VIII. Epistemological Issues in Mathematics
The following are a few reflections on the Philosophy of Mathematics, which I venture to offer although not a mathematician, having over time encountered treatments of issues that as a philosopher and logician I found questionable. The assault on reason throughout the 20th Century has also had its effects on the way philosophers of mathematics understood the developments in that subject. Having a different epistemological background, I can propose alternative viewpoints on certain topics, even while admitting great gaps in my knowledge of mathematics.
Attending lectures on the work of Jean Piaget, I was struck by the confusion between logic and mathematics in his identification of learning processes. Some that I would label as mathematical, he labeled as logical; and vice versa. This is of course due to the blurring of the distinction found in a lot of modern logic. There are two aspects to this issue, according to the direction of viewing.
a) Mathematics is used in logic. Mathematics, here, refers mainly to arithmetic and geometry; for instances, in considerations of quantity (or more broadly, modality) in the structure of propositions or within syllogistic or a fortiori arguments.
b) Logic is used in mathematics. Logic is here intended in a broad sense, including the art (individual insights) and the science (concepts, forms and process) of logic; for instance, logic is used to formulate conditions and consequences of mathematical operations.
For example, the statement “IF there are 100 X at time t1 AND there are 150 X at time t2, THEN the rate of change in number of X was (150 – 100)/(t2 – t1) per unit time.” Here mathematical concepts (the numbers 100, 150, t1 and t2) are embedded in the antecedent (if) of a hypothetical proposition (implication), and additionally a formula (viz. (150 – 100)/(t2 – t1)) for calculating a new quantity is embedded in the consequent (then), derived from the given quantities.
The logical part of that statement here is the “if-then-“ statement. What makes it logical is that it is a form not limited to mathematics, but which recurs in other fields of knowledge (physics, psychology, whatever). It is a thought process (the act of understanding and forming a proposition) with wider applicability than mathematical contexts; it is more general.
The mathematical part of said statement is the listed numerical concepts involved and the calculation based on them – the operations involved (in the present case, two subtractions and a division. The insight that the proposed formula indeed results in the desired knowledge (the resulting quantity) belongs to mathematics. Logic here only serves to conceptually/verbally express a certain relation (the implication) established by mathematical reasoning.
We should also note the mathematical elements found in defining the “if-then-“ form – notably appeal to a geometrical example or analogy of overlapping circles (Euler or Venn diagrams). Nevertheless, there clearly remains in such forms a purely logical, in the sense of non-mathematical, element; such explanations cannot fully express their meaning. The quantitative part is merely the visible tip of the iceberg of meaning; the qualitative – more broadly conceptual – part is a more difficult to verbalize and so relatively ignored aspect.
Of course, we can also say that in the largest sense of the term logic – discourse, thought process – even mathematical reasoning is logic. The division is ultimately artificial and redundant. Nevertheless, these subjects have evolved somewhat separately, with specialists in mathematics and specialists in more general (or the rest of) logic. It is also probable, judging by the work of Jean Piaget and successors in child learning processes, that different logical or mathematical concepts and processes are learned at different ages/periods of early childhood, and there are variations in temporal order from one child to another.
Historically, it is a fact that we have adopted the separation of these investigations and a division of labor, so that logic and mathematics have been considered distinct subjects of study. Of course, there has been much communication and intertwining between these two fields, and indeed attempts at merger. Here, I merely want to indicate where the boundaries of the distinction might lie. Specifically quantitative concepts and operations are mathematics; whereas logic deals with thought processes found in other fields besides. In this view, mathematics is quantitative discourse, whereas logic is (also) non-quantitative discourse.
By making such fine distinctions, we can for instance hope to better study human mental development.
The idea that mathematical systems such as Hilbert’s are “axiomatic” – that is, pure of any dependence on experience is a recurring myth, which is based on an erroneous view of how knowledge of this field has developed. I have discussed the source of this fallacy at length in my Future Logic (see chapter 64, among others); here I wish to make some additional, more specific remarks.
I do not deny that Hilbert’s postulates are mutually consistent and by themselves sufficient to develop geometrical science. My objection is simply to the pretentious claim that his words and propositions are devoid of reference to experience. We need only indicate the use of logical expressions like “exists,” “belonging,” “including,” “if – then –,” etc., or mathematical ones like “two,” “points” “line,” etc., to see the dependence.
Take for example the concept of a group (to which something “belongs” or in which something is “included”). The concept is not a disembodied abstract, but has a history within knowledge. The idea of grouping is perhaps derived from the practice of herding animals into an enclosure or some such concrete activity. The animals could all be cows – but might well be cows mixed with goats and sheep. So membership in the group (presence in the enclosure) does not necessarily imply a certain uniformity (a class, based on distinctive similarity – e.g. cows), but may be arbitrary (all kinds of animals, say). Thus, incidentally, the word group has a wider, less specific connotation than the word class (which involves comparison and contrast work). Without such a physical example or mental image of concrete grouping, the word would have no meaning to us at all. So, genetically, the word grouping – and derived expressions like belonging or including, etc. – presupposes a geometrical experience of some sort (a herding enclosure or whatever). We cannot thereafter, after thousands of years of history of development of the science of geometry, claim that the word has meaning without reference to experience. Such a claim is guilty of forgetfulness, and to claim that geometry can be built up from it is circular reasoning and concept-stealing.
It would be impossible for us to follow Hilbert’s presentation without bringing to mind visual images of points, successions of points, lines crisscrossing each other, this or that side of a line, etc. Those images at least are themselves mental objects in internal space, if not also end products of our past experiences of physical objects in external space. The value and justification of Hilbert’s work (and similar attempts, like Euclid’s) is not that is liberates geometry from concrete experiences of objects in space, but merely that it logically orders geometrical propositions so that they are placed in order of dependence on each other (from the least to the most).
Geometrical “axioms” are thus not absolutes somehow intuited ex nihilo, or arbitrary rules in a purely symbolic system, but hypotheses made comprehensible and reasonable thanks to experience. That experience, as I argue below, need only be phenomenal (it does not ultimately matter whether it is “real” or “merely illusory”) but it needs to be there in the first place. That experience does not have to give us the axioms ready-made – they remain open to debate – but it gives us the concepts underlying the terms we use in formulating such axioms. In this sense, geometry – and similarly all mathematics – is fundamentally empirical (in a phenomenological sense) – even if much rational work is required beyond that basic experience to express, compare and order geometrical propositions.
It is futile to attempt to avoid this observation by talking of succession of symbolic objects, A, B, C. Even here, I am imagining the symbols A, B, C in my mind or on paper as themselves concrete objects placed in sequence next to each other! I am still appealing to a visual – experiential and spatial – field. Thus, any claim to transcend experience is naïve or dishonest. Experience is evidently a sine qua non for any axiomatization, even though it is clearly not a sufficient condition. The experiences make possible and anchor the axioms, but admittedly do not definitely prove them – they remain hypotheses. Geometry is certainly not as some claim a deductive science, but very much an inductive one, and the same is true of other mathematical disciplines.
3.1 The so-called axioms of geometry have changed epistemological status in history as follows:
a) At first, they seemed obvious, i.e. immediately proved by experience (naïve view). But the naïve view, not being based on reflection, is rejected as such once reflection begins.
b) Then they were regarded as axioms, i.e. theses without possible credible alternatives (axiomatic view). But this view, which is a worthy attempt to justify the preceding, suffers upon further reflection from an apparent arbitrariness. The label “axiom” is found to be a pretentious claim to an absolute – when denial of it does not result in any contradiction.
c) Then it was considered that they were merely credible hypotheses among other possibilities, i.e. that alternative hypotheses were conceivable and possibly credible (hypothetical view). One can even imagine that different geometries might be applicable in different contexts, and regard the Euclidean model as approximately representative on the human everyday scale of things, and thus consider that all or many of these alternative hypotheses are equally credible.
d) Then they were thought to be pure inventions of the human mind, incapable of either verification or falsification (speculative view). This view may at first sight seem epistemologically unacceptable, since it claims to transcend the hypothetical view and posits to know a truth that is by definition beyond our testing abilities. However, it must be understood in the context of the doubt in the existence of geometrical points, lines or surfaces. That is, it is a denial of geometrical science as such.
However, as we shall see, these latter criticisms can themselves be subjected to rebuttal, especially on phenomenological grounds.
3.2 The arguments put forward against geometrical science as such are indeed forceful. We have considered the main ones in the section on ‘Unity In Plurality,’ pointing out that physical objects do not, according to modern physical theories based on scientific experiments, have precise corners or edges or surfaces, but fuzzy, arbitrarily defined limits, so that we are forced to admit all things as ultimately just ripples in a single world-wide entity.
There might be a fundamental weakness in such argumentation – a logical fault it glosses over. If the whole of modern physical science is itself based on the existence and coherence of geometrical science (by which I of course do not mean only Euclidean geometry, but all the discipline developed and accepted over time by mathematicians), can it then turn around and draw skeptical conclusions about that Geometry? Remember, all the mathematics of waves and particles, of space and time, were used as premises, together with empirical results of physical experiments, to inductively formulate and test the physical theories we currently adhere to – can the latter physical conclusions then be used to argue against these very mathematical premises?
Logically, there is no real self-contradiction in this. The sequence is “Math theory” (together with empirical findings) implies “Physical theory” that in turn implies doubt on initial “Math theory.” So what we have in fact is denial of (part of) the antecedent by the consequent, which is not logically impossible, though odd. The consequent is not denying itself, although it puts its own parent in doubt.
Thus, a more pondered and moderate thesis about geometry has to be formulated, which avoids such difficulties while taking into account the aforesaid criticisms regarding points, lines and surfaces. Waves and particles (which are presumably clusters of waves) may somehow be conceivable and calculable, without heavy reliance on the primary objects of our current geometry (points, lines and surfaces), which apparently have no clear correspondence in nature. In the meantime, our current geometry can legitimately be used as a working hypothesis, since it gives credence to our physical view.
3.3 Let us now consider where the extreme critics of geometry may have erred. We can accept as given the proposition that no dimensionless points, no purely one-dimensional lines, no purely two-dimensional surfaces (Euclidean or otherwise) can be pointed to in natural space-time accessible to us.
This is granting that to exemplify such primary objects of geometry we would need to find material objects with definite tips, edges or sides – whereas we know that all material objects are made of atoms themselves made of elementary particles themselves very fuzzy objects, apparently subject to Heisenberg’s Uncertainty principle.
Nevertheless, we tend to regard the ultimate nature of these nondescript bodies to be clusters of “waves of energy”. This is of course a broad statement, which ignores the particle-wave predicament and which rushes forth in anticipation of a unified field theory; furthermore, it does not address the question regarding what it is that is being waved, since the Ether assumed by Descartes has since the experiments of Michelson and Morley and Einstein’s Relativity theory been (apparently definitively) discredited.
But my purpose here is not to affirm this wave view of matter as the ultimate truth, but rather to consider the impact of supposing that everything is waves on our question about the status of geometry. For if particles are eventually decided to be definitely not entirely reducible to waves, then geometry would be justified by the partial existence of particles alone; so the issue relates to waves.
If we refer to the simplest possible wave, whatever it be, a gravitational field or a ray of light – it behaves like a crease or dent in the fabric of the non-ether where waves operate (to use language which is merely figurative). Such hypothetical simplest fractions of waves surely have a geometrical nature of some sort. That is to say, if we could look that deep into nature, we would expect to discern precise points, lines and surfaces – even if at a grosser level of matter we admittedly cannot.
Thus, I submit, the possible wave-nature of all matter is not really a forceful argument against geometry. Even if we can never in practice precisely discern points, lines and surfaces, because there may be no material bodies of finite shape and size, geometry remains conceivable, as a characteristic of a world of waves.
All the above is said in passing, to clear out side issues, but is not the main thrust of my argument in defense of geometry. We admittedly can perhaps never hope to perceive waves directly, i.e. our assumption of their geometrical nature is mere speculation. But that is not an argument of much force against geometry as such, in view of its existence and practical successes, which mean that geometry is not speculation in the sense of a thesis incapable of verification or falsification, a pure act of faith, but more in the way of a hypothesis that is repeatedly confirmed though never definitely proved. Simply an inductive truth – like most scientific truths about nature!
But let us consider more precisely how geometry actually arises in human knowledge. It has two foundations, one experiential (in a large sense) and the other conceptual.
3.4 The experiential aspect of geometrical belief is that there seems to be points, lines (straight or curved), surfaces (flat or warped) and volumes (of whatever shape) in the apparently material world we sense around us as well as in the apparently mental world of our imaginings. This seeming to be is enough to found a perfectly real and valid geometry. The justification of geometry is primarily phenomenological, not naturalistic!
Seeming is (I remind you) the appearance, or (in this case) phenomenal, level of existence, prior to any judgment as to whether such phenomenon is a reality or an illusion. In other words, geometrical objects do not have to be proven to be realities – in the sense of things actually found in an objective physical nature – they would be equally interesting if they were mere illusions! Because illusions, too, be they mere ‘physical illusions’ (like reflection or refraction) or mental projections, are existents, open to study like realities.
The study of phenomena prior to their classification as realities or illusions is called phenomenology. At the phenomenological level, ‘seeming to be’ and ‘being’ are one and the same copula. Only later, on the basis of broad, contextual considerations, is a judgment properly made as to the epistemological status of particular appearances, some being pronounced illusions, and the remainder being admitted as realities. If, therefore, geometrical science has a phenomenological status, i.e. if it is a science that can and needs be constructed already at the level of phenomena, it is independent of ultimate discoveries about the physical world.
The mere fact, admitted by all, including radical critics of geometry, that we get the impression, at the human everyday level of perception, that a table has four corners and sides and a flat top, suffices to justify geometry. This middle-distance depth of perception, even if it is ultimately belied at the microscopic level of atoms or the macroscopic level of galaxies, still can and has to be considered and analyzed. A science of geometry only requires apparent points, lines and surfaces.
And even if this last argument were rejected, saying that the points, lines and surfaces we seem to see in our table are just mental projections by us onto it, we can reply that even so, mental projections of points, lines and surfaces are themselves real-enough objects existing somehow in this world. They may be illusions, in the sense that they wrongly inform us about the external world, they may be purely internal constructs, but they still even as such exist. A subjective existent is as much an existent as an objective one – in the sense that both are equally well phenomena.
The mental matrix of imagination, at least, must therefore be capable of sustaining such geometrical objects. And if this restricted part of the world – our minds – displays points, lines and surfaces – then geometry is fully justified, even if the rest of the world – the presumed material part – turns out to be incapable of such a feat and geometry turns out to be inapplicable to it.
But the latter prospect thus becomes very tenuous! As long as geometry could be rejected in principle, by the elusiveness of its claimed objects under the microscope, there was a frightening problem. But once we realize that the very existence of Geometry requires the possibility somewhere of the concretization of its objects – even if only as a figment of our imaginations – the problem is dissolved. In short, our very ability to discuss geometrical objects, if only to doubt their very existence, is proof of our ability to at least produce them in the mind, and therefore of their ability to exist somewhere in this world. And if all admit that geometrical objects can exist in some part of the world (the mental part at least), then it is rather inductively difficult and arbitrary to deny without strong additional evidence that they exist elsewhere (in the material part). The onus of proof reverts to the deniers of material geometry.
3.5 The conceptual aspect of geometrical belief must however be emphasized, because it moderates our previous remarks concerning the experiential aspect. Conceptualization of geometrical objects has three components, two positive ones and a negative one.
a) The primary positive aspect of geometrical conception consists of rough observation, abstraction and classification, (i) refers to the above mentioned concrete samples of points, lines, surfaces and volumes, apparent in the material and mental domains of ordinary experience – this is phenomenological observation; and (ii) observes their distinctive similarities (e.g. that this and that shape are both lines, even though one is straight and short and the other is long and curved, say) – this is abstraction; and (iii) groups them accordingly under chosen names – this is classification.
b) The negative aspect of geometrical conception is the intentional act of negation, reflecting the inadequacy of mere reference to raw experience. Unlike their empirical inspirations, a theoretical point has no dimension (no length, no breadth, no depth); a theoretical line is extended in only one dimension – it has no surface; a theoretical surface in only two dimensions – it has no volume. Each theoretical geometrical object excludes certain empirical extensions. It is thus an abstraction (based on concretes, of course) rather than a pure concrete.
As I have explained elsewhere, negation is a major source of human concepts, allowing us to form them without any direct experience of their objects. That is, while the concrete referents of “X” may be directly perceivable; those of “Non-X” need not be so. We consider defining them by negation of X as sufficient – since every thing (except the largest concept “thing”, or existent) has to have a negation, since every thing within the universe is limited and leaves room for something else.
Such negative definition of the geometrical objects is not, however, purely verbal or a mere conjunction of previous concepts (“not” + “X”). There is an active imaginative aspect involved. I mentally, or on paper, draw a point or a line, and mentally exclude or rub-off further extensions from it. Thus, even if my mental matrix, or my pencil and paper, may be in practice unable to exemplify for me a truly dimensionless point or fine line or mere surface, I mentally dismiss all excessive thickness in my sample. This act may be viewed as a perceptual equivalent of conceptual negation.
c) Another, more daring positive conceptual act may be called assimilation, which we can broadly define as: regarding something considerably different as considerably similar. This a more creative progression by means of somewhat forced simile or analogy, through which we expand the senses of terms.
For example, the concept of a “dimension” of space is passed on to time. The Cartesian fourth dimension is at first perhaps thought up as a convenient tool, but eventually it is reified and in Einstein we find it cannot be dissociated from space. Our initial concept of dimension has thus shifted over into something slightly different, since the time extension of bodies is distinctively one-directional and not as visible as their space extensions (see more on this topic in earlier chapters).
Another example is the evolution from Euclidean geometry, the first system that comes to mind from ordinary experience (and in the history of geometrical science), to the later Non-Euclidean systems. A shape considered as “curved” in the initial system is classed as “straight” or “flat” in another system. We have to assimilate this mentally – i.e. say to ourselves, within this new geometrical system, straightness or flatness has another concrete meaning than before, yet the role played by these previously curved shapes in it is equivalent to that played by straight lines or flat surfaces Euclidean system.
Note well how ordinary experience of everyday events and shapes are repeatedly and constantly appealed to by the mind in all three of the above conceptual acts. It is important to stress this fact, because some mathematicians try to ignore such experiential grounding and cavalierly claim that what they do is independent of any experience. The whole of the present essay is intended to belie them, by increasing awareness of the actual genetic processes underlying the development of mathematical sciences.
The academic exercise of formulating the starting assumptions (“axioms”) of the various geometrical systems does not occur in a vacuum. In order to understand whether “parallels” meet or not, I visualize ordinary (Euclidean) parallels, then imagine them curving towards each other or curving apart; then I say “even though they meet or spread apart, I may still call them parallel within alternative geometrical systems”. Without some sort of concretization, however forced, the words or symbols used would be meaningless.
3.6 Finally, I’d like to mention here in passing that many of the remarks made here about geometry apply to other fields of mathematics. Thus, arithmetic should also be viewed as a phenomenological science. That is, its primary objects – the unit (“1”) and growing collections of such units (“2”, “3”, etc.) – that is, natural, whole, positive, real numbers – do not require any reference to an established “reality,” but could equally be constructed from a sense field (visual or other) composed entirely of illusory events or entities. It is enough that something appears before us to concretely grasp a unit, and many things, to concretely grasp the pluralities.
By arithmetic entities, we initially mean units and pluralities (the natural numbers). These objects, which are not unrelated to geometrical objects, need only be phenomenal. One can conceptualize a unit and pluralities of units equally well from an illusory or imaginary field of perception as from a real one. The sense-modality involved is also irrelevant: shapes, sounds, touch-spots, items smelt or tasted – any of these can be units.
What is the epistemological status of novel arithmetical entities? Some mathematicians apparently claim that a concept like the negative number –1 or the imaginary number √-1 is a “new entity” incapable of being reduced to its constituent operations (-, √) and numbers (1, etc.). The definitions of such abstract entities are given in series of equations like:
Where –1 + 1 = 0, –2 + 2 = 0, etc….
Where √-1 • √-1 = –1, √-2 • √-2 = –2, etc….
However, this means that the signs used (- , + , = , √ , • , ⁄ , etc.) are each in turn a new thing in each definition, even though presented to us in the same physical form (symbol-shape and name) as existing entities. Here, the sign that was originally an operator (a relational concept between two terms) has become attached to a term (making of it a new term) – so that the sign itself has changed nature.
It seems clear to me that this doctrine of irreducibility and newness, while a good-faith try at explaining the leaps of imagination involved in such mathematical concepts, in fact involves some dishonesty since such definitions tacitly rely on the implicit meanings of the building blocks that are their sources both logically and in the progression and history of thought.
Rather we should, in my view, look at these leaps as indefinite stretching of meaning, i.e. we say: “let this concept (-,√, whatever) be widened somewhat (to an undecided, undetermined extent) so that the following analogy be possible….” This extending of meaning (or intention) is itself imaginary, in that we cannot actually trace it (just we cannot concretize the concept of infinity by actually going to infinity, but accept a hazy non-ending).
(Such development by analogy is nothing special. As I have shown throughout my work, all conceptualization is based on grouping by similarity, of varying precision or vagueness – or the negation of such. Terms are rarely pre-definable, but are usually open-ended entities whose meaning may evolve intuitively as more referents are encountered.)
We thus produce doubly imaginary hypothetical entities. And here an analogy to the concrete sciences is possible, in that the properties of such abstract entities are tested (in accordance with adductive principles), not only logically in relation to conventions and arbitrary laws initially set up by our imagination (as the said mathematicians claim), but also empirically in relation to the properties known to be obtained for natural numbers.
Natural numbers, therefore, do not merely constitute a small segment of the arsenal of mathematical entities (as they claim), but have the status of limiting cases for all other categories of numbers (negatives, imaginaries, etc.). If any proposed new abstract formula does not work for natural numbers, it is surely rejected.
This is evident, for instance, in William Hamilton’s attempted analogy from couples to triplets. He found that though complex numbers expressed as couples (with one imaginary number i2 = –1) could readily be multiplied together, in the case of triplets (using two imaginary numbers i2 = j2 = –1) results inconsistent with expectations emerged when natural numbers were inserted in the formula.
Note particularly this reference to two (or more) different imaginary numbers, namely i and j whose squares are both equal to –1. Here, we introduce j as an imaginary extension of the concept of i that has no distinguishing mark other than the symbolic difference applied to it! We simply imagine that the meaning of j might somehow differ from that of i so that although i2 = j2 = –1 it does not follow that i = j = √-1 (or even that ij = –1). An unstated and unspecified differentia is assumed but never in fact provided. This is yet another broadening of mathematics “by stretching” (i.e. by unsupported analogy, as above explained).
The example here referred to clearly shows that, however fanciful its constructs (by definition and analogy), mathematics undergoes an occasional empirical grounding with reference to natural numbers, which limits the expansiveness of its imagination and ensure its objectivity. New mathematical entities, although initiated by mere conventions or arbitrary postulates, must ultimately pass the test of applicability to natural numbers, i.e. consistency with their laws, to be acceptable as true mathematics. Natural numbers thus fix empirical restrictions on the development of theoretical mathematics.
If I may be allowed some far-out, unorthodox, amateur reflections consider the following concerning fractions of natural numbers.
A physical body can only really be divided into n parts, say, if it has a number of constituents (be these molecules or atoms or elementary particles or quarks or whatever) divisible exactly by n – otherwise, the expression 1/n has no realistic solution!
For example, a hydrogen atom cannot be divided by two, unless perhaps its constituent elementary particles contained an even number of quarks. Or again, if I wanted to divide (by volume or weight) an apple fairly among three children, it would have to have a number of identical apple molecules precisely divisible by three. Otherwise, each child would get 0.333… (recurring) part of an apple – which we have no experimental proof is practically possible and indeed we know is not!
The concept of an infinitely recurring decimal is a big problem – consider the debates about Π (pie) in the history of mathematics. How can I even imagine going on adding digits to infinity, when I know my life, and that of humanity, and indeed of the Universe are limited in time, and when I know that space is physically limited so that there would not be place enough for a real infinity of digits even if there were time enough? Surely, such a concept may be viewed as an antinomy.
What this means is that arithmetic as we know it is not necessarily a thoroughly “empirical” science – it is an ideal assuming infinite divisibility of its objects. The mere fact that I can imagine an apple or atom as divisible at will, does not make it so in the real world. Though in some cases the number ½ or 1/3 may have a real object, a realistic solution, in many cases this is in fact a false assumption. 
Even in the mental domain, although we can seemingly perfectly divide objects projected in the matrix of imagination (whatever its “substance” may be), it does not follow that viewed on a very fine level (supposing we one day find tools to do so) such division is always in fact concretely possible.
These thoughts do not invalidate the whole of arithmetic, but call for an additional field or system of arithmetic where the assumption of infinite divisibility of integers is not granted. That is, in addition to the current “ideal” or a-priori arithmetic (involving “hypothetical” entities, like improper fractions or recurring decimals), we apparently need to develop a thoroughly “empirical” or a-posteriori – one might say positivist – arithmetic, applicable to contexts where division does not function.
The same may of course be said of the related field of geometry. Infinite divisibility is a mere postulate, which may stand as an adopted axiom of a restricted system, but which should not at the outset exclude alternative postulates being considered for adjacent systems. The mathematics based on such postulate may be effective – it seems to work out okay, so perhaps its loose ends cancel each other out in the long run – but then again, the development of other approaches may perhaps result in some new and important discoveries in other fields (e.g. quantum mechanics or unified field theory).
Why should mathematics be exempt from the pragmatic considerations and norms of knowledge used in physics? Can it, like alchemy or astrology were once, be uncritically based partly on fantasies? Surely, every field of knowledge must ultimately be in perfect, holistic accord with every other field and with all experience – to be called a “science” at all. The division of knowledge into fields is merely a useful artifice, not intended to justify double standards and ignorance of seemingly relevant details. Once philosophy has understood the inductive nature of knowledge, it demands severe scrutiny of all claims to a-priori truth and strict harmony with all a-posteriori truths.
We could get even more picky and annoying, and argue that no material (or mental) body is as finite as it appears, as we did in the section on ‘Unity In Plurality.’ Since the limits of all material or mental entities are set arbitrarily, it follows that everything is one and the same thing, and that nothing is at all in fact divisible. However, such (almost metaphysical) reflections need not (and won’t) stop us from pursuing mathematical knowledge, since they gloss over issues to do with causality.
That mathematical science is like all knowledge inductive, and not merely deductive, is evident from any reading of the history of the subject. Mathematicians understand the word induction in a limited sense, with reference to leaps from examples or special cases to generalities (abstractions or generalizations) or to analogies (“as there, so here” statements). But I am referring here to many more processes. Individual mathematicians, as they develop mathematics, use trial and error (adduction), putting forward hypotheses and analyzing their consequences, rejecting some as inadequate. Initially accepted mathematical propositions have often been found mistaken by other or later mathematicians, due for instances to vagueness in definitions or to short-circuits in processing, and duly criticized and corrected.
Mathematicians are well aware of the breadth of their methodology in practice. Mathematics is a creative enterprise for them, quite different from the learning process students of the subject use. The latter have the end-results given them on a platter, so that their approach is much more deductive. Mathematicians do not merely recycle established techniques to solve problems and develop new content; to advance they have to repeatedly innovate and conceive of new techniques.
 Notably in 1998, when I attended certain courses at Geneva University, such as lectures (I forget by whom) on the work of Jean Piaget and others given by Prof. J.-C. Pont on the History of Mathematics. Many (but not all) of the notes in this essay date from those encounters.
 I write this looking at a university handout listing the “axioms (or plan) of Hilbert’s system”, in four groups (belonging, order, congruence and parallels). I was struck with the numerous appeals to “stolen concepts” in it (see Future Logic, chapter 31.2)
 Even purely “logical” if-then- statements depend for their understanding on geometrical experience. When I define “if P, then Q” as “P and nonQ cannot coexist” – I visualize a place and time where P and nonQ are together (overlapping) and then negate this vision (mentally cross it off). One cannot just ignore that aspect of the ideation and claim a purely abstract knowledge.
 As Cantor claims.
 Euclid’s axioms were the first attempted hypotheses, Hilbert and others later attempted alternative hypotheses.
 Note well, this is not a discussion of space and time, but of the discipline called Geometry.
 See chapter IV.5, above.
 Of course, such looking would have to be independent of a Heisenberg effect. A pure act of consciousness without material product. Clearly, this assumes that consciousness is ultimately a direct relation to matter, which transcends matter. Heisenberg’s argument refers to experimental acts, interactions of matter with matter, which we use to substitute consciousness of an effect for that of its cause. The Uncertainty principle is not a principle about consciousness modifying its objects, but about the impossibility of unobtrusive experiment.
 At which stage “is” acquires a more narrow and ambitious meaning than “seems to be”.
 Personally, with reference to terminology used in formal logic, I would say that negative numbers or irrational numbers or imaginary numbers are compounds of copula and predicate. They are artificial predicates, consisting of a normal predicate (final term) combined with the relational factor (copula) to any eventual subject (first term). They “hold-over” or “carry-over” a potential operation – that of subtracting or finding a root or both – until the unstated term (the subject) is specified. Such expressions give rise to a predicate in the original sense (i.e. a number), and disappear, when the operation is actually effected. Their status as effective predicates is only utilitarian. It is interesting to note, in this context, that within general logic, such permutation (as it is called) is not always permissible (see my treatment of the Russell Paradox, in Future Logic chapter 45, for example). For this reason, one should always be careful with such processes.
 Natural numbers have, and thus retain, an exceptional ontological status. Their derivatives are thus inductively adapted to the previously established algebraic properties of natural numbers. The point of all this is, of course, to develop a universally effective algebra – processes and rules that function identically for natural numbers and all their derivatives, uniform behavior patterns.
 Later, he showed that quadruplets or quarternions – involving three imaginary numbers i, j, k – could however be multiplied together. Similarly with an eight-element analogy.
 At a later stage, these different imaginary numbers i, j, k etc. are associated with geometrical dimensions – but such application is not relevant at the initial defining stage.
 I should here repeat that this mental process is not limited to the mathematical field. For instance, in psychology, when we speak of “mental feelings”, as distinct from physical feelings (experienced viscerally, in the chest or stomach or rest of the body, whether of mental or purely physical source), we are engaging in such analogy. By definition, mental feelings (e.g. I like you) have no concrete manifestation that we can point to; we introduce them into our thinking by positing that they are somehow, somewhat similar to feelings experienced in the physical domain, but they occur in the mental domain and are much less substantial (more abstract). The word “feeling” thus takes on a new wider meaning, even though we have no clear evidence (other than behavioral evidence of certain values) for the existence of a mental variety of it. Thus, Mathematics should not be singled out and scolded for using such processes – they are found used in all fields – but it is important to notice where such leaps of imagination occur and acknowledge them for what they are, so that we remain able to test them empirically as far as possible. Incidentally, such leaps are comparatively rare in Logic.
 I spoke of these ideas once, back in April 1998, at a round-table at the Archives Piaget in Geneva.
 We should also perhaps make a distinction between divisibility and separability. Even if I may distinguish a number of equal parts in a body, I may not in fact (by some natural or conventional law) be able to actually isolate these constituents from each other. In which case, what would division of that number by itself factually mean? Would say 5/5 equal 1, or would it be a meaningless formula, without solution? Is 5/5=1 a universal equation or is it only true in specific situations? (By conventional law, I mean for example, when farthings or halfpennies were withdrawn from circulation, a penny could no longer be subdivided in accounting.)
 Clearly, I am using the word “empirical” here in a specific sense. Even “ideal” arithmetic has an empirical basis, in the sense that at least its primary objects – the natural numbers 1, 2, 3, … – are phenomenological givens. But it does not follow that further processes, such as division, always have an empirical basis – hence my use of the adjective thoroughly empirical.
 For all I know, such alternative mathematics already exist. I do not claim to know the field, nor have any desire to seem original or revolutionary. These are primarily philosophical reflections.
 See chapter IV.5, above.
 Which issues I will be dealing with in my forthcoming work on the subject.