Metaphysics and Geometry

[Boydstun]

[Boydstun]

Metaphysics and Geometry

Let us return through the mists of time to high school geometry. Our text would be one crafted from geometry as in Euclid’s The Elements, down from about BCE 300, a couple of decades after the life of Aristotle. Aristotle well knew proofs of geometric fact such as are posed and proven in Elements. Aristotle did not get to see the economical organization of all geometry that is provided in Elements, wherein, from a set of Definitions, Postulates, and Common Notions (also called Axioms); there are deduced certain constructions, and with constructions established as are needed, then, from all that battery of bases: truths of geometry are deduced as theorems.

Until the nineteenth century, it was naturally assumed that Euclid’s was the only possible basic geometry and that it was the geometry of the physical space in which we act. So in proving new theorems in Euclidean geometry, one was finding truths of the concrete physical world, notwithstanding the circumstance that access to this character among concretes was by way of very abstract extended thought. Although there are now other geometries that have been discovered and appropriate physical applications have been found for them, the local physical space in which we build locomotives for transportation or draperies for the home is Euclidean. To this day, it is still the case that in discovering truths in Euclidean geometry (and in the newer geometries), we are still discovering truths (e.g. new theorems within the last century in Euclidean geometry) about physical space by the abstract method of Euclid in his geometry.

Recall that in high school Euclidean geometry, our instruments were only paper, pencil, straight edge, and compass (no measurements with ruler or protractor). Straight lines and circles are the only sort of lines being contemplated in the Euclidean plane. The first of Euclid’s five Postulates says that any two points are connectible by a unique straight line. In my class, we learned this as: any two points determine a line, where line did not mean only some determinate length of line, but line of any length. Line, unqualified, means an infinite one in that sense, and the two points mark a particular segment of that line. Points have no parts, and lines have no breadth, by Euclid’s Definitions. By Postulate 1, Euclid is implicitly asserting that such lines and points exist, and he is countering the earlier derogation of geometers by philosophers, including Plato and Aristotle, that because the lines geometers draw are imperfect, not ideal, geometers use false premises. I should defend the geometers, transplanting and putting to use an epistemological leitmotif of Rand: a line drawn in a geometrical construction is a representation of any one of the perfect lines (infinite in number) contained in the volume of the line drawn using straight-edge or compass.

Postulate 3 asserts that around any point, circles of any particular radius can be scribed (with compass). Postulates 3 and 1 are among the assumptions used in the first demonstration of a construction in The Elements, Proposition 1, which shows how to construct with one’s two instruments an equilateral triangle incorporating any given line segment as one side of the triangle. Perfect and everywhere are circles and such triangles, with sides of equal length in any of the Euclidean planes of Euclidean space. (Not only I, but Descartes and Newton, adopt this way of thinking about space.) Another assumption relied on in the demonstration of Proposition 1 is the Common Notion that things equal to the same thing are equal to each other. 

Proposition 1, having been established, is used as one of the assumptions of the construction that is Proposition 11: to draw a straight line at right angles to a given straight line from a given point on it. Proposition 11 is among the assumptions used in demonstrating the theorem that is Proposition 13: Any straight line intersecting a straight line with which it is not coincident yields two angles to one side (either side) of the base line which are either two right angles or equal to two right angles. Proposition 13 is among the assumptions used in demonstrating the theorem that is Proposition 31: The three interior angles of any triangle are equal to two right angles. Let us give Proposition 31 the honor of a label, the label 2R.

One philosophical puzzle about geometry in Aristotle’s time was this: Geometric truths such as 2R are demonstrated by a discursive deduction, where assertions are not about perceived particulars such as pencil lines or acts of drawing; rather, about absolute perfect lines and circles and constructions made with them within what one draws, where what one draws is their representation. Such constructions bearing and representing such elements and their joins are necessary to the discursive deductions of geometry (a glory of firm Greek thought) such as 2R.

The rationality of such constructions and theorems cannot be rationally impugned; to try to do so would not be to contradict oneself generally speaking, but to enter a course for merely blinding oneself to part of reality and the human rational power to access it.

Geometry is knowledge, and on the face of it, knowledge with deliveries from its procedures, not only knowledge of its procedures. What powers of mind join the logical powers in the deliveries of geometry such as 2R? Aristotle’s answer is: a power of imagination (Humphreys 2023; Webb 2006; Franklin 2014).

Kant understood Euclid’s geometry as extending knowledge in its theorems and as radically independent of the uptake of any empirical information. For him the puzzle becomes How is such synthetic a priori knowledge possible? His answer is: a power of human intuition ineluctably having the form we know as Euclidean space (Pippin 1982; Friedman 1992; Falkenstein 1995; Carson 1996; Shabel 2003; Folina 2006; Parsons 2012, Chapter 1; Carson and Shabel 2016; Posy and Rechter 2020; Sutherland 2022).

As I mentioned in Interlude of “Space Not Relative to Its Discernment,” a big embarrassment eventuated for Kant’s doctrines on the nature of space in his formal idealism. Kant’s characterization of space and our cognitions concerning it expressly entailed that Euclidean geometry was the only possible sort of geometry. Several years after Kant, the hyperbolic and elliptic geometries were discovered. Something was amiss in Kant’s theory, we know by that later development alone. I want to uncover Kant’s erroneous thinking on geometry directly from consideration of his arguments. Some of the axioms for hyperbolic and elliptic geometries are also common to Euclidean geometry, and I should like to find what Kant’s account can look like if restricted to only that core of synthetic geometry.

To be continued.

[Boydstun]

Beyond geometry in Euclid’s The Elements and the writings of Aristotle and Kant concerning that geometry, my detail and assessment, in the sequel, will be greatly assisted by these resources:

Auxier, R.E. and L.E. Hahn, editors. 2006. The Philosophy of Jaakko Hintikka. Chicago: Open Court.

Carson, E.J. 1996. Mathematics, Metaphysics and Intuition in Kant. Ph.D. dissertation. Ann Arbor: ProQuest.

Carson, E.J. and L. Shabel, editors, 2016. Kant: Studies on Mathematics in the Critical Philosophy. New York: Routledge.

Falkenstein, L. 1995. Kant’s Intuitionism – A Commentary on the Transcendental Aesthetic. Toronto: University of Toronto Press.

Folina, J. 2006. Poincaré’s Circularity Arguments for Mathematica Intuition. In Friedman and Nordmann 2006.

Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics. New York: Palgrave Macmillan.

Friedman, M. 1992. Kant and the Exact Sciences. Cambridge, MA: Harvard University Press.

Friedman, M. and A. Nordmann, editors, 2006. The Kantian Legacy in Nineteenth-Century Science. Cambridge, MA: MIT Press.

Humphreys, J. 2023. The Invention of Imagination – Aristotle, Geometry, and the Theory of the Psyche. Pittsburgh: University of Pittsburgh Press.

Parsons, C. 2012. From Kant to Husserl. Cambridge, MA: Harvard University Press.

Pippin, R.B. 1982. Kant’s Theory of Form. New Haven: Yale University Press.

Posy, C. and O. Rechter, editors, 2020. Kant’s Philosophy of Mathematics. Vol. I. Cambridge: Cambridge University Press.

Shabel, L.A. 2003. Mathematics in Kant’s Critical Philosophy. New York: Routledge.

Southerland, D. 2022. Kant’s Mathematical World – Mathematics, Cognition, and Experience. Cambridge: Cambridge University Press.

Webb, J.C. 2006. Hintikka on Aristotelian Constructions, Kantian Intuitions, and Peircean Theorems. In Auxier and Hahn 2006.

Edited December 13, 2024 by Boydstun

[KyaryPamyu]

Kant’s Views on Non-Euclidean Geometry -- Michael E. Cuffaro

ABSTRACT. Kant’s arguments for the synthetic a priori status of geometry are generally taken to have been refuted by the development of non-Euclidean geometries. Recently, however, some philosophers have argued that, on the contrary, the development of non-Euclidean geometry has confirmed Kant’s views, for since a demonstration of the consistency of non-Euclidean geometry depends on a demonstration of its equi-consistency with Euclidean geometry, one need only show that the axioms of Euclidean geometry have ‘intutive content’ in order to show that both Euclidean and non-Euclidean geometry are bodies of synthetic a priori truths.
    Friedman has argued that this defence presumes a polyadic conception of logic that was foreign to Kant. According to Friedman, Kant held that geometrical reasoning itself relies essentially on intuition, and that this precludes the very possibility of non-Euclidean geometry. While Friedman’s characterization of Kant’s views on geometrical reasoning is correct, I argue that Friedman’s conclusion that non-Euclidean geometries are logically impossible for Kant is not. I argue that Kant is best understood as a proto-constructivist and that modern constructive axiomatizations (unlike Hilbert-style axiomatizations) of both Euclidean and non-Euclidean geometry capture Kant’s views on the essentially constructive nature of geometrical reasoning well.

[Boydstun]

@KyaryPamyu – Thanks, of related interest: Janet Folina – ". . . modifying Kant’s conception of intuition in order to accommodate the increasing abstractness of mathematics . . ."

First, I'll nail down precisely and completely what was Kant's conception of intuition at work in KrV (A, B). But the main study will be the Euclidean geometry in the form it was known to us in high school (USA) and to Kant (Kästner, Wolff) and to what Kant, and Aristotle, had to say about the amazing fruitfulness of the method and the ontological status of the fruit.

How old were you when you had your first course in geometry (Euclid)? I would have been 15. If ever I had taught a course in theoretical philosophy, or for that matter history of philosophy, first we would do some Euclid, say enough to produce 2R.

I'm now 76, and I must keep to completing certain specific philosophy writings with limits for a reasonable expectation for end of my powers. My explorations and vistas attained in philosophy are known to a few of the readers here and readers of a couple of academic journals (and of course intellectual lurker-absorbers who have plenty of press, in which they will never mention any gem from me with credit, as ever), but that is about it and I expect how it will end. So more and more, winding to the end, I do it mostly for myself, for the vista I was after. Loyalty to that. I won't stop before I have to, and it is nice to see interest of others along the way (as in counts ).

Edited December 14, 2024 by Boydstun

[Boydstun]

METAPHYSICS and GEOMETRY – Introduction

– Aristotle – A Cognitive Imagination (Part 1 of 2)

The subject-figures in plane geometry are composed of lines having no breadth, lines perfectly straight or perfectly circular. There is some rough sense to calling such lines and figures platonic objects of thought because thinking of such items as abstractions and counting them as particulars existing independently of our thought reminds one of  ultra-realism in theory of universals. We should not prejudge, however, what process of abstraction is at hand in thinking of lines with no breadth, and even were this process of abstraction of the sort used to know electrons, it would not follow that such lines and figures composed of them were not particulars.

Indeed, that something is reached crucially by abstraction is no showing that what is reached is not a concrete particular (or potentially a concrete, such as any patentable invention in the making or any empirically testable implication of a scientific model). Part of what we mean by “concrete” is that such a thing is in space and time. It would be a bit strange to say that space is in space, so it would be a bit strange to say that space is a concrete. Concretes are in space, and that spatiality is an inseparable condition of concretes. I’ll say then that space and its formalities, such space that is a potential situation of concretes, is concrete-biding and not merely an abstraction.

There are, anyway, two ways in which platonic would be a misnomer for such lines without breadth and the figures they compose. I and others think these items as physically real relations in space (at least down to Planck scale: 6.3 billionths of a trillionth of a trillionth of an inch), whereas platonic objects of thought would not be something physical. Secondly, though these items are formal, physical objects (items themselves void of mass-energy, but biding with objects of mass-energy) accessible to intellect, they are not platonic Forms, which can only be singular, such as  LINE or CIRCLE. Platonic Forms cannot themselves bear relations such as intersection and parallelism.

Geometric proofs at the time of Aristotle and as we know in Euclid’s Elements require engagement of discursive reasoning concerning an intelligible figure. Aristotle discerned that our power to perform such proofs requires more than our powers of perception and discursive thought. An example of such discursive thought would be: All rabbits have hearts, therefore the rabbits in our neighborhood have hearts. Discursive thought that is reasoning (viz., immediate deductive inferences or syllogistic deductive inferences) is a crucial part of our proofs in geometry, but more is required for the feat. The required supplemental power for geometry, in Aristotle’s determination, is a certain sort of cognitive imagination. As with powers of perception and thought, in engaging this cognitive imagination, “the soul discriminates and is aware of something which exists” (DA 427a21–22, translation of Fred D. Miller).

The particular technical sort of imagination that is of concern here is set out principally in the third book of Aristotle’s On the Soul (DA). Before coming to his specifications of cognitive imagination and my assessment of its supplement-adequacy in accounting for our ability for proofs in geometry (Part 2, following the present installment), let us inventory the various kinds of imagination, by our own lights, inspecting some specific timbers having to be set up for Euclid’s proof of 2R. 

I said, in the preceding installment, Postulate 3 asserts that around any point, circles of any particular radius can be scribed (with compass). I should take this as assertion that although a particular circle has a particular radius, I can take a circle of particular radius as standing for circle-radius as an algebraic variable. The Greeks did not have algebra, with its concept variable, of course. But they could understand the Rand sort of idea that a particular circle with its particular radius can stand for and be a member of the whole open-ended class of circle radii above zero. Animated visual imagination, through drawing constructs in geometry, can aid in holding in mind the concept circle of any radius (which on Rand’s measurement-omission analysis of concepts is implicit already in the concept circle).

So, making it easier to hold a concept in mind, specifically, being an iconic representation joining the symbolic representation that is the word circle is one service and kind of imagination in the full assembly supporting Euclid’s proof of 2R. Notice that this imagination is not the imagination at play in dreams. This is an imagination deliberately, actively directed to the task of gaining understanding.

I said in the preceding installment that Proposition 1 is a step in proving 2R. Prop. 1 shows how to construct from straight-edge and compass an equilateral triangle incorporating any line segment that is given for one side of the triangle. That is, this Proposition shows how to construct a triangle with sides all three equal to each other.

A year before taking high school geometry, were I asked to draw a triangle whose sides were of equal length, I’d perhaps take out my little finely-scaled metal ruler from my X-Acto kit and use it to draw a horizontal line segment, say, two inches long. Label the endpoints of this line segment A and B, and refer to this line segment as AB. I’d have known a triangle needs to have two more sides, an endpoint of one them staked on A, and an endpoint of the other staked on B. Name those two sides CD and EF, and close the figure by having CD and EF meet each other at a single point D,F. That is, stake endpoint C at A, stake endpoint E at B, and land F at D. 

So, using my ruler, I’d draw a two-inch line CD to a visually estimated location for join with EF at join-point D,F above AB. Then draw the third side EF, from the endpoint B of AB to that elevated endpoint D of CD. Measure EF to see if it is 2 inches. Likely not. I’d keep drawing new locations of the join-point D,F until EF measures 2 inches, like AB and CD.

A science or math teacher should point out to me that whenever you make a measurement of 2 inches, what you really know, with my fine ruler and eyesight, is that the line segment is 2 inches plus or minus a 64th of an inch. So I have drawn a triangle whose sides are not known to be exactly equal. Proposition 1 will teach me, the following year, how to construct a triangle perfectly equilateral, represented by a construction using a straightedge and compass, which are unscaled, taking the visible line segments as iconically representing invisible line segments of perfectly determinately exact lengths and of zero volume. We shall then not know how long are the constructed visible line segments or how long are the invisible segments, perfectly without any breadth, represented by visible line segments. But we shall know the perfect constructed segments and such triangles in physical Euclidean space have sides equal each other exactly, and not by our mere say-so.

The type of imagination we found at work in taking in Postulate 3 is also at work in the constructions, both the resulting equilateral triangle and the auxiliary circles, for  establishing Proposition 1. Call this “act-exhibit-of-a-concept imagination.” For short, AEC imagination.

An additional sort of imagination is at work in Euclid’s proof of Proposition 1. One inventing proof-enabling constructions such as Prop. 1 must use a cognitive imagination to find auxiliary constructions in space that make possible the target construction, and provably so, by true statements of relationships between the auxiliary figures and the target figure. Call this Aux imagination.

I observed in the preceding installment of “Geometry and Metaphysics” (the present study of philosophy of geometry in Aristotle and Kant and formulation of my replacement for theirs) that the proven Prop. 1 is invoked in proving the correctness of Prop. 11, which prescribes how to construct at any point on a given line L a line that is truly perpendicular to L. Establishing Prop. 11 invokes from its discoverer again both AEC and Aux imagination, as one can see by looking up Euclid’s proof of Prop. 11. Likewise for Prop. 13 feeding into Prop. 31, which is the theorem 2R. Prop. 13, recall, is the theorem that any straight line set on any line L will form either two right angles on L or angles equal to two right angles.

Through all these chained propositions proven for correctness of constructions and theorems for any plane in Euclid’s space, the physical space around us, I find in play exactly these two powers of cognitive imagination: AEC and Aux. It is important to understand that Aux is not identical with a kindred cognitive imagination of concocting some sort of diagram or array on paper to answer a question of a certain unstated relationship implicit in a thicket of stated relationships, such as problems that used to come up in the analytic section of the Graduate Record Exam. Diagrammatic positioning of the given relationships immediately shows one the asked for relationship. Aux is not that sort of helper. It does not bring one an immediate “I see.” Any such immediate seeing of relationships would be only with tolerances, such as were the case with my measurements with my steel ruler. That is not geometry, the sure discipline. What Aux enables is the proof, and the proof is necessary for each of the Propositions. 

(To be continued.)

 

 

[Boydstun]

On 12/31/2024 at 6:02 PM, Boydstun said:

METAPHYSICS and GEOMETRY – Introduction

– Aristotle – A Cognitive Imagination (Part 1 of 2)

. . .

Aux is not identical with a kindred cognitive imagination of concocting some sort of diagram or array on paper to answer a question of a certain unstated relationship implicit in a thicket of stated relationships,* such as problems that used to come up in the analytic section of the Graduate Record Exam. Diagrammatic positioning of the given relationships immediately shows one the asked-for relationship. Aux is not that sort of helper. It does not bring one an immediate “I see.” Any such immediate seeing of relationships would be only with tolerances, such as were the case with my measurements with my steel ruler. That is not geometry, the sure discipline. What Aux enables is the proof, and the proof is necessary for each of the Propositions. 

* (Such as this one.)

 

[Boydstun]

 

This author on Aristotle and Mathematics is great.

Henry Mendell's Making Sense of Aristotelian Demonstration is hard, but also great.

[Boydstun]

Before completing the Aristotle portion, I'd like to highlight in its own post an error concerning his view which I made in Part 1 of "A Cognitive Imagination."

I said earlier that Aristotle proposed a cognitive imagination in which “the soul discriminates and is aware of something which exists” (DA 427a21–22), as do the powers of perception and thought. That is an inaccurate report of Aristotle. He writes:

Quote

Imagination is different from perception and cognition; and it does not occur without perception, and without it there is no judgement.

It is evident that it is not the same process of thinking as judgement. For it is an affection which is up us to bring about whenever we wish . . . but it is not under our control to hold a belief; for we must either arrive at falsehood or truth. (427b15–21; translation of Fred D. Miller, Jr.)

Aristotle’s cognitive imagination is a tool for the mind's discriminations and discovery of what is, but this power of imagination does not do those things, unlike perception and judgment. Additionally, I had implied previously that this power of cognitive imagination was not a genre of thought. Aristotle evidently regarded it as a type of thought. In calling it an Aristotelian cognitive imagination, I do not mean to ascribe to it of itself power for assessing truth or falsehood, which cognitive suggests. This power of imagination is a supporter of cognition in that sense, including cognition in geometry.

Edited January 31 by Boydstun

[Boydstun]

On 12/31/2024 at 6:02 PM, Boydstun said:

METAPHYSICS and GEOMETRY – Introduction

– Aristotle – A Cognitive Imagination (Part 1 of 2)

. . .

I said, in the preceding installment, Postulate 3 asserts that around any point, circles of any particular radius can be scribed (with compass). I should take this as assertion that although a particular circle has a particular radius, I can take a circle of particular radius as standing for circle-radius as an algebraic variable > any circle of any radius. The Greeks did not have algebra, with its concept variable, of course. But they could understand the Rand sort of idea that a particular circle with its particular radius can stand for and be a member of the whole open-ended class of circle radii above zero (cf. Mendell 1998, 182–84). Animated visual imagination, through drawing constructs in geometry, can aid in holding in mind the concept circle of any radius (which on Rand’s measurement-omission analysis of concepts is implicit already in the concept circle).

Aristotle nears the Rand measurement-omission analysis of concepts in On Memory and Recollection:

Quote

Grant that imagination has been discussed in On the Soul, and one cannot think without an image. For the same experience takes place in thinking as in drawing a diagram, since in the latter case we make no use of the fact that a triangle has a determinate quantity [area], yet we draw it with a determinate quantity. And in the same way someone who thinks, even if he does not think of an object with quantity, sets out an object with quantity before the mind's eye, although he does not think of it as having a quantity. And even if it has a quantity by nature, though an indeterminate one, one sets out an object with a determinate quantity but thinks of it as having only quantity. (449b32–450a8; translation of Fred Miller)

So, making it easier to hold a concept in mind, specifically, being an iconic representation joining the symbolic representation that is the word "circle" is one service and kind of imagination in the full assembly supporting Euclid’s proof of 2R. . . .

. . .

 

Edited February 15 by Boydstun

[Boydstun]

Pause on Rand – An Interlude

This interlude between the two Parts on Aristotelian cognitive imagination in geometry traces some ramifications of Rand’s measurement-omission analysis of concepts for characterization of geometry. I shall add remarks on the necessity in geometry. I’ll locate synthetic geometry as well as single dimensions, such as time or hydrostatic pressure, within my own metaphysics, born of Rand’s. In the remaining Parts of “Geometry and Metaphysics,” this interlude will be convenient for contrasting this modern Objectivist cast of geometry (and my own) with Aristotle’s and Kant’s views in this area, including their conceptions of and roles for imagination (Aristotle) and intuition (Kant).

Thomas Heath, translator of Euclid’s Elements, points out that the definition of Circle (Definition 15) states what it is. Let me put it in a customary modern way: a circle is a figure in a plane consisting of points equidistant from some point in the plane. Heath observes that this definition does not tell us whether circles exist, and this definition has no genetic aspect, that is, it does not say how a circle can be brought about.

Heron of Alexandria (d. 70 C.E.; also known as Hero) does give a genetic definition: a circle is the figure scribed by the free endpoint of a line segment rotated completely with its other endpoint of the segment fixed to a point in a plane. That is what we are doing when we draft a circle with a compass. Heron also defines a straight line in a rather kinematical way as a line stretched to the utmost. Putting actions and motions into fundamentals in geometry like that may have caused a tremor in the garden where Plato was buried.

Heath understands Euclid (c. 300 B.C.E.) as proving the existence of circles by his Postulate 3 asserting the ability to draw circles anywhere and with any radius. Earlier in this study, I urged that constructions in geometry be understood as iconic representations (by resemblance, that is) of the infinitely thin lines whose character and relations are topics of plane geometry. Those iconic representations can aid working memory as symbolic representations by words are stated of those topic objects.

I pointed out that a measurements-omitted concept of circle would have Postulate 3 already within the concept in its definition: a circle is a locus of points in a plane equidistant from some point in that plane, where that distance must have some (non-zero) value, but can have any value. It is the act, the construction capability, in Postulate 3 that is the existence-warranty of the concept CIRCLE in Euclid. That does not go for my Rand-style concept of a circle as well, for the indexing of the instances as existents was there from the beginning of the ontogeny of the concept: shape of the rim of a glass or bottle or bowl; shape of tires or merry-go-rounds or hula-hoops; shape of rinds of a watermelon slice or imagined cross-section of a soap bubble. CIRCLE is firstly a concept of existential spatial form, a form available for perception and action.

A cognitive imagination, distinct from, but not separate from (i) thinking of geometric objects and (ii) thinking of form-characteristics of perceptual bodies would be the cognitive imagination I’ve called AEC (act-exhibit-of-a-concept) imagination in the preceding Aristotle installment. This power of imagination is a genuine power facilitating geometric thought.

In Rand’s epistemology, I should not call it a cognitive imagination (nor a concept by intuition; Seddon 1993, 40–47), but a measurement-omission concept of spatial forms. We should notice that the same measurement-omission characterization of concepts is applied to empirical concepts.

Indeed, I am the one who has applied it to geometrical concepts at all. Well, Rand broached application of her concept formula to the concept SHAPE in ITOE 11, 14. It failed because she did not understand how shape is measured. I gave the correct account of how SHAPE can be brought under the measurements-omitted analysis in Boydstun 2004.*

Yet the methods of reaching universal truths pertaining to the concretes covered by empirical concepts is very different than the methods for reaching universal truths pertaining to the particulars covered by geometric concepts. I’ll suggest in a moment what is the difference in their objects that gives rise to this difference in method.

The geometry that Rand knew (c. 1920)—supposing reasonably that she did know (Sciabarra 2013, 67, 382)—was the same as the geometry known to Aristotle and to Kant (late 1730’s via a Wolff text). Geometry as the “science of spatial magnitudes” will do for all of them (EN 1143a3). Magnitude is interchangeable with quantity in this context. These are unscaled in the context of synthetic geometry, such as we have in Euclid. Magnitude, or quantity, are in the world, and are that to which we add measurement scales (ITOE App. 199–200).

Philosophers have sometimes attributed the iron solidity in geometric truths to a simplicity of geometric objects and attendant exhaustive deductive accessibility of their character under rather constant geometric concepts (Phys. 200a16–17; Kant 1764, 2:282–83; Rand ITOE App. 201–203; Peikoff 2012, 218–19; further, Franklin 2014, 160–62). I have suggested, rather, that the necessity of geometric truths in Euclidean space is heir of the absoluteness that existence exists and is identity, applied to spatial form. It is the same kind and level of necessity applying to empirical truths, say, the existence and character of oxygen.

The difference in method for discovering truths mathematical and truths empirical is from of their subject matter, I say: a formality of a situation of concretes in the former; a concrete with its formalities in the latter. The difference for the two methods is not from comparative simplicity of their subject matters. The history of mathematics does not bear out such comparative simplicity. The method of Euclid’s geometry is concordant with the nature of its subject and exposes necessities in that subject (cf. Metaphy. 1094b24–27).

Geometry is of spatial structure. Homogeneity of two or more dimensions of spatial structure is the type of object for which the methods of geometry are suitable. The necessity of geometric truths are from the facts of the objects, not from the method of their discovery.

Longitude and latitude lines are homogenous dimensions. Both consist in being subsets of the concrete-biding entity that is space. Character of those breadth-less, physical lines in the surface of a glass globe can be discovered by the methods of geometry. Pressure, temperature, and humidity of the air in my study are not homogenous dimensions of that volume of air. Pounds per square inch of the air at some point is physically different in kind from degrees Kelvin of that air. Geometry-like method is not the path to discovery of relations between pressure and temperature of air, and the laws of nature—specific covariation of the physical quantities they relate—are no necessity-inferiors to truths of geometry such as 2R.

Ontologically, pure space, whose inherent structure is the topic of elementary geometry, is formal structure in situation of concretes. Pure space is a formal structure and is given in broad stroke within perception. Space perceptually empty is perceived every day in one’s locomotion. We employ the pure space of geometry saliently and precisely in geometric optics (cf. Phys. 194a10–11). Physical traits of space such as electric susceptibility, magnetic permeability, and vacuum energy are discovered by empirical-science methods, not by plain perception and geometry-like methods. That last is because, I suggest, the various dimensions they possess are not homogeneous with each other. Within each such physical dimension are the situation formalities of a linear order, but that’s it (see Rosenstein 1982).

Form in my metaphysics is in opponent-contrast to the concrete, but is present in concretes and is perceptible and thinkable in and among perceptible concretes (entities, situations, passages, and characters). (There is a similar standing of potentials in relation to actualities.) Formal structure of concrete-biding, physical space coexists with those classical electromagnetic traits and quantum-field traits of vacuum, physical space.

Topics in this interlude “Pause on Rand” will be developed further in the remainder of this study “Metaphysics and Geometry.”

 

References

Aristotle c. 345–322 B.C.E. The Complete Works of Aristotle. J. Barnes, editor. 1984. Princeton: Princeton University Press.

Boydstun, S. 2004. Universals and Measurement. The Journal of Ayn Rand Studies. 5(2):271–305.

Franklin, J. 2014. An Aristotelian Realist Philosophy of Mathematics. New York: Palgrave Macmillan.

Heath, T. L., translator, 1925. Euclid – The Thirteen Books of The Elements. 2nd ed. New York: Dover.

Kant 1764. Inquiry Concerning the Distinctness of the Principles of Natural Theology and Morality. In Walford1992.

Miller, F. D. , translator, 2018. Aristotle On the Soul and Other Psychological Works. New York: Oxford University Press. 

Peikoff, L. 2012. Understanding Objectivism. 1983 Lectures edited by M. Berliner. New York: New American Library. 

Rand, A. 1966–1967. Introduction to Objectivist Epistemology. Expanded 2nd ed. 1990. H. Binswanger and L. Peikoff, editors. New York: Meridian.

Rosenstein, J. 1982. Linear Orderings. New York: Academic Press.

Seddon, F. 1993. On Newtonian Relative Space. Objectivity 1(6):37–53.

Sciabarra, C.M. 2013. Ayn Rand – The Russian Radical. 2nd ed. University Park, PA: Penn State University Press.

Walford, D., translator, 1992. Immanuel Kant – Theoretical Philosophy 1755–1770. New York: Cambridge University Press.

 

 

Edited February 17 by Boydstun

[Boydstun]

METAPHYSICS AND GEOMETRY

Introduction

Aristotle – A Cognitive Imagination – Part 1 of 2

Aristotle – A Cognitive Imagination – Part 2 of 2

 (I’ve divided the remainder of the Aristotle Part of this study into two sections 2a and 2b. References for both follow the latter.)

~2a~

“While the move toward formalization in mathematics antedates Plato’s philosophical reflections, scientific geometry emerges in Plato’s time as a privileged field for a philosophical battle between sophistry and science” (Humphreys 2023, 19). Aristotle’s conceptions of a demonstrative science are likely spun in part from practice of the geometers of his time (Knorr 1986, Chapters 1–4; McKirahan 1992, Chapters 11–12). Euclid continues that practice in his organization of geometry. Aristotle did not live so long as to see Euclid’s Elements. It is reported by ancient sources that Euclid twice visited the Academy of Plato; there is no report that Euclid ever visited Aristotle’s school or that he ever read Aristotle (Heath 1925, 116–24; Netz 1999).

Aristotle conceived of theorem 2R as holding for all triangles because in the proof (e.g. in Euclid I.32) we are conceiving of triangle qua triangle. The triangle we have drawn at the start stands for triangle qua triangle. Triangle is a figure shape, and shape is independent of size. A triangle one draws in geometry stands for triangle as a shape regardless of the size of the drawn triangle.

Geometry examines lines and figures with the material composition of the lines also regarded as irrelevant. Lines connecting three stars reflected in still water are not fundamentally lines of water molecules. The water is irrelevant to the lines one envisions connecting those points and forming a triangle. They are lines in space—lines actual in my own view—which happen to be occupied by water molecules. Furthermore, having such a triangle in mind does not mean for Aristotle (or me) that in each occasion of having it in mind one has to have abstracted it from some particular matter attending the occasion. (On this last, I contradict Humphreys 2023, 75–76).

Beyond abstraction from matter and size, to assure that we have found a truth of all triangles, we should demonstrate the truth for right triangles, acute ones, and obtuse ones, which is one exhaustive division of the kind of triangles there are. We’d reach a theorem for all triangles even if for each of the kinds of triangles exhausting the kinds we needed to invent a new proof of the theorem.

The triangle drawn by Euclid for Prop. I.32 (2R) is an acute one, but one can also draw a right triangle or an obtuse one, and, following Euclid’s same instructions for labeling and auxiliary construction, the result 2R will again follow by the very same steps of reasoning. Then Euclid’s proof of 2R suffices for all triangles.

Aristotle refers to geometers’ proof of 2R (e.g. APr. 4a17–74b4; 84b6–9). He does not include a diagram, although he alludes to the auxiliary constructions required in geometry, and I suppose he expected his students would know such proofs. 

“Why is the sum of the interior angles of a triangle equal to two right angles? Because the angles about one point [in a straight line] are equal to two right angles. If the [auxiliary] line parallel to the side had been already drawn [rather than being only a potential before its construction], the answer would have been obvious at sight” (Metaph. 1051a24–27). Where H. Tredennick has “obvious at first sight,” W. D. Ross has “evident to any one as soon as he saw the figure.” C.D.C. Reeve has “immediately clear on seeing it.” If Aristotle thought that the finalization of the proof was an immediate “seeing” from the figure with its auxiliary constructions, without further discursive steps of argument, then he was not an A-student of geometry, and the sort of imagination he is enlisting as element in geometric proofs is merely insight-imagination. In that case, Aristotle is failing to notice the imagination-tool of coming up with a proof-enabling auxiliary construction in geometry.

If Aristotle were being literal about “obvious at sight” or “evident as soon as seeing the figure” or “immediately clear on seeing the figure,” then he would as well miss the need for discursive proof that the geometer’s procedure for constructing a figure such as an equilateral triangle to certify that the procedure does indeed deliver what it promises. That is, if we take Aristotle literally in the quotation at Metaphysics 1051, he did not discern the power of and need for what I’ve called Aux imagination.

Proofs of 2R and some other theorems of Euclid were found by other, somewhat later geometers that do not require introduction of auxiliary constructions in the proofs. There is no showing, however, that such Aux-free proofs are possible for all the theorems. Additionally, proofs of constructions, such as construction of an equilateral triangle (Prop. I.1), employed in proofs of theorems (e.g. 2R), demonstrations of solutions to what are known as Problems in Euclid, would have to be free of auxiliary constructions in order to deny any need for Aux in Euclid’s geometry.

Aristotle had some ambivalence as to the auxiliary constructions in geometry (PrAn. 50a1–4). This was not plausibly due to prescience of Hilbert’s profound redesign of Euclidean geometry in 1899. More likely Aristotle was only suspecting that, in some way never shown, the essence of geometric proofs was first-figure syllogism (infra. further, Webb 2006, 197–215).

The proof of 2R shown by Euclid calls for two auxiliary lines: one extending side BC of the triangle, the second being a line through C and parallel to the side BA of the triangle. In the passage from Aristotle, he mentions only one line: a line parallel a side of the triangle. There is another proof of 2R besides the one landing in Euclid’s Elements, a proof also good for all triangles. It is thought to be from Pythagoras. It has only a parallel line in the auxiliary construction. It is quite possible Aristotle and his audience were handy with this proof and not so familiar with the one to be used by Euclid a bit later. The Pythagorean proof of 2R (see p. 320 of Heath vol. 1) also requires dispositive discursive argument, not simply insight-imagination upon viewing the diagram with the auxiliary line. 

I rather suppose Aristotle in this passage and others is only skittering a stone across the pond in mention of 2R proof, that he is an A-student of geometry, and that he is after (in de An.) a cognitive imagination tuned tightly to Euclid-style proof in geometry, which would not be mere insight-imagination. Imagination is the regular translation of Aristotle’s phantasia in passages not sensibly translated as merely appearance. In our contemporary terminology, Aristotle’s term phantasia as cognitive imagination is a presentation to consciousness, whether immediate or in memory and whether true or illusory. My talk of cognitive imagination addressed in De Anima is Aristotle’s phantasia in its relation to conceptual, discursive, assertive thought. Aristotle did not conceive of this phantasia as “one of a number of distinct ‘inner senses’ or as a specific faculty located in the brain [which is from Galen]” (White 1985, 484). Phantasia in Aristotle does not include our notion of imagination in an “extraordinary power of insight” (ibid.).

Aristotle claims that geometry proofs have the form of first figure syllogisms (APo. 79a17–32; 94a20–35; APr. 41b13–31; Lear 1980, 12–14; Webb 2006, 198; Mendell 1998, 183–86). In Prior Analytics, Aristotle had used the plain correctness of first-figure inference for base, and, by showing the reducibility of the other figures to first, he was able to show the correctness of all the syllogistic inferences. He set on that base also proofs per impossibile (APr.  29b1–25; 40b17–41b5). Perhaps Aristotle’s impression that geometry proofs fit the form of first figure syllogisms also raised the possibility that first figure—first figure which he alleges to be greatest figure in explanatory power—was the source of the manifest correctness in geometry proofs. Nowhere did Aristotle show or attempt to show that any proof of the geometers could be put into merely a syllogistic form.

Mendell 1998 advances reasons that would enable Aristotle to at least maintain that geometric proofs can be syllogistic under an expansion of Aristotle’s theory of the syllogism, which Aristotle could approve. Still, Mendell concludes, such a rendering of the geometric-proof way, as in Euclid, to telling the why of geometric truths as entirely demonstrative syllogisms is not attainable (note Salmieri 2016, 302n16; for the whole story, from Aristotle to de Morgan and beyond, see Mancosu and Mugnai 2023).

Roger Bissell attempts rendition of Euclid’s Proposition I.2 using diagram elements as terms in his syllogism (2017, 72–76). Bissell does not remark on whether a proof of this Proposition, which is a construction problem with use of auxiliary constructions in demonstrating that the given construction problem has been solved, is also possible by a proof that does not require auxiliary constructions (and whether the proof of Prop. I.1 on which I.2 relies can also be proven without auxiliary constructions). So far as I know, no such auxiliary-free proofs have been found. Bissell does not remark on the questions of whether auxiliary constructions licensed under existential Postulates and Common Notions (Axioms) are co-sources of manifest correctness of inferences in geometry proofs alongside manifest correctness of first-figure logical inference. And whether the freedoms in allowed constructions are sources of universality independently of the logical universal quantifications. And whether the licensed constructions contribute to the necessity and explanatory power of the geometric proof. The material in Bissell 2017 was issued earlier as an e-book which was reviewed in Seddon 2014, but the review does not address this part of Bissell 2017. Evaluation of the Bissell 2017 effort in the relationship of syllogism to geometry needs to be assessed in light of Mancosu and Magnai 2023.

In Rand-Peikoff Objectivism, and in my own view as well, the plain correctness of first figure syllogisms and of the Euclid-type proofs and the plain incorrectness of contradiction are inherited from the ultimate absolute necessity-that: Existence exists and is Identity, this being frame for all getting of truth. I suggest that the universality character of first figure as well as the universality character of Euclid’s proofs is self-same as Rand’s some-any with respect to which-ness (not with respect to the more sophisticated some-any of measure values). For universality in validated construction of an equilateral triangle, we rely on the arbitrariness in the outset: Mark any two points on a paper (some-any of pairs of points). Draw the line segment between them with a straightedge (Postulates 1 and 2). This is the “any given line segment” at the start of certifying the construction method of an equilateral triangle which Euclid will later use demonstrating 2R for all triangles. (See further Netz 1999, Chapter 6, on generality and Chapter 5 on necessity in deduction in Greek mathematics.)

I strikes me that Aristotle’s contention that demonstrations of the geometers can be reduced to demonstrations in the syllogistic is in friction with his discernment of a basic need to posit a new cognitive power, cognitive imagination (what I’ve called Aux imagination), to accomplish geometry proofs such as in Euclid. Nevertheless, Aristotle does so posit a cognitive imagination to this purpose, to which power we return in 2b. 

(Continued.)

[Boydstun]

(Continued.)

~2b~

Aristotle situates a cognitive imagination between perception and judgment. He writes:

“Since there is no separate thing apart from perceptible magnitude (as it seems), it is in perceptible forms that objects of thought exist, both those spoken of in abstraction and as many as are states and affections of perceptible objects. And for this reason one could not learn or comprehend anything without perceiving something; and whenever one contemplates, one must at the same time contemplate a sort of image, for images are like percepts, except that they are without matter.

“But imagination is different from assertion and denial; for the true or false is an interweaving of thoughts. In what way, then, will primary thoughts differ from images? Rather even these will not be images, though they will not exist without images” (de An. 432a4–14).

Translator Fred D. Miller, Jr. explains that the “spoken of in abstraction” would include geometrical figures, which are things having no perceptible matter. “States and affections” refers to perceptible characteristics of bodies. Miller suggests that Aristotle’s qualification “as it seems” may be in consideration of Aristotle’s view that the Prime Mover has no magnitude (which, I add, is fitting for something that does not exist). It is debated whether “at the same time contemplate a sort of image” means image as object of thought or a vehicle of thought. In dreaming imagination acts very like sensory perception in waking hours, and these sensory-like elements are drawn from sensory perceptions of waking hours (de An. 429a1–2; Insomn. 459a17). 

Imagination is appearance, “but not everything that appears is true,” unlike contents of sensory perception (Metaph. 1010b3). In Aristotle’s view, sensory perceptions receive form of their objects (de An. 424a17–20). For this sense of form, I concur for vision, and, like Aristotle, I take infirm form in dreams to be from determinate form in waking sensory perception. 

I mentioned that Aristotle thought of lines we might draw as already in space as potentials, and our drawing of them makes them actual. In contrast I held forms in elementary geometry to be actuals only. What is made actual from potential in drawing a line for geometry is only the making of an iconic representation of the perfect lines and figures of space.

In Aristotle’s idea of sensation, forms of things are received without their material (de An. 429a16–17). Today we might cash such received form without matter as the spatial elements from a visual scene that the pre-attentive visual system processes, which are then assembled automatically into percepts with spatial layout. Our wide contemporary difference with Aristotle is that whereas Aristotle has form of objects set in activity of mind free of all material of one’s body, we have object form (standings in spatial and temporal relations) captured in shadowing neural activity patterns.

Furthermore, Aristotle did not know of the nervous system, its activities, and its functions. He thought only a form component of objects in the world, not their matter components, could be brought into mind by the senses (de An. 431b27–432a). Aristotle speculated that two contraries in the world, hot-cold and wet-dry, in their various combinations yield the elements earth, water, air, and fire (GC 330a30–330b5). (Sense organs are composed of air and water, and they have vital heat [de An. 424b30–31; 425a6–7].) He took heat to be in matter, but because we sense heat, it is a form in material, not itself material and not matter in his metaphysical hylomorphism. 

We now know that heat is a type of physical energy, and just as heat passes from one object to another, so it passes from objects to one’s body. In the skin are neural receptors that respond to the various rates of heat flow into or out of one’s body. Those nervous patterns of activations are sent to the brain where we experience them as level of warmth or coolness, according to direction and rate of heat flow. (Other receptors respond to burning or freezing yielding pain.) There is not some formal aspect of physical heat in objects needed for us to feel heat. Our perceptual system takes in formalities belonging to a perceptible object, but some matter as well (viz., varieties and patterns of energy). Whether formalities or matter in the world, the mind takes them in transduced to neuronal, physical activities, the very stuff we are made of as mind.

All the same, we can join Aristotle in the idea that mental acts of imagination and judgment concerning an object, unlike sensory perception of that object, proceed independently of whether the object remains physically present for interaction. Aristotle suggests that 

“receptions and phantasiai remain in the sense organs even when the sensible objects have gone (de An 425b23–25). This lingering, resonating, echoing presence of sensible forms freed from their original matter appears to be the essence of phantasia for Aristotle. The movement in the sense-power caused by the sensible object itself sets up a second movement which continues after the sensation has ceased and the object has gone. The animal soul, therefore, is not merely one which is in touch with its environment through cognition of sensation, but one which carries the reverberating effects of its past encounters with sensible objects; it is not only a receiver, but also a preserver and a storehouse, of the sensible forms of things.” (White 1985, 498)

Straightness of a board is something we discern in sensory perception in sighting from an end of the board down the length of the board. Aristotle rightly understood that in pure geometry we are working with straightness itself apart from any material possessing some degree of straightness. Engagement of straightness (a form) in geometry is work not of sensory perception, but of cognitive imagination and discursive thought (de An. 429b10–23; White 1985, 500). 

Aristotle realized that higher animals possess higher levels of memory and some low levels of imagination for their way of life in the world (Nussbaum 1979). He saw human reason as requiring sensory perception and high cognitive imagination. Why does reason require the imagination cohort, not only the sensory perception cohort? (I’ll be asking this question also of Kant with respect to intuition in place of imagination in the Kant portion of this study.) Cognitive imagination entails ability to lift away from the inherent passing and altering in time of the objects of perception their essences and mathematical characters. Such have the relative timelessness suitable to be objects in the workshop of reason (White 1985, 501).

A compass, I say, gives us the power to transport iconic representations of identical line segments from one location to another. Together such iconic representations yield iconic representation of spatial relations of the two distinct perfect lines. Holding the opening of compass constant, the line segment that is the radius of circles drawn by the compass remains constant wherever in the plane a circle is drawn with that set compass. Then too, using compass and straightedge, one can construct a line parallel to a given line. That is, with these tools, there are operations by which lines having the same orientation of a given line in a plane can be represented by these tools constructing icons in their relationships to one another. With these tools, iconic representations of line segments of same length or of same orientation in the plane are enabled. Geometry in Euclid’s manner, I say, is a system of sounding for the structure of pure space, structure between the perfect natural lines of physical space.

By such transports, situations of points and lines in a plane with other points and lines in that plane are specified. Theorems derived using such transports further specify character of the plane, which is to say in my metaphysics, theorems further specify formal structure of the situation of concretes.

(Next will be the Kant Part [intuition in Kant’s special sense] of this study, which, like the Aristotle Part, will require a few months of work.)

(To be continued.)

 

References

Aristotle c. 345–322 B.C.E.

    Prior Analytics – Barnes 1984.

    Posterior Analytics – Barnes 1992. 

    Coming to Be and Passing Away – Reeve 2023.

    De Anima (On the Soul) – Miller 2018. 

    De Insomniis (On Dreams) – Miller 2018.

    Metaphysics – Reeve 2016.

Auxier, R.E. and L.E. Hahn, editors, 2006. The Philosophy of Jaakko Hintikka. Chicago: Open Court.

Barnes, J., editor, 1984. The Complete Works of Aristotle. Princeton: Princeton University Press.

Barnes, J., translator, 1992. Aristotle – Posterior Analytics. 2nd ed. New York: Oxford University Press.

Bissell, R.E. 2017. How the Martians Discovered Algebra: Explorations in Induction and the Philosophy of Mathematics. CreateSpace.

Gotthelf, A. and G. Salmieri, editors, 2016. A Companion to Ayn Rand. Chichester, West Sussex: Wiley-Blackwell.

Heath, T.L., translator, 1925. Euclid – The Thirteen Books of The Elements. 2nd ed. New York: Dover. 

Humphreys, J. 2023. The Invention of Imagination – Aristotle, Geometry, and the Theory of the Psyche. Pittsburgh: University of Pittsburgh Press. 

Knorr, W.R. 1986. The Ancient Tradition of Geometric Problems. New York: Dover.

Lear, J. 1980. Aristotle and Logical Theory. New York: Cambridge University Press.

Mancosu, P. and M. Mugnai 2023. Syllogistic Logic and Mathematical Proof. New York: Oxford University Press.

McKirahan, R.D. Jr. 1992. Principles and Proofs – Aristotle’s Theory of Demonstrative Science. Princeton: Princeton University Press.

Mendell, H. 1998. Making Sense of Aristotelian Demonstration. In Taylor 1998.

Miller, F.D. Jr., translator, 2018. Aristotle On the Soul and Other Psychological Works. New York: Oxford University Press.

Netz, R. 1999. The Shaping of Deduction in Greek Mathematics. New York: Cambridge University Press.

Nussbaum, M.C. 1979. The Role of Phantasia in Aristotle’s Explanation of Action. In Aristotle’s De Motu Animalium. Princeton: Princeton University Press.

Reeve, C.D.C., translator, 2016. Aristotle Metaphysics. Indianapolis: Hackett.

——. translator, 2023. Aristotle’s Chemistry – On Coming to Be and Passing Away and Meteorology 1.1–3, 4.1–12. Indianapolis: Hackett.

Salmieri, G. 2016. The Objectivist Epistemology. In Gotthelf and Salmieri 2016.

Seddon, F. 2014. E-Book Enthusiasm. Journal of Ayn Rand Studies 14(2):275–81.

Taylor, C.C.W., editor, 1998. Oxford Studies in Ancient Philosophy, vol. XVI. Oxford: Clarendon Press.

Webb, J.C. 2006. Hintikka on Aristotelian Constructions, Kantian Intuitions, and Peircean Theorems. In Auxier and Hahn 2006.

White, K. 1985. The Meaning of Phantasia in Aristotle’s De Anima, III, 3–8. Dialogue 24:483–505.

Edited March 14 by Boydstun