. The ideal to aim at is to bring your concepts into exactly that kind of precision. At least those concepts you know—you cannot have omniscience, and you cannot guarantee that you will not expand your knowledge (as I explain in Chapter 5) and change a concept’s defining characteristic. (ITOE Appendix 201)

Gotthelf and Rand are here in accord with Aristotle’s thought that our knowledge in geometry has such great accuracy due to its simplicity and cognitive self-sufficiency when we have abstracted its objects from their embodiments. In the next two installments, I shall consider what Locke and Leibniz, then Kant say about sources of the exactitude of geometric truth, as well as its certainty, inferential validity, generality, and immutability. Here I want to pause over what Gotthelf and Rand remarked informally concerning immutability of mathematical concepts.

Where Rand answered “That’s right” to Gotthelf, I answer “That’s roughly right.” The basic elements for arithmetic and for geometry are set in the beginning of those disciplines and are occasioned every day all around us. New observations sometimes stimulate introduction of new or revised concepts in mathematics. But for the most part, changes in mathematical concepts come by way of creative resolutions of tensions within mathematics itself. (We should notice too that what were the elementary concepts in Euclidean geometry as Euclid wrote it are not the complete set of elements we identify for that geometry today; when doing it rigorously, at an advanced level, we now know there are further, less obvious elements on which that geometry is logically based.)

Concepts can rationally change in the referents they subsume when it becomes evident that the concept should be (i) narrowed, recognizing that not all referents previously subsumed are of the same type at the level of the concept in the (possibly shifting) conceptual hierarchy or (ii) broadened, recognizing that referents previously known but excluded or referents previously not conceived should be included under the concept.

Furthermore, in the development of science and mathematics, new concepts are introduced. Sometimes that is because although an old concept was picking out a kind, it was conceived within a theory later seen to be false. An example from chemistry would be replacement of the concept dephlogisticated air with the concept oxygen. An example from mathematics would be replacement of the concept number-whose-square-is-minus-one, where the concept of number really allowed no such number, with the concept imaginary number, where the concept of number had been broadened such that complex numbers were included (Kitcher 1984, 175–77).

A new mathematical concept may be introduced as a distinction of subspecies under a current concept. An example would be introduction of the concept uniformly continuous function under the concept function. An example of broadening a mathematical concept would be the history of the concept function (Kline 1972, 338–40, 403–6, 505–7, 677–78). An example of narrowing would be the history of the concept integrability (Kline 1972, 959–61).

In sum mathematical concepts and definitions do change as the discipline advances. What concepts are most basic in an area of mathematics can also change, though we are able to locate the old basics in the new framework.

Euclid’s geometry is in the class we today call synthetic geometry. If one took geometry in high school, it was probably Euclid’s geometry, and one knows some synthetic geometry. It is geometry as synthetic relations that can be geometry as concrete relations in the world independently of mind.

When we see the word geometry without qualification, it is fairly safe to suppose the reference is to synthetic geometry, rather than analytic geometry. When we see accounts of how geometry is rooted in sensory experience, it is the empirical origin of some of the concepts in synthetic geometry that is being proposed. Analytic geometry uses algebraic methods and equations to study geometric problems (Boyer 1956; Kline 1972, 302–24, 544–66; Netz 2004). Experiential origins of concepts in the arithmetic, the algebraic, and the mathematically analytic are issues for the epistemologist, but they are distinct from the issue of experiential origins of concepts in (synthetic) geometry.

With the discovery of non-Euclidean geometries elliptical and hyperbolic, which like Euclidean geometry are geometries with planes and spaces of constant curvature (Euclidean has constant curvature of zero), we can awaken to possible empirical sources in some of the concepts underlying all three of these synthetic geometries and thereby wake to possible empirical sources for Euclidean geometrical concepts additional to volume, surface, line, and point. Helmholtz proclaimed this new, specific, and very plausible possibility (Friedman 2000, 200–202; DiSalle 2006). There can be unidentified empirical sources for unidentified elements implicit in our mathematical thought.

Physical concepts are much more than their mathematical characters. Establishing new truth in physical science requires observation and experimental tests. In the history of mathematics, there have been some episodes in which finding a physical exemplification of a mathematical innovation has drawn the mathematics community into taking the innovation more seriously. But physical exemplification is unnecessary for, and empirical testing is irrelevant to establishing new mathematical truth. The deliberate simplicity and delimitation of mathematical concepts, which Aristotle noted, are surely some part of the story of why that is so.

(To be continued. This essay is a step needed for completing the next installment in “Objective Analyticity”* and the next installment in “Randian Axioms and Postulates in Metaphysics.”*)

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Bolte, J., and F. Steiner, editors, 2012. Hyperbolic Geometry and Applications in Quantum Chaos and Cosmology. Cambridge.

Boyer, C. B. 1956. History of Analytic Geometry. 2004. Dover.

DiSalle, R. 2006. Kant, Helmholtz, and the Meaning of Empiricism. In The Kantian Legacy in Nineteenth Century Science. MIT.

Euclid c. 300 B.C. The Thirteen Books of The Elements. T. L. Heath, translator. 2nd ed. 1956 [1908, 1925]. Dover.

Friedman, M. 1992. Kant and the Exact Sciences. Harvard.

——. 2000. Geometry, Construction, and Intuition in Kant and His Successors. In Between Logic and Intuition. G. Sher and R. Tieszen, editors. Cambridge.

Kitcher, P. 1984. The Nature of Mathematical Knowledge. Oxford.

Kline, M. 1972. Mathematical Thought – From Ancient to Modern Times. Oxford.

Lear, J. 1980. Aristotle and Logical Theory. Cambridge.

——. 1988. Aristotle and the Desire to Understand. Cambridge.

Lennox, J. G. 1986. Aristotle, Galileo, and “Mixed Sciences.” In Reinterpreting Galileo. W. A. Wallace, editor. Catholic University of America.

Martin, G. E. 1975. The Foundations of Geometry and the Non-Euclidean Plane. Springer.

Mumma, J. 2011. The Role of Geometric Content in Euclid’s Diagrammatic Reasoning. Les Ètudes Philosophiques 97:243–58.

Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. Cambridge.

——. 2004. The Transformation of Mathematics in the Early Mediterranean World – From Problems to Equations. Cambridge.

Norman, J. 2006. After Euclid – Visual Reasoning & the Epistemology of Diagrams. CSLI.

Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.

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Truth of Geometry

Part 2 – Locke and Leibniz

John Locke thought that extension, the terminations of it, and figure are primary qualities of nature and are among our perfectly simple ideas. They are things “really in the world as they are, whether there were any sensible being to perceive them or no” (EU 2.31.2).

Though they exist in nature, Locke thought of the exact figures of geometry as not simple primary qualities of nature. They are not objects simply given objectively to the mind. They are our voluntary assemblies, “without reference to any real archetypes, or standing patterns existing anywhere” (EU 2.31.3). Such assemblies, Locke calls ideas of modes. A mode is a complex not self-subsistent, but depending on self-subsistent things, which is to say, depending on substances. A triangle is a complex, as is any mode, but among modes, it is relatively simple. And it is an idea clear, distinct, and certain (EU 2.12.4, 2.31.3, 3.3.18, 3.9.19, 4.4.6, 4.7.9).

Locke calls an idea adequate if it does not lack anything. It is complete, perfect. The figures of geometry are adequate ideas.

By having the idea of a figure with three sides meeting at three angles, I have a complete idea, wherein I require nothing else to make it perfect. That the mind is satisfied with the perfection of this its idea is plain, in that it does not conceive that any understanding hath, or can have, a more complete or perfect idea of that thing it signifies by the word triangle, supposing it to exist, than itself has, in that complex idea of three sides and three angles, in which is contained all that is or can be essential to it, or necessary to complete it, wherever or however it exists. (EU 2.31.3)

It is never a triangle per se that is in nature. “The general idea of a triangle . . . must be neither oblique nor rectangle [right-angled], neither equilateral, equicrural [isosceles], nor scalon; but all and none of these at once” (EU 4.7.9). The general idea of triangle is a tool in communication and in the enlargement of knowledge, according to Locke.

The knowledge we have of mathematical truths is not only certain, but real knowledge; and not the bare empty vision of vain, insignificant chimeras of the brain: and yet, if we will consider, we shall find that it is only of our own ideas. . . . The knowledge [the mathematician] has of any truths or properties belonging to a circle, or any other mathematical figure, are . . . true and certain, even of real things existing: because real things are no further concerned, nor intended to be meant by any such propositions, than as things really agree to those archetypes in his mind. Is it true of the idea of a triangle, that its three angles are equal to two right ones? It is true also of a triangle, wherever it really exists. . . . Intending things no further than they agree with those his ideas, he is sure what he knows concerning those figures, when they have barely an ideal existence in his mind, will hold true of them also when they have a real existence in matter: his consideration being barely of those figures, which are the same wherever or however they exist. (EU 4.4.6)

There are some strains of Aristotle in that. Locke, however, does not base certainty ultimately on certainty of the existence of the world and what it contains, as would Aristotle or Rand. He finds the ultimate ground of certainty in some of the agreements of our ideas with each other. “A man cannot conceive himself capable of a greater certainty than to know that any idea in his mind is such as he perceives it to be; and that two ideas, wherein he perceives a difference, are different and not precisely the same” (EU 4.2.1; see also 3.8.1; 4.7.4, 19). Such immediate knowledge is called intuitive by Locke.

Is the sum of the angles of a triangle the same or variable from one triangle to another? Seeing the sameness of that sum and the sameness of that sum to the angle of a half-circle for all triangles in the Euclidean plane is not immediately evident. It is not a single perceptive act of intuition, rather it requires demonstration (EU 4.2.2). Each step of a demonstration, in Locke’s view, requires an intuitive knowing (EU 4.2.6). The mind can perceive immediately the agreement or disagreement of each step in the demonstration (EU 4.2.5).

Locke thought the reason mathematics has demonstrative certainty is that in mathematics the mind can perceive the immediate agreement and difference between its ideas, which are the ideas of “extension, figure, number, and their modes”(EU 4.2.9).

Though in extension every the least excess is not so perceptible, yet the mind has found out ways to examine, and discover demonstratively, the just equality of two angles, or extensions, or figures: and both these, i.e. numbers and figures, can be set down by visible and lasting marks, wherein the ideas under consideration are perfectly determined; which for the most part they are not, where they are marked only by names and words. (EU 4.2.10)

Locke goes on to remark that simple ideas other than those given for mathematics—say the non-mathematical simple ideas of color or brightness—cannot be compared in quantity, but only in degree, which is to say not so thoroughly as with mathematical simple ideas (EU 4.2.11–13). Locke is using the concept quantity to indicate what we would call today a magnitude affording ratio scaling (which is more narrow than what Rand, Gotthelf, or I take for the class determinate magnitude, or quantity.)

Locke does allow that non-mathematical ideas such as color or brightness could be entered into effective systematic reasoning, though those ideas are precise only to the level of the least differences we can perceive, unlike ideas of line or figure in geometry. There are degrees of difference in our intuitive perceptions of a simple secondary quality, such as brightness, and such degrees of difference suffice to found inferential knowledge beyond subjects such as geometry or mechanics. Where the difference in discerned difference in a secondary quality “is so great as to produce in the mind clearly distinct ideas, whose differences can be perfectly retained, there these ideas or colors, as we see in different kinds, as blue and red, are as capable of demonstration as ideas of number and extension” (EU 4.2.13). Where Locke has written “as capable of demonstration,” I think he means “as capable of use in demonstration.”

Locke is sensitive to the work of constructing figures and auxiliary lines in Euclid’s proofs. That much is good. Our way of learning geometric possibilities for geometries possibly physical (definitely physical in Euclid) is not our everyday way or scientific way of learning non-geometric physical possibilities. That much of Locke’s account is also right. However, in opposition to Locke’s account, it should be stressed that the difference between geometric and non-geometric objects is not that the former are quantitative, whereas the latter are not. Coolness to the touch is registration of a rate of heat flow, and that is a quantitative object, in Locke’s sense, and a non-geometric object of knowledge. We do not proceed in thermodynamics as we proceed in geometry, notwithstanding the circumstance that for both the objects are quantitative, in the elementary sense of Locke and his era. That would be acceded by Locke, but I add that it is not getting anywhere to say the precision of geometry is on account of its objects being amenable to quantification. Rather, precise determinateness is one of the requirements of the quantifiable. More deeply, his theory is defective in basing the certain truth of Euclid’s geometry, derived in part from constructions, ultimately on certainty in the agreement between our ideas rather than on certainty of some possibilities for acts in the world along with possibilities of the world affording those acts.

Locke’s attempt at accounting for the extent to which demonstrative certainty has been attained in mathematics has the merit of not foreclosing an essential justificative role of the lettered diagram that is appealed to in most of Euclid’s demonstrations. It allows, at least implicitly, for the possibility that we think with the lettered diagram—though the thought is of the exact form there within the lettered diagram—and that this kind of thinking forms part of the justification of the certain truths of Euclid’s geometry.

Gottfried Wilhelm Leibniz responded to Locke’s Essay concerning Human Understanding (EU – 1690)* with New Essays on Human Understanding (NEU – 1704).* Leibniz opposes much of the Lockean view of what is going on in Euclidean demonstration. Admittedly, the Euclidean diagrams are “helping judgment to gain demonstrative knowledge” (NEU 385; further, 352–53). Sensory experience can aid in the geometric proof, but only as a crutch for our progression of thought. It is not essential to thought and is not part of the justification for the geometric truth. Sensory experience is confused perception, not distinct perception, and it is not essential to thought. Distinct ideas may accompany sensory ideas, but it is only the former that serve for demonstrations (NEU 137, 487).

Leibniz thinks the ideas of extension and figure come from “the common sense, that is, from the mind itself; for they are ideas of the pure understanding (though ones which relate to the external world and which the senses make us perceive), and so they admit of definitions and of demonstrations” (NEU 128; also 1682–84, 286). Ideas are not images, and imagination is not thought (NEU 261–62; c. 1691). “One can have the angles of a triangle in one’s imagination without thereby having clear ideas of them. Imagination cannot provide us with an image common to acute-angled and obtuse-angled triangles, yet the idea of triangle is common to them” (NEU 375; further, 451–53 and De Risi 2007, 35–39).

Concerning necessary, geometric truths:

The senses are inadequate to show their necessity, and . . . therefore the mind has a disposition (as much active as passive) to draw them from its own depths; though the senses are necessary to give the mind the opportunity and the attention for this, and to direct it towards certain necessary truths rather than others. . . . The fundamental proof of necessary truths comes from the understanding alone, and other truths come from experience or from observations of the senses. Our mind is capable of knowing truths of both sorts, but it is the source of the former; and however often one experienced instances of a universal truth, one could never know inductively that it would always hold unless one knew through reason that it was necessary. (NEU 80; also 1714, §5)

In geometry, as in syllogistic inference, thought is apart from sense (NEU 370–72). Geometry’s elementary ideas are innately, implicitly within us, and we expose them and their relations in the discipline (NEU 50, 77, 392). “Neither a circle, nor an ellipse, nor any other line we can define exists except in the intellect, nor do lines exist before they are drawn, nor parts before they are separated off” (1689, 34). Nevertheless, “number and line are not chimerical things . . . for they are relations that contain eternal truths, by which the phenomena of nature are ruled” (1695, 146–47; further, Garber 2009, 158–62).

[Extension and] the continuum in general, as we understand them in mathematics, are only ideal things—that is they express possibilities, just as do numbers. . . . Space and time . . . relate not only to what actually is but also to anything that could be put into its place, just as numbers are indifferent to the things which can be enumerated. . . . [in nature] is never found . . . actual figures which possess in full force the properties which we learn in geometry . . . . Yet the actual phenomena of nature are arranged, and must be, in such a way that nothing ever happens which violates . . . any of the . . . most exact rules of mathematics. (1702, 583; further, Belot 2011, 173–78)

“As for the proposition The square is not a circle, . . . in thinking it one applies the principle of contradiction to materials which the understanding itself provides” (NEU 83). Yes and no. That building blocks stack and balls roll is learned by the infant prior to language. Incompatible shapes are available to see and handle. Leibniz will allow that. He allows also that a child having language can know what is a square and its diagonal without yet knowing a square’s diagonal is incommensurable with its side (NEU 102). To grasp the perfectly exact figures and relations that enter geometry—such as the incommensurability of a square’s diagonal with its side or the equality of the sum of angles in a triangle to two right angles—requires a high level of conceptual understanding. Leibniz errs, however, in thinking perfectly exact figures and relations are only from abstract thought. They are partly taken from the world, they may obtain perfectly in physical space, and without mind.

Unlike Locke and like the other rationalists, Leibniz takes Euclid’s figures and their specific natures to be objective givens in the mind (1675, 1–2). In the view of Leibniz, extension and its geometric modes, though they are related to physical objects, come forth in the mind and are therefore definite objects and relations suited for entering demonstrations. No material objects have shapes so precise and determinate as geometric objects (1687, 86–87; 1689, 34; 1704 or 1705, 183). The boundaries of figures we draw on paper in a geometric proof are not exactly the boundaries and figure in mind for the proof (further, NEU 360; Norman 2006, Ch. 6; Azzouni 2004*), and while the former, as with all matter, are compositions, the latter are not (1695, 146–47).

Leibniz observes that even where we have distinct physical knowledge, as in a definition of gold, we yet have knowledge incomplete, for we know not much (in Leibniz’ day) about the processes yielding the traits in the definition of gold. Not knowing much, that is, in comparison to knowledge in geometry, wherein ideas are so distinct that all their components are distinct (NEU 266–67, 308–9, 346–48). Such completeness is perfect knowledge, which Locke and many others called adequate knowledge. In geometry “we can prove that closed plane sections of cones and cylinders are the same, namely ellipses; and we cannot help knowing this if we give our minds to it, because our notions pertaining to it are perfect ones” (NEU 267). The inner natures of geometrical figures can be reached by the human mind; not nearly so swiftly might the inner natures of “the incomparably more composite species in corporeal nature” be reached (NEU 348).

Leibniz thought possible and hoped for an algebraic rendering of geometry. He thought possible and hoped both could be brought under an art of formal deduction, one subsuming syllogistic logic (NEU 478–79; 1666-67; 1678; 1679; c. 1691; c. 1692; De Risi 2007, 40–41, 63–80, 85–89, 95–98, 569–75; Capozzi and Roneaglia 2009; Sutherland 2010, 155–63). Some of his attempts to achieve this program proposed a philosophical definition of similarity, from which he hoped both similarity in Euclidean geometry and similarity in its non-mathematical occasions might logically stem. Rand, on the other hand, proposed a mathematical, mensural account of similarity in general, one applicable to similarity in geometry and, she hoped, to similarity everywhere else (ITOE 13–14; ITOE Appendix 139–40; cf. Heath 1956, 132–33).

(To be continued.)

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Ariew, R., and D. Garber, translators, 1989. G. W. Leibniz – Philosophical Essays. Hackett.

Belot, G. 2011. Geometric Possibility. Oxford.

Capozzi, M., and G. Roncaglia 2009. Logic and Philosophy of Logic from Humanism to Kant. In The Development of Modern Logic. L. Haaparanta, editor. Oxford.

De Risi, V. 2007. Geometry and Monadology – Leibniz’s Analysis Situs and Philosophy of Space. Birkhäuser.

Garber, D. 2009. Leibniz: Body, Substance, Monad. Oxford.

Euclid c. 300 B.C. The Thirteen Books of The Elements. T. L. Heath, translator. 2nd ed. 1956 [1908, 1925]. Dover.

Leibniz, G. W. 1666. Dissertation on the Art of Combinations. In Loemker 1969 (L).

——. Letter to Foucher. In Ariew and Garber 1989 (AG).

——. 1678. Letter to Tschirnhaus. (L)

——. 1679. Studies in a Geometry of Situation with a Letter to Huygens. (L)

——. 1682–84. On the Elements of Natural Science. (L)

——. 1687. Letter to Arnauld, 30 April. (AG)

——. 1689. Primary Truths. (AG)

——. c. 1691. Ars Representatia. V. De Risi, translator. Leibniz Review 15:134–39.

——. c. 1692. Uniformis Locus. V. De Risi, translator. Leibniz Review 15:140–51.

——. 1695. Note on Foucher’s Objection. (AG)

——. 1702. Reply to Bayle. (L)

——. 1704. New Essays on Human Understanding. P. Remnant and J. Bennett, translators. Cambridge.

——. 1704 or 1705. Letter to de Volder. (AG)

——. 1714. Principles of Nature and Grace, Based on Reason. (AG)

Locke, J. 1690. Essay Concerning Human Understanding. 1959. Dover.

Loemker, L. E. 1969 [1954]. Gottfried Wilhelm Leibniz – Philosophical Papers and Letters. 2nd ed. Kluwer.

Norman, J. 2006. After Euclid – Visual Reasoning & the Epistemology of Diagrams. CSLI.

Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.

Sutherland, D. 2010. Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.

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Euclid c. 300 B.C. The Thirteen Books of The Elements. T. L. Heath, translator. 2nd ed. 1956 [1908, 1925]. Dover.

A marvelous book! Heath is such a top notch scholar. In the first of the three dover books, Heath wrote a nearly 200 page introduction even before he gets to Book I. He covers the existing manuscripts, provides background on Euclid and quotes Proclus a lot. Also he goes into great detail on the nature of definitions and he quotes from Aristotle heavily especially Topics, Categories and Posterior Analytics.

And on top of it the cost is quite reasonable.

Ba'al Chatzaf

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"Euclid alone hath looked on beauty bare"

--ESV Millay

A bit of an exaggeration. However Eucid's Elements is one of the early monuments to the axiomatic method. Up until the middle of the 19th century it was taken to be leading treatise on geometry. Think about it: a technical treatise or textbook in continuous publication for nearly 2300 years. Only the Bible has a longer run.

The most important part of The Elements is the incorporation of Eudoxus' handling of ratio. He was able to define ratios of both commensurable and incommensurable quantities. If Eudoxus had a decent algebraic scheme he might have invented real numbers 2300 years ago.

Ba'al Chatzarf

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Truth of Geometry

Part 3 – Kant, Precritical

We have seen that Locke had a keen appreciation of the profound place of construction in Euclid’s geometry. We have seen also his alignment with Aristotle in conceiving as aspects these geometric figures and relationships in the physical world; they are not only in the mind. Leibniz denied the first point. Thought alone, free of sensory perception, reveals geometry. Euclid’s constructions are not essential to justifying the truth of the geometric propositions, according to Leibniz. He opposed Aristotle and Locke on the second point as well. Nature conforms to principles of geometry because, like logical principles, those principles are necessary principles of possibility, coeternal with the author of nature.

We have seen Leibniz arguing that truths of geometry could not be reached by induction from experience, for then they could not have the formality, necessity, and universality we know them to have. That charge is not fair to Locke or Aristotle. If their idea was that geometric figures and their natures are arrived at by induction from experience, it was surely by the sort of induction known as abstractive or intuitive, not by the sort one first thinks of, which is called incomplete, problematic, or ampliative. The latter sort is “a passage from the individuals to universals” (Topics 105a12) and a passage “from the known to the unknown” (Top. 156a5). The former sort is induction as “exhibiting the universal as implicit in the clearly known particular” (Posterior Analytics 71a8; Boydstun 1991, 36; further, Peikoff 1985). So we can agree with Leibniz’ point against ampliative induction to geometric figures and their natures, while leaving open the possibility of a role for abstractive induction antecedent the process of thought that is Euclid’s Elements (further, Grosholtz 2007, 53–55; Stekeler-Weithofer 1992).

In Kant’s mature philosophy of transcendental idealism (which is also known as critical or formal idealism), he was rightly sensitive, like Locke, to the profound role of construction in Euclidean geometry. Contrary to Leibniz, discursive thought is not enough in the cognition that is geometry. True, as Leibniz stated, the results of geometry are not attained by (ampliative) induction. Contrary to Locke and Hume, the necessity in starting points and conclusions in Euclid is not a kind of feeling arising from a comparison of our ideas. It is not, as Locke had it, from an intuitive sense of agreement or disagreement in our ideas. According to Kant, it is from a kind of intuition, a kind of content for concepts, one that tells us some constraints of form on sensory experience, namely the constraint of spatial relations.

We begin before Kant had arrived at those positions. We begin in Kant’s precritical period with his 1764 essay “Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality.” Kant observes that in geometry definitions are arrived at synthetically. One can define clearly a category of figures, and thereby they are so. This is reminiscent of Leibniz saying that lines and defined figures of geometry do not exist (outside the divine understanding) before we define and draw them. Kant sees it this way: “Whatever the concept of a cone may ordinarily signify, in mathematics the concept is the product of the arbitrary representation of a right-angled triangle which is rotated on one of its sides. In this and in all other cases the definition [Erklärung] obviously comes into being as a result of synthesis” (2:276; cf. Spinoza c. 1662, 2.27.15–25). By synthesis, Kant here means a free combination of elements into a concept.

Recall the attempts of Aristotle, Locke, and Rand to account for exactitude and certainty of mathematical concepts in comparison to ordinary and philosophical concepts. Here is how Kant contrasted mathematical definitions with philosophical definitions. Mathematical definitions are synthetic; philosophical ones are analytic.

In philosophy the concept of a thing is always given, albeit confusedly, or in an insufficiently determinate manner. The concept has to be analyzed; the characteristic marks which have been separated out and the concept which has been given have to be compared with each other in all kinds of contexts; and this abstract thought must be rendered complete and determinate. (2:276)

Compare with Rand’s analysis of justice or reason* (ITOE 51) .

According to Kant in this essay, had we defined a philosophic concept synthetically, “it would have been a happy coincidence indeed if the concept, thus reached synthetically, had been exactly the same as that which completely expresses the idea . . . which is given to us” (2:277). Granted some philosophers have “defined” philosophic concepts synthetically. Consider synthetic proposals concerning the concept substance.

A case in point would be that of a philosopher arbitrarily thinking of a substance endowed with the faculty of reason and calling it a spirit. My reply, however, is this: such determinations of the meaning of a word are never philosophical definitions. If they are to be called definitions at all, then they are merely grammatical definitions. For no philosophy is needed to say what name is to be attached to an arbitrary concept. Leibniz imagined a simple substance which had nothing but obscure representations, and he called it a slumbering monad. But, in doing so, he did not define the monad. He merely invented it, for the concept of monad was not given to him but created by him. (2:277)

In the case of Leibniz’ concept of monad, I should take some issue with Kant. It was partly synthetic, but it was partly analytic of the concept substance, which was given ordinarily and was given to Leibniz in the philosophic formulas prior to his own. Still, Kant’s point is a very good one. Fundamentally and pervasively, philosophical concepts are analytic in the present sense.

In Inquiry Kant argues against the programs of Leibniz and Wolff to rewrite Euclidean geometry using philosophical definitions of similarity in place of Euclid’s; philosophical concepts do none of the work distinctive to geometry. He argues also against their efforts to assimilate geometry into an overarching philosophical, logical formalism.

Leibniz proposed that “the theory of similarities or of forms lies beyond mathematics and must be sought in metaphysics” (1679, 254–55). Geometers, in his view, could make greater use of similarity, but heretofore philosophers had not produced a definition of the concept clear, distinct, and adapted to mathematical investigation.

Leibniz offered a philosophical definition of similarity and employed it to derive some of Euclid’s theorems. “We call two presented figures similar if nothing can be observed in one viewed in itself, which cannot be equally observed in the other” (1679, 255). With this definition, a related lemma, and an axiom, Leibniz composed a direct proof for Elements XII.2, which says the ratio of the areas of two circles equals the ratio of the squares of their diameters. In Euclid the result had been obtained only by reductio ad absurdum, not directly.

The concept of similarity is defined in Elements, which is what one likely learned in high school: Two rectilinear figures are similar if they have corresponding angles equal between the figures and the ratio between corresponding sides of the two figures are the same for all those pairs of sides.

Daniel Sutherland explains why Leibniz’ definition of similarity is what it is.

Leibniz’s search for primitive ideas that correspond to reality was influenced by Aristotle’s Categories and its long history in western philosophy, from Porphyry to Melanchthon. . . . By late scholasticism, a treatment of categories was not limited to the particular Aristotelian categories such as substance, quality and quantity, but encompassed the ordering of fundamental concepts under a highest genus. Leibniz’s own efforts to identify primitive ideas, or at least relatively primitive ideas, included quality and quantity. . . .

As is well known, the concept of identity is central to Leibniz’s logic and metaphysics: Leibniz took the logical statements that reveal the containment relations among ideas to be identity statements. It is therefore unsurprising that the concept of identity plays a role in Leibniz’s attempts to reform the Aristotelian categories. In particular, he defined similarity as identity of quality and equality as identity of quantity, thereby making similarity and equality dependent upon his account of quality and quantity. (2010, 161)

Prof. Sutherland points to the tradition of Aristotle on that last point. Aristotle had defined similarity and equality in terms of quality and quantity. Those things “are like whose quality is one; those are equal whose quantity is one” (Metaph. 1021a11–12).

Leibniz observed a further distinction of quantity and quality.

Quantity can be grasped only when the things are actually present together or when some intervening thing can be applied to both: But quality presents something to the mind which can be known in a thing separately and can then be applied to the comparison of two things without actually bringing the two together either immediately or through the mediation of a third object as a measure. (1679, 254)

With this understanding of quantity and quality, Leibniz’ metaphysical definition of similarity follows, and Euclid’s definition is seen to be an instance of that metaphysical definition. Contrary his hopes, however, Leibniz never delivered a presentation of Euclidean geometry with similarity ascendant over congruence nor geometry subordinate metaphysics.

Early in the eighteenth century, Christian Wolff, preeminent follower of Leibniz, had written two geometry texts. One was in German, and it came to be the standard introductory student text in Germany, replacing Euclid’s Elements. It was a practical text, permitting use of scaled rulers and protractors. The second text was in Latin and was intended for a more scholarly audience. Its title is Elements of Universal Mathematics (Elementa Matheseos Universae). Wolff thought that all human knowledge would be derivable if concepts were properly defined and organized according to their logical relations. He thought the axioms of geometry could in principle be replaced with definitions and derivations from them. In Elementa Wolff dropped Euclid’s reliance on congruence. Inspired by Leibniz’ proposal to put a philosophical definition of similarity to work directly in geometry, he attempted just that, though with his own, related, but more elaborate, definition of similarity (Sutherland 2010, 163–69). I note in passing that part of one of Wolff’s two corollaries of his definition of similarity was quite close to Rand’s definition of similarity.* Wolff’s second corollary includes the clause “similar things, without loss of similarity, are able to differ in quantity” (quoted in Sutherland 2010, 165).

In Elementa Wolff radically rewrote Euclid’s geometry, resting the new deductions on his philosophical definition of similarity, joined with a definition of things “determined in the same manner.” The deductions of the theorems of geometry he sets forth are in fact fallacious. The results do not follow from the premises. Sutherland thinks Wolff’s lack of genuine rigor stems from Wolff’s preconceptions about the way human knowledge should be organized.

In his view, all knowledge can ideally be deduced from proper definitions. The search for correct definitions is a search for fundamental concepts from which all knowledge can be derived by means of definitions. . . . All concepts fit into a single hierarchy of concepts beginning with the most general, and the uniqueness of the hierarchy requires that mathematics be integrated into human knowledge more broadly. Geometry in particular was to be integrated by means of the philosophical definitions of similarity and equality in terms of quality and quantity and the further definitions of quality and quantity. (2010, 168)

We have noted Aristotle’s mistaken idea that the theorems of Euclidean geometry, with their necessity and universality, can be deduced purely by syllogistic demonstration. However, unlike Wolff, Aristotle did not suffer the further misconception that mathematics and demonstrative science more generally requires no axioms among its starting points, only definitions.

We have seen that in Kant’s “Inquiry” philosophy’s distinctive method and task is to render complete and determinate certain concepts that are given in a confused manner in general usage. This task, which is called analysis, is in contrast to that of mathematics: “combining and comparing given concepts of magnitudes, which are clear and certain, with a view to establish what can be inferred from them” (2.278). Wolff’s attempt to bring the concept similarity in an analytic version into geometry and make it a base of geometry was a big mistake. Euclid’s conception of similarity is sufficient for geometry, and similarity under more general philosophic definitions fills no gap and does no real work in geometry. As a matter of fact, not trading in concepts as analytic is why geometry is not in “the same wretched discord as philosophy itself” (2: 277; further, Sutherland 2010, 177–88).

Kant observes, furthermore, in geometry:

To discover the properties of all circles, on circle is drawn; and in this one circle, instead of drawing all the possible lines which could intersect each other within it, two lines only are drawn. The relations which hold between these two lines are proved; and the universal rule, which governs the relations between intersecting lines in all circles whatever, is considered in these two lines in concreto (2:278; also 291–92)

Proposition 35 of Book III would be an example.* This would seem to be an act of abstractive induction, though not for analytic concepts. I say it is abstractive induction for synthetic concepts and their interrelations. Bear in mind that abstractive induction is also called intuitive induction.

Kant begins to correct the error of Wolff, Leibniz, and Aristotle, who had not recognized that proofs in Euclid have additional legitimate resources beyond deductions with words marking general concepts having not concrete, perceptual, combinatorial indication of the interrelations of these concepts (2:278–79). Kant at this stage, like Locke before him, has not smoothly embedded his theory of mathematical cognition into a general theory of cognition. Like Locke, his account of cognition peculiar to proofs of Euclid does little more than say we have such intellectual capabilities.

Philosophers before Kant, as we have seen, had things to say about the greater exactitude and certainty of mathematical knowledge over philosophical knowledge, such as knowledge of the category of substance. Rand and Gotthelf touched on the issue too. Kant joins that chorus, mentioning factors along the lines of their factors, in addition to his original notice that the mathematical concepts are synthetic, not analytic, and mathematical proof includes handling mathematical universal concepts in concreto.

The basic concepts and starting propositions of geometry are few in comparison to such concepts and propositions in philosophy (2:279–82). The object of mathematics is magnitude, which is easy in comparison to the object of philosophy. The latter object is difficult and involved.

The relation of a trillion to unity is understood with complete distinctness, whereas even today the philosophers have not yet succeeded in explaining the concept of freedom in terms of its element, that is to say in terms of the simple and familiar concepts of which it is composed. In other words, there are infinitely many qualities which constitute the real objects of philosophy . . . . It is far more difficult to disentangle complex and involved cognitions by means of analysis than it is to combine simply given cognitions by means of synthesis and thus to establish conclusions. (2:282)

Nevertheless, in the area of philosophy that is metaphysics, as much certainty is possible as in geometry. The object of mathematics is magnitude, the object of metaphysics is encountered in a thicket of various, numerous qualities. Grasp of the object is in principle attainable in either case.

In all disciplines, the formal elements in judgments rely on the indubitable “laws of agreement and contradiction” (2:296).

[The proposition] which expresses the essence of every affirmation and which accordingly contains the supreme formula of all affirmative judgments, runs as follows: to every subject there belongs a predicate which is identical with it. This is the law of identity. The proposition which expresses the essence of all negation is this: to no subject does there belong a predicate which contradicts it. This proposition is the law of contradiction. . . . These two principles together constitute the supreme universal principles, in the formal sense of the term, of human reason in its entirety. (2:294)

In metaphysics, as in mathematics, there are material concepts and principles that are indemonstrable and foundational. The number of these is greater in metaphysics than in mathematics. Metaphysics is more difficult than Euclidean geometry, though not in principle less secure in its truths. The grounds of metaphysical truths are objective. They are not subjective criteria of conceivability or feeling of certainty (2:294; also 285–86).

In both metaphysics and geometry, the formal element of the judgments exists in virtue of the laws of agreement and contradiction. In both sciences, indemonstrable propositions constitute the foundation on the basis of which conclusions are drawn. But whereas in mathematics the definitions are the first indemonstrable concepts of the things defined, in metaphysics, the place of these definitions is taken by a number of indemonstrable propositions which provide the primary data. Their certainty may be just as great as that of the definitions of geometry. They are responsible for furnishing either the stuff, from which the definitions are formed, or the foundation, on the basis of which reliable conclusions are drawn. . . . Mathematics is easier and more intuitive in character. (2:295–96)

(To be continued.)

References

Aristotle c. 348–322 B.C. The Complete Works of Aristotle. J. Barnes, editor. 1983. Princeton.

Boydstun, S. 1991. Induction on Identity. Objectivity 1(2):33–46.

Grosholtz, E. R. 2007. Representation and Productive Ambiguity in Mathematics and Science. Oxford.

Jetton, M. 1991. Philosophy of Mathematics. Objectivity 1(2):1–32.

Kant, I. 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. In Theoretical Philosophy 1755–1770. D. Walford, translator. Cambridge.

Leibniz, G. W. 1679. On Analysis Situs. In Loemker 1969.

Loemker, L. E. 1969 [1954]. Gottfried Wilhelm Leibniz – Philosophical Papers and Letters. 2nd ed. Kluwer.

Rand, A. 1966–67. Introduction to Objectivist Epistemology. Expanded 2nd edition. 1990. Meridian.

Spinoza, B. c. 1662. Emendation of the Intellect. In The Collected Works of Spinoza. E. Curley, translator. Princeton.

Stekeler-Weithofer, P. 1992. On the Concept of Proof in Elementary Geometry. In Proof and Knowledge in Mathematics. M. Detlefsen, editor. Routledge.

Sutherland, D. 2010. Philosophy, Geometry, and Logic in Leibniz, Wolff, and the Early Kant. In Discourse on a New Method. M. Domski and M. Dickson, editors. Open Court.

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Truth of Geometry
Part 1 – Aristotle
Part 2 – Locke and Leibniz
Part 3 – Kant, Precritical
Part 4 – Kant, Critical (forthcoming)

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"accuracy= i.e. simplicity."

Apparently not, from this intriguing new presentation.

Thanks for it from an amateur geometry/algebra fangirl.

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