I wrote this series a few years ago. This philosophical look covers Aristotle, Locke, Leibniz, Wolff, and precritical Kant. I hope to continue it in the next few years, bringing it up to present-day mathematics. This series is a strand to be used in pulling together the 3-ply cord of mathematics, logic, and metaphysics, as well as in formulating a concept of objective analyticity. It will figure into completion of my thread Peikoff’s Dissertation as well as into completion of my book.
**Truth of Geometry – Necessity in 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](http://www.objectivistliving.com/forums/index.php?showtopic=6263&#entry61242); 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)[\*](http://farside.ph.utexas.edu/euclid/Elements.pdf). 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 justificatory 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.
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> 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)
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> 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)
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> 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[\*](http://www.johnmumma.org/Writings_files/Role%20of%20geometric%20content.pdf)) 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). Leonard Peikoff took the proviso “in the present context of knowledge” as 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).
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> _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. . . .
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> _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.
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> _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.
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> _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_.)
<u><span>References</span></u>
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. 2<sup>nd</sup> 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 2<sup>nd</sup> edition. 1990. Meridian.