## Mathematics Philosophy Physics Metaphysics of Mathematics

### Mathematics and Infinitesimal Calculus Uniting One as both Continuous-Infinite and Discrete-Finite On Truth, Reality and Godel's Incompleteness Theorem

Although I am one of those who have done much work on mathematics, I have constantly meditated on philosophy from my youth up, for it has always seemed to me that in philosophy there was a way of establishing something solid by means of clear proofs.
(Gottfried Leibniz)

Godel's Theorem constituted a major challenge not only to the generally held assumption that basic systems in mathematics are complete in that they contain no statements that can be either proved or disproved, but also to Hilbert's view that proofs of the consistency of such a system can be formulated within the system itself. Briefly, the theorem states that in a formal system S of arithmetic, there will be a sentence P of the language S such that if S is consistent neither P nor its negation can be proved within S. The impact of this on Principia Mathematica (Whitehead, Russell) was to undermine the latter's project of providing a set of logical axioms from which the whole of pure mathematics, as well as the non-axiomatic residue of logic, were deducible, since the theorem showed that mathematics contains propositions that are neither provable nor disprovable from the axioms. ....
This shows that a consistency proof of ordinary arithmetic is not possible using finite procedures.

Mathematical knowledge appeared to be certain, exact and applicable to the real world; moreover it was obtained by mere thinking, without the need of observation. Consequently, it was thought to supply an ideal, from which everyday empirical knowledge fell short. It was supposed, on the basis of mathematics, that thought is superior to sense, intuition to observation. If the world of sense does not fit mathematics, so much the worse for the world of sense. ... Mathematics is, I believe, the chief source of the belief in eternal and exact truth, as well as a super-sensible intelligible world. (Bertrand Russell)

What we call physics comprises that group of natural sciences which base their concepts on measurements; and whose concepts and proportions lend themselves to mathematical formulation. Its realm is accordingly defined as that part of the sum total of our knowledge which is capable of being expressed in mathematical terms. (Albert Einstein, Ideas and Opinions, 1954)

No human investigation can be called true science without passing through mathematical tests. (Leonardo da Vinci, 'Treatise on Painting')

NOTE: I have moved the mathematics section - it is good / well worth reading. See:

http://www.spaceandmotion.com/mathematical-physics/logic-truth-reality.htm

Geoff Haselhurst

## Introduction to Philosophy of Mathematics

Mathematics, being a source of exact logic and truth, is very interesting to Philosophy, Physics and Metaphysics. Yet the Metaphysics of Mathematics, why mathematics works, how mathematics is connected to Reality, has remained a mystery. The purpose of this website is to explain this new metaphysical foundation, not only for Mathematics, but for all the Sciences. It is founded on the metaphysics of Space and Motion (Wave Structure of Matter) rather than the Metaphysics of Space and Time (particle structure of matter) which Kant showed to be only Ideas. There is a brief Introduction to the Wave Structure of Matter (WSM) below, at the end of this Philosophy of Mathematics page you will find links to Science Articles which explain and solve many problems of modern physics, philosophy and metaphysics from this new foundation.

This page on the Philosophy and Metaphysics of Mathematics requires substantial work. I am currently in the stage of collecting quotes and getting some structure into the arguments. The central argument is that mathematics is founded on ONE. Thus if One is both Finite / Discrete (Matter, Universe) and Infinite / Continuous (Space), then to understand mathematics without contradiction requires clear understanding of when One is Finite and when One is Infinite. This would obviously have central ramifications to all of mathematics, particularly in solving the problems of Calculus, how Mathematics connects to Reality, and Infinite Set Theory. Basically the solution is to know reality, that Space exists as a Wave Medium, and thus know what this One thing is that connects and causes the many things, (Matter as Spherical Wave Motion of Space).

Please feel free to browse the philosophy of mathematics quotes below. I also strongly recommend that you read on the Wave Structure of Matter - if you have good knowledge of the Philosophy of Mathematics then I am sure this knowledge will explain and solve many of your problems.
Geoff Haselhurst, January, 2005

## Mathematics Philosophy Physics Metaphysics of Mathematics

### Mathematics Calculus - Uniting One as both Continuous-Infinite and Discrete-Finite On Truth, Reality and Godel's Incompleteness Theorem

Currently mainstream modern physics and philosophy does not understand reality: how we can see a clock or feel the warmth of sunlight on our skin, how matter exists in space and interacts with other matter in the space around it, how matter experiences light and gravity, thus they do not know how we exist in space and sense the world around us. Humanity does not actually know what it means to be 'human'
This lack of understanding of reality and how it founds our senses has frustrated philosophers for many thousands of years. It now seems that most modern physicists and philosophers no longer believe that knowledge of a fundamental truth and reality is possible. The philosopher is inclined to believe, as Popper argues, that truth is evolving and that no final truth will ever be found. Only successively closer approximations to truth are possible due to the limitations of our language and our minds.
The physicist further argues that the inherent particle/wave duality of light and matter is an example of the paradoxical properties of physical reality that we cannot understand. Thus the mathematical physicist tends to believe that their language of mathematics is the closest possible approximation to describing this strange reality, and is limited to describing the numerical relationships between things, rather than describing the things themselves.

This belief is well summarised by the famous English physicist/philosopher Sir James Jeans;

.. the progress of science has itself shown that there can be no pictorial representation of the workings of nature of a kind that would be intelligible to our limited minds. The study of physics has driven us to the positivist conception of physics. We can never understand what events are, but must limit ourselves to describing the pattern of events in mathematical terms: no other aim is possible .... the final harvest will always be a sheaf of mathematical formulae. These will never describe nature itself, but only our observations on nature. (Jeans, 1942)

Unfortunately, this failure to solve the puzzle that is true reality has allowed Post Modern Physics to become quite absurd and paradoxical. In fact, Reality must be simple, as Leibniz correctly argues;

Reality cannot be found except in One single source, because of the interconnection of all things with one another. (Leibniz, 1670)

Thus it seems to me that It is humans who are paradoxical and confusing, reality, is simple and logical (as science invariably discovers).

To be completed ...

### On Mathematics and Why is it is Both Useful and Deceptive for Describing Reality Founded on the Metaphysics of Space and Motion and the Wave Structure of Matter

Mathematics is able to describe the Relationships between the Motion of Matter in Space because the Fundamental Measurements of Physics; Time (Frequency of the Spherical Standing Wave - SSW), Length (Wavelength of the SSW), and Mass (Change in Velocity of In-Waves relative to Change in 'Apparent Velocity' of Wave-Center) can be given Number Quantities.

Reality is Logical (necessary and deterministic) because of the Absolute Properties of an Absolute Space. i.e. The Necessary Connection between Wave Velocity, Wave-Amplitude and mass-energy density of Space. Mathematical Physics (Einstein's Relativity) confirms this precise Logical / Necessary Structure of Reality.

Mathematics is deceptive because the Objects (Concepts) which Humans construct can produce Mathematical (Logically Consistent) Theories and yet the Objects may not actually Exist in Space. (e.g. Feynman - Reflection of Light from the 'Surface' of Glass, Light and Matter as 'Particle' Interactions.) See below.

### Mathematical Truth and Physical Truth

I wish to talk briefly about mathematics and its relationship to reality, as it is important to appreciate both the strengths and weaknesses of mathematics as a tautology, or logical relationship language. The great power of mathematics is that once we develop the exact logical relationships between objects, then we can do away with the objects and simply consider the logical (mathematical) relationship between these things. Hence mathematics has a remarkable power which people did not understand, and as Bertrand Russell explains, further enhanced its mystical aspect.
Mathematics was associated with a more refined type of error. Mathematical knowledge appeared to be certain, exact, and applicable to the real world; moreover it was obtained by mere thinking, without the need of observation. Consequently, it was thought to supply an ideal, from which everyday empirical knowledge fell short. It was supposed on the basis of mathematics, that thought is superior to sense, intuition to observation. If the world of sense does not fit mathematics, so much the worse for the world of sense.
This form of philosophy begins with Pythagoras. (Bertrand Russell)

Herein lies the great weakness, and the great strength of mathematics. It is possible to evolve more and more complex relationships between things, which shed light on ideas far beyond the original relationships. Unfortunately, it is also possible that these things do not actually exist, except as evolved complex mathematical constructions.

The skeptic will say: "It may well be true that this system of equations is reasonable from a logical standpoint. But this does not prove that it corresponds to nature." You are right, dear skeptic. Experience alone can decide on truth. (Albert Einstein)
.. some things that satisfy the rules of algebra can be interesting to mathematicians even though they don’t always represent a real situation. (Richard Feynman)
From this we can logically conclude that there are two types of logical truths; Physical Truths, and Mathematical or Relational Truths.
Once we understand this we can solve many problems of mathematics and physics. e.g. The dual particle/wave truth of the photon, which initially appears so contradictory.

i) Mathematical Truths Only - For an example of a simple mathematical truth only, let us consider the partial reflection of light by glass of varying thickness. If we assume that the light is either reflected by the front surface of the glass or the back surface of the glass, then by summing Feynman’s probability arrows for both paths we can correctly calculate the probability of light reflecting from any thickness of glass.
But you may rightly ask, what are surfaces, and how do they reflect light?
And you would of course be wasting your time, because light does not reflect from the surface of glass.
This is what Feynman says; Thus we can get the correct answer for the probability of partial reflection by imagining (falsely) that all reflection comes from only the front and back surfaces. In this intuitively easy analysis, the "front surface" and "back surface" arrows are mathematical constructions that give us the right answer, whereas .... a more accurate representation of what is really going on: partial reflection is the scattering of light by electrons inside the glass. (Richard Feynman)

This is a fundamental limitation of mathematics.
It is quite possible to have a true mathematical relationship, that suggests a particular spatial model, and yet the spatial model may be completely wrong. This can (and does!) make mathematical physics very confusing.

### Light and the Photon as a Mathematical Truth

Light is mathematically consistent with the idea of photons as particles with discrete energy. But the behaviour of light is also consistent with the idea of waves in space. This has lead to a dual theory for light as a wave as well as a photon. How can this inconsistent relationship between light waves and photons be true?
The obvious answer is as follows;
Light as a photon or particle (photoelectric effect) is a mathematical relationship and is true.
Light as a Standing Wave Interaction (e.g. photoelectric effect, interference, diffraction, two slit experiment, etc.) is a physical relationship (Space as a wave Medium) and is true.
(i.e. The photon as a particle, is equivalent to light reflecting of the surface of glass. They are both mathematical constructions only.)
For this to be true, then the photoelectric effect must be able to be explained by a wave theory. This has been simply shown by Professor Milo Wolff and the Wave Structure of Matter
Logically, the effect of standing wave interactions must be discrete, which enables us to treat the effect as a particle with a discrete energy and momentum. This is logically acceptable, and hence has a mathematically acceptable solution. The result of this argument, is that we explain light’s dual nature in terms of a wave theory, while accepting the mathematical truth of its particle effect.

ii) Mathematical Truths, which are also Physical Truths. An obvious example of a mathematical truth, which is also spatially true, is Pythagoras’s Theorem. This is the reason for this relationship’s great power, and its use in Einstein’s metrics.

## Quotes on Mathematics - Albert Einstein

What place does the theoretical physicist's picture of the world occupy among all these possible pictures? It demands the highest possible standard of rigorous precision in the description of relations, such as only the use of mathematical language can give.

Geometry and Experience (1921)
One reason why mathematics enjoys special esteem above all other sciences is that its propositions are absolutely certain and indisputable while those of all other sciences are to some extent debatable and in constant danger of being overthrown by newly discovered facts.
These axioms are free creations of the human mind. Schlick in his book on epistemology has therefore characterised axioms very aptly as 'implicit definitions'. This view of axioms advocated by modern axiomatics purges mathematics of all extraneous elements, and thus dispels the mystic obscurity which formerly surrounded the basis of mathematics.
------------
The skeptic will say: "It may well be true that this system of equations is reasonable from a logical standpoint. But this does not prove that it corresponds to nature." You are right, dear skeptic. Experience alone can decide on truth.
-----------
What we call physics comprises that group of natural sciences which base their concepts on measurements; and whose concepts and proportions lend themselves to mathematical formulation. Its realm is accordingly defined as that part of the sum total of our knowledge which is capable of being expressed in mathematical terms. (Albert Einstein, Ideas and Opinions, 1954)

## Quotes on Mathematics - Aristotle

For it is the measure by which quantity is known, the knowledge of quantity qua quantity arising either through one or through some number and the knowledge of number arising through one. Hence all quantity qua quantity is known through one, and One Itself is that through which quantities are primarily known. Hence one is the principle of number qua number. (Aristotle p287)
Iota 3
Now there are several ways in which the one and the many are in opposition. One of these lies in the fact that the one and the many are opposed as indivisible and divisible. What is either divided or divisible is accounted for as a kind of plurality, whereas what is indivisible or not divided is said to be a unity. (p293)
Iota 6
After all, to speak of one or many is similar to speaking of one and ones or a white thing and white things and to setting the objects of measurement against the measure. And it is in this way too that we speak of things being manifold. The reason for saying of each number that it is many is just that it is ones and that each number is measured by the one. (p304)
The opposition consists in the fact that one is the measure and the other thing measured. Hence it is not the case that whatever is one is number.. (p305)
..plurality is number and unity is its measure. (p305)
. If, on the other hand, the accounts are based on a common feature, (SSWs in space) then being would form the domain of a single science. (p324)
..what is one is in a way also being and what is being is in a way also one. (p325)
And just as the mathematician is conducting a study into things in abstraction (for his study commences after the removal of all perceptible features, such as weight and lightness and hardness and the contrary of hardness, as also heat and cold and the other perceptible contrarieties, leaving only quantity and continuity, in one, two or three dimensions, and the affections of things qua quantitative and continuous, not contemplating them relative to anything else, and examines, on the one hand, the mutual relations of some and the features of those relations, and, on the other, the commensurabilities and incommensurabilities of others, and of yet others the proportions- but for all that we suppose geometry to be one and the same science for all these), so do things also stand with being.
It is for philosophy, and for philosophy alone, to study the accidents of being in so far as it is being, the contrarieties of being qua being. To physics is ascribed the study of things not qua things that are but qua participants in process. And dialectics and sophistics have, indeed, to do with the accidents of things that are, but not qua things that are, nor about that which is just in so far as it is that which is. It is left, then, to the philosopher to study the items we have mentioned, to the extent that they are as we have said.
And since, despite the plurality of accounts, everything that is is said to be by virtue of one common feature, and the contraries in the same fashion (they are referred to as the primary contrarieties and differentiae of being), and since it is possible for these to fall under a single science, the puzzle originally cited is resolved, that namely of how there is to be a single science of a plurality of things differing in kind. (p325-6)
Kappa 4
The text here recapitulated in Gamma 3, which explains that the fundamental axioms of logic fall within the domain of metaphysics, being peculiarly suitable to the generality of the interest in being which is distinctive of metaphysics. (p327 intro)
Also, since the mathematician himself applies common axioms, but does so in a special way, it would seem to fall first to philosophy to examine the principles of these too.
For instance, that when equals are subtracted from equals remain is common to all quantities, but mathematics discriminates and conducts a study into a certain part of its proper matter, i.e. into lines, or angles, or numbers or some one of the other quantities, not qua things that are but just qua each as a continuity in one, two or three dimensions. Philosophy, by contrast, does not examine some portion of what is, in respect of the accidents of each such group of things, but contemplates being, as the being of each of such things.
And physics is in the same boat as mathematics. It studies the accidents and principles of entities, qua participating in process and not qua being. And in contrast we have said that primary science is the science of these things in so far as they, its subjects, are things that are, and not in regard to any other feature.
Hence both physics and mathematics are to be considered mere parts of total understanding. (p327)

### David Hume

It seems to me, that the only objects of the abstract science or of demonstration are quantity and number, and that all attempts to extend this more perfect species of knowledge beyond these bounds are mere sophistry and illusion. (Hume, p163)
Does it contain any abstract reasoning concerning quantity or number? No. Does it contain any experimental reasoning concerning matter of fact and existence? No. Commit it then to the flames: for it can contain nothing but sophistry and illusion. (Hume, p165)

### Heisenberg. "Quantum Theory" University of Chicago, 1930

From these experiments it is seen that both matter and radiation possess a remarkable duality of character, as they sometimes exhibit the properties of waves, at other times those of particles. Now it is obvious that a thing cannot be a form of wave motion and composed of particles at the same time - the two concepts are too different.
The solution of the difficulty is that the two mental pictures which experiment lead us to form - the one of the particles, the other of the waves - are both incomplete and have only the validity of analogies which are accurate only in limiting cases.

Light and matter are both single entities, and the apparent duality arises in the limitations of our language.

It is not surprising that our language should be incapable of describing the processes occurring within the atoms, for, as has been remarked, it was invented to describe the experiences of daily life, and these consist only of processes involving exceedingly large numbers of atoms. Furthermore, it is very difficult to modify our language so that it will be able to describe these atomic processes, for words can only describe things of which we can form mental pictures, and this ability, too, is a result of daily experience. Fortunately, mathematics is not subject to this limitation, and it has been possible to invent a mathematical scheme - the quantum theory - which seems entirely adequate for the treatment of atomic processes; for visualisation, however, we must content ourselves with two incomplete analogies - the wave picture and the corpuscular picture. (Werner Heisenberg, 1930)

On being asked what he meant by the beauty of a mathematical theory of physics, Dirac replied that if the questioner was a mathematician then he did not need to be told, but were he not a mathematician then nothing would be able to convince him of it.

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It's like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.
('Freeman Dyson: Mathematician, Physicist, and Writer'. Interview with Donald J. Albers, The College Mathematics Journal, vol 25, no. 1, January 1994.)

## Immanuel Kant, Prolegomena to Any Future Metaphysics

### First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?

http://www.abu.nb.ca/Courses/GrPhil/Modphil/KantReading.htm

Here is a great and established branch of knowledge, encompassing even now a wonderfully large domain and promising an unlimited extension in the future. Yet it carries with it thoroughly apodictical certainty, i.e., absolute necessity, which therefore rests upon no empirical grounds. Consequently it is a pure product of reason, and moreover is thoroughly synthetical. [Here the question arises:] "How then is it possible for human reason to produce a cognition of this nature entirely a priori?"

Does not this faculty [which produces mathematics], as it neither is nor can be based upon experience, presuppose some ground of cognition a priori, which lies deeply hidden, but which might reveal itself by these its effects, if their first beginnings were but diligently ferreted out?

Sect. 7. But we find that all mathematical cognition has this peculiarity: it must first exhibit its concept in a visual form [Anschauung] and indeed a priori, therefore in a visual form which is not empirical, but pure. Without this mathematics cannot take a single step; hence its judgments are always visual, viz., "Intuitive"; whereas philosophy must be satisfied with discursive judgments from mere concepts, and though it may illustrate its doctrines through a visual figure, can never derive them from it. This observation on the nature of mathematics gives us a clue to the first and highest condition of its possibility, which is, that some non-sensuous visualization [called pure intuition, or reine Anschauung] must form its basis, in which all its concepts can be exhibited or constructed, in concrete and yet a priori. If we can find out this pure intuition and its possibility, we may thence easily explain how synthetical propositions a priori are possible in pure mathematics, and consequently how this science itself is possible. Empirical intuition [viz., sense-perception] enables us without difficulty to enlarge the concept which we frame of an object of intuition [or sense-perception], by new predicates, which intuition [i.e., sense-perception] itself presents synthetically in experience. Pure intuition [viz., the visualization of forms in our imagination, from which every thing sensual, i.e., every thought of material qualities, is excluded] does so likewise, only with this difference, that in the latter case the synthetical judgment is a priori certain and apodictical, in the former, only a posteriori and empirically certain; because this latter contains only that which occurs in contingent empirical intuition, but the former, that which must necessarily be discovered in pure intuition. Here intuition, being an intuition a priori, is before all experience, viz., before any perception of particular objects, inseparably conjoined with its concept.

Sect. 8. But with this step our perplexity seems rather to increase than to lessen. For the question now is, "How is it possible to intuit [in a visual form] anything a priori?" An intuition [viz., a visual sense perception] is such a representation as immediately depends upon the presence of the object. Hence it seems impossible to intuit from the outset a priori, because intuition would in that event take place without either a former or a present object to refer to, and by consequence could not be intuition. Concepts indeed are such, that we can easily form some of them a priori, viz., such as contain nothing but the thought of an object in general; and we need not find ourselves in an immediate relation to the object. Take, for instance, the concepts of Quantity, of Cause, etc. But even these require, in order to make them understood, a certain concrete use-that is, an application to some sense-experience [Anschauung], by which an object of them is given us. But how can the intuition of the object [its visualization] precede the object itself?

Sect. 9. If our intuition [i.e., our sense-experience] were perforce of such a nature as to represent things as they are in themselves, there would not be any intuition a priori, but intuition would be always empirical. For I can only know what is contained in the object in itself when it is present and given to me. It is indeed even then incomprehensible how the visualizing [Anschauung] of a present thing should make me know this thing as it is in itself, as its properties cannot migrate into my faculty of representation. But even granting this possibility, a visualizing of that sort would not take place a priori, that is, before the object were presented to me; for without this latter fact no reason of a relation between my representation and the object can be imagined, unless it depend upon a direct inspiration.

Therefore in one way only can my intuition [Anschauung] anticipate the actuality of the object, and be a cognition a priori, viz.: if my intuition contains nothing but the form of sensibility, antedating in my subjectivity all the actual impressions through which I am affected by objects.

For that objects of sense can only be intuited according to this form of sensibility I can know a priori. Hence it follows: that propositions, which concern this form of sensuous intuition only, are possible and valid for objects of the senses; as also, conversely, that intuitions which are possible a priori can never concern any other things than objects of our senses. 7

Sect. 10. Accordingly, it is only the form of sensuous intuition by which we can intuit things a priori, but by which we can know objects only as they appear to us (to our senses), not as they are in themselves; and this assumption is absolutely necessary if synthetical propositions a priori be granted as possible, or if, in case they actually occur, their possibility is to be comprehended and determined beforehand.

Now, the intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodictic and necessary are Space and Time. For mathematics must first have all its concepts in intuition, and pure mathematics in pure intuition, that is, it must construct them. If it proceeded in any other way, it would be impossible to make any headway, for mathematics proceeds, not analytically by dissection of concepts, but synthetically, and if pure intuition be wanting, there is nothing in which the matter for synthetical judgments a priori can be given. Geometry is based upon the pure intuition of space. Arithmetic accomplishes its concept of number by the successive addition of units in time; and pure mechanics especially cannot attain its concepts of motion without employing the representation of time. Both representations, however, are only intuitions; for if we omit from the empirical intuitions of bodies and their alterations (motion) everything empirical, or belonging to sensation, space and time still remain, which are therefore pure intuitions that lie a priori at the basis of the empirical. Hence they can never be omitted, but at the same time, by their being pure intuitions a priori, they prove that they are mere forms of our sensibility, which must precede all empirical intuition, or perception of actual objects, and conformably to which objects can be known a priori, but only as they appear to us.

Sect. 11. The problem of the present section is therefore solved. Pure mathematics, as synthetical cognition a priori, is only possible by referring to no other objects than those of the senses. At the basis of their empirical intuition lies a pure intuition (of space and of time) which is a priori. This is possible, because the latter intuition is nothing but the mere form of sensibility, which precedes the actual appearance of the objects, in, that it, in fact, makes them possible. Yet this faculty of intuiting a priori affects not the matter of the phenomenon (that is, the sense- element in it, for this constitutes that which is empirical), but its form, viz., space and time. Should any man venture to doubt that these are determinations adhering not to things in themselves, but to their relation to our sensibility, I should be glad to know how it can be possible to know the constitution of things a priori, viz., before we have any acquaintance with them and before they are presented to us. Such, however, is the case with space and time. But this is quite comprehensible as soon as both count for nothing more than formal conditions of our sensibility, while the objects count merely as phenomena; for then the form of the phenomenon, i.e., pure intuition, can by all means be represented as proceeding from ourselves, that is, a priori.

Sect. 12. In order to add something by way of illustration and confirmation, we need only watch the ordinary and necessary procedure of geometers. All proofs of the complete congruence of two given figures (where the one can in every respect be substituted for the other) come ultimately to this that they may be made to coincide; which is evidently nothing else than a synthetical proposition resting upon immediate intuition, and this intuition must be pure, or given a priori, otherwise the proposition could not rank as apodictically certain, but would have empirical certainty only. In that case, it could only be said that it is always found to be so, and holds good only as far as our perception reaches. That everywhere space (which (in its entirety] is itself no longer the boundary of another space) has three dimensions, and that space cannot in any way have more, is based on the proposition that not more than three lines can intersect at right angles in one point; but this proposition cannot by any means be shown from concepts, but rests immediately on intuition, and indeed on pure and a priori intuition, because it is apodictically certain. That we can require a line to be drawn to infinity (in indefinitum), or that a series of changes (for example, spaces traversed by motion) shall be infinitely continued, presupposes a representation of space and time, which can only attach to intuition, namely, so far as it in itself is bounded by nothing, for from concepts it could never be inferred. Consequently, the basis of mathematics actually are pure intuitions, which make its synthetical and apodictically valid propositions possible. Hence our transcendental deduction of the notions of space and of time explains at the same time the possibility of pure mathematics. Without some such deduction its truth may be granted, but its existence could by no means be understood, and we must assume II that everything which can be given to our senses (to the external senses in space, to the internal one in time) is intuitd by us as it appears to us, not as it is in itself."

Sect. 13. Those who cannot yet rid themselves of the notion that space and time are actual qualities inhering in things in themselves, may exercise their acumen on the following paradox. When they have in vain attempted its solution, and are free from prejudices at least for a few moments, they will suspect that the degradation of space and of time to mere forms of our sensuous intuition may perhaps be well founded.

If two things are quite equal in all respects ask much as can be ascertained by all means possible, quantitatively and qualitatively, it must follow, that the one can in all cases and under all circumstances replace the other, and this substitution would not occasion the least perceptible difference. This in fact is true of plane figures in geometry; but some spherical figures exhibit, notwithstanding a complete internal agreement, such a contrast in their external relation, that the one figure cannot possibly be put in the place of the other. For instance, two spherical triangles on opposite hemispheres, which have an arc of the equator as their common base, may be quite equal, both as regards sides and angles, so that nothing is to be found in either, if it be described for itself alone and completed, that would not equally be applicable to both; and yet the one cannot be put in the place of the other (being situated upon the opposite hemisphere). Here then is an internal difference between the two triangles, which difference our understanding cannot describe as internal, and which only manifests itself by external relations in space.

But I shall adduce examples, taken from common life, that are more obvious still.

What can be more similar in every respect and in every part more alike to my hand and to my ear, than their images in a mirror? And yet I cannot put such a hand as is seen in the glass in the place of its archetype; for if this is a right hand, that in the glass is a left one, and the image or reflection of the right ear is a left one which never can serve as a substitute for the other. There are in this case no internal differences which our understanding could determine by thinking alone. Yet the differences are internal as the senses teach, for, notwithstanding their complete equality and similarity, the left hand cannot be enclosed in the same bounds as the right one (they are not congruent); the glove of one hand cannot be used for the other. What is the solution? These objects are not representations of things as they are in themselves, and as the pure understanding would know them, but sensuous intuitions, that is, appearances, the possibility of which rests upon the relation of certain things unknown in themselves to something else, viz., to our sensibility. Space is the form of the external intuition of this sensibility, and the internal determination of every space is only possible by the determination of its external relation to the whole space, of which it is a part (in other words, by its relation to the external sense). That is to say, the part is only possible through the whole, which is never the case with things in themselves, as objects of the mere understanding, but with appearances only. Hence the difference between similar and equal things, which are yet not congruent (for instance, two symmetric helices), cannot be made intelligible by any concept, but only by the relation to the right and the left hands which immediately refers to intuition.

REMARK 1.

Pure Mathematics, and especially pure geometry, can only have objective reality on condition that they refer to objects of sense. But in regard to the latter the principle holds good, that our sense representation is not a representation of things in themselves but of the way in which they appear to us. Hence it follows, that the propositions of geometry are not the results of a mere creation of our poetic imagination, and that therefore they cannot be referred with assurance to actual objects; but rather that they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing else than the form of all external appearances, and it is this form alone in which objects of sense can be given. Sensibility, the form of which is the basis of geometry, is that upon which the possibility of external appearance depends. Therefore these appearances can never contain anything but what geometry prescribes to them.

It would be quite otherwise if the senses were so constituted as to represent objects as they are in themselves. For then it would not by any means follow from the conception of space, which with all its properties serves to the geometer as an a priori foundation, together with what is thence inferred, must be so in nature. The space of the geometer would be considered a mere fiction, and it would not be credited with objective validity, because we cannot see how things must of necessity agree with an image of them, which we make spontaneously and previous to our acquaintance with them. But if this image, or rather this formal intuition, is the essential property of our sensibility, by means of which alone objects are given to us, and if this sensibility represents not things in themselves, but their appearances: we shall easily comprehend, and at the same time indisputably prove, that all external objects of our world of sense must necessarily coincide in the most rigorous way with the propositions of geometry; because sensibility by means of its form of external intuition, viz., by space, the same with which the geometer is occupied, makes those objects at all possible as mere appearances.

It will always remain a remarkable phenomenon in the history of philosophy, that there was a time, when even mathematicians, who at the same time were philosophers, began to doubt, not of the accuracy of their geometrical propositions so far as they concerned space, but of their objective validity and the applicability of this concept itself, and of all its corollaries, to nature. They showed much concern whether a-line in nature might not consist of physical points, and consequently that true space in the object might consist of simple [discrete] parts, while the space which the geometer has in his mind [being continuous] cannot be such. They did not recognize that this mental space renders possible the physical space, i.e., the extension of matter; that this pure space is not at all a quality of things in themselves, but a form of our sensuous faculty of representation; and that all objects in space are mere appearances, i.e., not things in themselves but representations of our sensuous intuition. But such is the case, for the space of the geometer is exactly the form of sensuous intuition which we find a priori in us, and contains the ground of the possibility of all external appearances (according to their form), and the latter must necessarily and most rigidly agree with the propositions of the geometer, which he draws not from any fictitious concept, but from the subjective basis of all external phenomena, which is sensibility itself. In this and no other way can geometry be made secure as to the undoubted objective reality of its propositions against all the intrigues of a shallow Metaphysics, which is surprised at them [the geometrical propositions], because it has not traced them to the sources of their concepts.

REMARK II.

Whatever is given us as object, must be given us in intuition. All our intuition however takes place by means of the senses only; the understanding intuits nothing, but only reflects. And as we have just shown that the senses never and in no manner enable us to know things in themselves, but only their appearances, which are mere representations of the sensibility, we conclude that all bodies, together with the space in which they are, must be considered nothing but mere representations in us, and exist nowhere but in our thoughts.' You will say: Is not this manifest idealism?

Idealism consists in the assertion, that there are none but thinking beings, all other things, which we think are perceived in intuition, being nothing but representations in the thinking beings, to which no object external to them corresponds in fact. Whereas I say, that things as objects of our senses existing outside us are given, but we know nothing of what they may be in themselves, knowing only their appearances, 1. e., the representations which they cause in us by affecting our senses. Consequently I grant by all means that there are bodies without us, that is, things which, though quite unknown to us as to what they are in themselves, we yet know by the representations which their influence on our sensibility procures us, and which we call bodies, a term signifying merely the appearance of the thing which is unknown to us, but not therefore less actual. Can this be termed idealism? It is the very contrary.

Long before Locke's time, but assuredly since him, it has been generally assumed and granted without detriment to the actual existence of external things, that many of their predicates may be said to belong not to the things in themselves, but to their appearances, and to have no proper existence outside our representation. Heat, color, and taste, for instance, are of this kind. Now, if I go farther, and for weighty reasons rank as mere appearances the remaining qualities of bodies also, which are called primary, such as extension, place, and in general space, with all that which belongs to it (impenetrability or materiality, space, etc.)-no one in the least can adduce the reason of its being inadmissible. As little as the man who admits colors not to be properties of the object in itself, but only as modifications of the sense of sight, should on that account be called an idealist, so little can my system be named idealistic, merely because I find that more, nay, A11 the properties which constitute the intuition of a body belong merely to its appearance.

The existence of the thing that appears is thereby not destroyed, as in genuine idealism, but it is only shown, that we cannot possibly know it by the senses as it is in itself.

I should be glad to know what my assertions must be in order to avoid all idealism. Undoubtedly, I should say, that the representation of space is not only perfectly conformable to the relation which our sensibility has to objects-that I have said- but that it is quite similar to the object,-an assertion in which I can find as little meaning as if I said that the sensation of red has a similarity to the property of vermilion, which in me excites this sensation.

REMARK III

Hence we may at once dismiss an easily foreseen but futile objection, "that by admitting the ideality of space and of time the whole sensible world would be turned into mere sham." At first all philosophical insight into the nature of sensuous cognition was spoiled, by making the sensibility merely a confused mode of representation, according to which we still know things as they are, but without being able to reduce everything in this our representation to a clear consciousness; whereas proof is offered by us that sensibility consists, not in this logical distinction of clearness and obscurity, but in the genetical one of the origin of cognition itself. For sensuous perception represents things not at all as they are, but only the mode in which they affect our senses, and consequently by sensuous perception appearances only and not things themselves are given to the understanding for reflection. After this necessary corrective, an objection rises from an unpardonable and almost intentional misconception, as if my doctrine turned all the things of the world of sense into mere illusion.

When an appearance is given us, we are still quite free as to how we should judge the matter. The appearance depends upon the senses, but the judgment upon the understanding, and the only question is, whether in the determination of the object there is truth or not. But the difference between truth and dreaming is not ascertained by the nature of the representations, which are referred to objects (for they are the same in both cases), but by their connection according to those rules, which determine the coherence of the representations in the concept of an object, and by ascertaining whether they can subsist together in experience or not. And it is not the fault of the appearances if our cognition takes illusion for truth, i.e., if the intuition, by which an object is given us, is considered a concept of the thing or of its existence also, which the understanding can only think. The senses represent to us the paths of the planets as now progressive, now retrogressive, and herein is neither falsehood nor truth, because as long as we hold this path to be nothing but appearance, we do not judge of the objective nature of their motion. But as a false judgment may easily arise when the understanding is not on its guard against this subjective mode of representation being considered objective, we say they appear to move backward; it is not the senses however which must be charged with the illusion, but the understanding, whose province alone it is to give an objective judgment on appearances.

Thus, even if we did not at all reflect on the origin of our representations, whenever we connect our intuitions of sense (whatever they may contain), in space and in time, according to the rules of the coherence of all cognition in experience, illusion or truth will arise according as we are negligent or careful. It is merely a question of the use of sensuous representations in the understanding, and not of their origin. In the same way, if I consider all the representations of the senses, together with their form, space and time, to be nothing but appearances, and space and time to be a mere form of the sensibility, which is not to be met with in objects out of it, and if I make use of these representations in reference to possible experience only, there is nothing in my regarding them as appearances that can lead astray or cause illusion. For all that they can correctly cohere according to rules of truth in experience. Thus all the propositions of geometry hold good of space as well as of all the objects of the senses, consequently of all possible experience, whether I consider space as a mere form of the sensibility, or as something cleaving to the things themselves. In the former case however I comprehend how I can know a priori these propositions concerning all the objects of external intuition. Otherwise, everything else as regards all possible experience remains just as if I had not departed from the vulgar view.

But if I venture to go beyond all possible experience with my notions of space and time, which I cannot refrain from doing if I proclaim them qualities inherent in things in themselves (for what should prevent me from letting them hold good of the same things, even though my senses might be different, and unsuited to them?), then a grave error may arise due to illusion, for thus I would proclaim to be universally valid what is merely a subjective condition of the intuition of things and sure only for all objects of sense, viz., for all possible experience; I would refer this condition to things in themselves, and do not limit it to the conditions of experience.

My doctrine of the ideality of space and of time, therefore, far from reducing the whole sensible world to mere illusion, is the only means of securing the application of one of the most important cognitions (that which mathematics propounds a priori) to actual objects, and of preventing its being regarded as mere illusion. For without this observation it would be quite impossible to make out whether the intuitions of space and time, which we borrow from no experience, and which yet lie in our representation a priori, are not mere phantasms of our brain, to which objects do not correspond, at least not adequately, and consequently, whether we have been able to show its unquestionable validity with regard to all the objects of the sensible world just because they are mere appearances.

Secondly, though these my principles make appearances of the representations of the senses, they are so far from turning the truth of experience into mere illusion, that they are rather the only means of preventing the transcendental illusion, by which metaphysics has hitherto been deceived, leading to the childish endeavor of catching at bubbles, because appearances, which are mere representations, were taken for things in themselves. Here originated the remarkable event of the antimony of Reason which I shall mention by and by, and which is destroyed by the single observation, that appearance, as long as it is employed in experience, produces truth, but the moment it transgresses the bounds of experience, and consequently becomes transcendent, produces nothing but illusion.

Inasmuch therefore, as I leave to things as we obtain them by the senses their actuality, and only limit our sensuous intuition of these things to this, that they represent in no respect, not even in the pure intuitions of space and of time, anything more than mere appearance of those thin-s, but never their constitution in themselves, this is not a sweeping illusion invented for nature by me. My protestation too against all charges of idealism is so valid and clear as even to seem superfluous, were there not incompetent judges, who, while they would have an old name for every deviation from their perverse though common opinion, and never judge of the spirit of philosophic nomenclature, but cling to the letter only, are ready to put their own conceits in the place of well-defined notions, and thereby deform and distort them. I have myself given this my theory the name of transcendental idealism, but that cannot authorize any one to confound it either with the empirical idealism of Descartes, (indeed, his was only an insoluble problem, owing to which he thought every one at liberty to deny the existence of the corporeal world, because it could never be proved satisfactorily), or with the mystical and visionary idealism of Berkeley, against which and other similar phantasms our Critique contains the proper antidote. My idealism concerns not the existence of things (the doubting of which, however, constitutes idealism in the ordinary sense), since it never came into my head to doubt it, but it concerns the sensuous representation of things, to which space and time especially belong. Of these [viz., space and time], consequently of all appearances in general, I have only shown, that they are neither things (but mere modes of representation), nor determinations belonging to things in themselves. But the word "transcendental," which with me means a reference of our cognition, i.e., not to things, but only to the cognitive faculty, was meant to obviate this misconception. Yet rather than give further occasion to it by this word, I now retract it, and desire this idealism of mine to be called critical. But if it be really an objectionable idealism to convert actual things (not appearances) into mere representations, by what name shall we call him who conversely changes mere representations to things? It may, I think, be called "dreaming idealism," in contradistinction to the former, which may be called "visionary," both of which are to be refuted by my transcendental, or, better, critical idealism.

## Godel, Kurt (Mathematical logician and philosopher of mathematics) 1906-1978

Between 1930 and 1940 Godel was responsible for three significant developments in mathematical logic. These were, first, the completeness proof relating to the first-order functional calculus; second, the theorem known as Godel's Theorem, (the first incompleteness theorem); and third, a demonstration that the system of Russell's and Whitehead's Principia Mathematica, which the incompleteness theorem showed to be apparently inconsistent, could be rendered consistent.
Godel's Theorem constituted a major challenge not only to the generally-held assumption that basic systems in mathematics are complete in that they contain no statements that can be either proved or disproved, but also to Hilbert's view that proofs of the consistency of such a system can be formulated within the system itself. Briefly, the theorem states that in a formal system S or arithmetic, there will be a sentence P of the language of S such that if S is consistent neither P nor its negation can be proved within S.
The impact of this on Principia Mathematica was to undermine the latter's project of providing a set of logical axioms from which the whole of pure mathematics, as well as the non-axiomatic residue of logic, were deducible, since the theorem showed that mathematics contains propositions that are neither provable nor disprovable from the axioms.
..it has been claimed that Godel's theorem demonstrates that human beings are superior to machines since they can know to be true, propositions that no machine programmed with axioms and rules can prove.
His 1949 paper includes a curious argument for the unreality of time. (p70-1)

A common interpretation of Godel (which I think is incorrect - I shall explain my reasons below)
There is no such thing as a theory of everything. Godel's theorem proves that if you have an axiomatic set there will always be statements that cannot be proven true or false by the given set. That means that even in the case we would find out the basic physical laws, there will be phenomena which we can't explain. This is a thing that most physicists are not aware of. This is one of the reasons that makes me believe that there is no fundamental law of nature, but rather an always increasing set of such laws.
Take the example of absolute geometry (as set by Hilbert). It goes fine until someone asks the question: "how many parallel lines one can draw through an external point to a straight line?"
The system cannot answer. Any answer you pick cannot be proven nor disproven by that set. What do you do? You simply add one more axiom to settle this and then the absolute geometry bifurcates in euclidian and non-euclidian. You may think it's over. But no. No matter what your axiomatic set is, you can always construct a question that the system cannot answer. It's the same in physics. We will found some fundamental laws (the axioms) and there will be questions (phenomena) that could not be answered by that set. We will enhance the set on and on, always improving it, but never exhausting the REALITY. It's funny but the starting point can be simple and yet what arises from this is infinitely complex: the arithmetic's uses a set of like 10 axioms describing the objects (natural numbers 0 and 1) plus operating rules (addition, or rule to generate new objects like 2, 3, 4 and so on. Yet, the operating rule is so powerful that this simple system can grow fast towards infinity and soon generate new operating rules and new objects like rational numbers, real numbers and complex numbers, then the transfinite infinities and so on. And it never stops, never exhaust itself. I strongly believe that this is the same stuff in physics: you start with "nothing" (vacuum) and "something" (electron) and a combination rule and this will generate on and on new objects (particles) and new combination rules and the system will get more and more complex and infinite. So, I don't believe in the Theory of Everything. It's a nonsense.

This last argument on the consequences of Godel's incompleteness theory is VERY interesting and important as it has been used by philosophers and physicists last century to argue similar as above - that we cannot completely describe reality with our human languages. (Also Nietzsche, Wittgenstein argue in similar ways about the relative meaning of words and that all logic is ultimately empty tautology.

This can only be solved by knowing the correct language (of the WSM) that then directly relates to what exists and does not depend upon other words. My knowledge of Godel is limited but I quote a few things from one of my philosophy books;
Godel's Theorem constituted a major challenge not only to the generally held assumption that basic systems in mathematics are complete in that they contain no statements that can be either proved or disproved, but also to Hilbert's view that proofs of the consistency of such a system can be formulated within the system itself. Briefly, the theorem states that in a formal system S of arithmetic, there will be a sentence P of the language S such that if S is consistent neither P nor its negation can be proved within S. The impact of this on Principia Mathematica (Whitehead, Russell) was to undermine the latter's project of providing a set of logical axioms from which the whole of pure mathematics, as well as the non-axiomatic residue of logic, were deducible, since the theorem showed that mathematics contains propositions that are neither provable nor disprovable from the axioms. ....
This shows that a consistency proof of ordinary arithmetic is not possible using finite procedures.

Now it occurs to me that;
1. Godel's incompleteness theory applies to Mathematics, thus it seems to me to suggest that mathematics can never completely describe the infinite possible relationships (between Wave-Centers) that can exist in an infinite space. I agree with this conclusion as Mathematical Physics does not in fact describe reality, but simply quantifies the number, wavelength, and relative motions of Wave-Centers.
2. Most significantly, I think that the Principles of the Wave Structure of Matter, because they describe how we exist (as the complex interactions of Wave-Centers) of finite Spherical Standing Waves (the size of our universe) within an infinite Space, then we overcome the problem of the finite and the infinite.
Of further interest from my Philosophy book;
In connection with this it has been claimed that Godel's theorem demonstrates that human beings are superior to machines since they can know to be true propositions that no machine programmed with axioms and rules can prove.
This is naive as it simply begs the question as to what we mean when we 'know something to be true'. Ultimately all things exist in space as the relative motions of Wave-Centers. We are all machines, and all knowledge (including our identity and our mind) exists as relative motions of Wave-Centers. (Logic exists because these Wave-Center interactions are logical / necessary as caused by the Properties of Space.)
Finally; His 1949 paper includes a curious argument for the unreality of time.
This is True - it is Space and Motion (not Time) which are the two fundamental 'Existents' in the universe. Godel was clearly a very clever philosopher (and very strange I suspect).

## Links for Philosophy / Metaphysics of Mathematics

http://www.objectivistcenter.org/articles/foundations_phil-of-mathematics.asp - The philosophy of mathematics is the philosophical study of the concepts and methods of mathematics. It is concerned with the nature of numbers, geometric objects, and other mathematical concepts; it is concerned with their cognitive origins and with their application to reality. It addresses the validation of methods of mathematical inference. In particular, it deals with the logical problems associated with mathematical infinitude. (By David S. Ross, Ph.D. A mathematician at Eastman Kodak Research Labs, David Ross has taught mathematics at New York University and the University of Rochester.)

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