Frege-Russell and Mathematical Identity

Frege considered it a principal task of his logical reform of arithmetic to provide absolutely determinate identity conditions for the objects of that science, i.e. for numbers. Referring to the contemporary situation in this discipline he writes:

How I propose to improve upon it can be no more than indicated in the present work. With numbers … it is a matter of fixing the sense of an identity.

Frege makes the following critically important assumption : identity is a general logical concept, which is not specific to mathematics. Frege says:

It is not only among numbers that the relationship of identity is found. From which it seems to follow that we ought not to define it specially for the case of numbers. We should expect the concept of identity to have been fixed first, and that then from it together with the concept of number it must be possible to deduce when numbers are identical with one another, without there being need for this purpose of a special definition of numerical identity as well.

In a different place Frege says clearly that this concept of identity is absolutely stable across all possible domains and contexts:

Identity is a relation given to us in such a specific form that it is inconceivable that various forms of it should occur.

Frege’s definition of natural number, as modified in Russell (Bertrand Russell – Principles of Mathematics) later became standard. Intuitively the number 3 is what all collections consisting of three members (trios) share in common. Now instead of looking for a common form, essence or type of trios let us simply consider all such things together. According to Frege and Russell the collection (class, set) of all trios just is the number 3. Similarly for other numbers. Isn’t this construction circular? Frege and Russell provide the following argument which they claim allows us to avoid circularity here: given two different collections we may learn whether or not they have the same number of members without knowing this number and even without the notion of number itself. It is sufficient to find a one-one correspondence between members of two given collections. If there is such a correspondence, the two collections comprise the same number of members, or to avoid any reference to numbers we can say that the two collections are equivalent. This equivalence is Humean. Let us define natural numbers as equivalence classes under this relation. This definition reduces the question of identity of numbers to that of identity of classes. This latter question is settled through the axiomatization of set theory in a logical calculus with identity. Thus Frege’s project is realized: it has been seen how the logical concept of identity applies to numbers. In an axiomatic setting “identities” in Quine’s sense (that is, identity conditions) of mathematical objects are provided by an axiom schema of the form

∀x ∀y (x=y ↔ ___ )

called the Identity Schema (IS). This does not resolve the identity problem though because any given system of axioms, generally speaking, has multiple models. The case of isomorphic models is similar to that of equal numbers or coincident points (naively construed): there are good reasons to think of isomorphic models as one and there is also good reason to think of them as many. So the paradox of mathematical “doubles” reappears. It is a highly non-trivial fact that different models of Peano arithmetic, ZF, and other important axiomatic systems are not necessarily isomorphic. Thus logical analysis à la Frege-Russell certainly clarifies the mathematical concepts involved but it does not settle the identity issue as Frege believed it did. In the recent philosophy of mathematics literature the problem of the identity of mathematical objects is usually considered in the logical setting just mentioned: either as the problem of the non-uniqueness of the models of a given axiomatic system or as the problem of how to fill in the Identity Schema. At the first glance the Frege-Russell proposal concerning the identity issue in mathematics seems judicious and innocent (and it certainly does not depend upon the rest of their logicist project): to stick to a certain logical discipline in speaking about identity (everywhere and in particular in mathematics).

Advertisement

Unconditional Accelerationists: Park Chung-Hee and Napoleon

Land’s Teleoplexy,

Some instance of intermediate individuation—most obviously the state—could be strategically invested by a Left Accelerationism. precisely in order to submit the virtual-teleoplexic lineage of Terrestrial Capitalism (or Techonomic Singularity) to effacement and disruption.

For the unconditional accelerationist as much as for the social historian, of course, the voluntarist quality of this image is a lie. Napoleon’s supposed flight from history can amply be recuperated within the process of history itself, if only we revise our image of what this is: not a flat space or a series of smooth curves, but rather a tangled, homeorhetic, deep-subversive spiral-complex. Far from shaping history like putty, Napoleon like all catastrophic agents of time-anomaly unleashed forces that ran far ahead of his very intentions: pushing Europe’s engagement with Africa and the Middle East onto a new plane, promulgating the Code Napoleon that would shape and selectively boost the economic development of continental Europe. In this respect, the image of him offered later by Marinetti is altogether more interesting. In his 1941 ‘Qualitative Imaginative Futurist Mathematics’, Marinetti claimed that Futurist military ‘calculations are as precise as those of Napoleon who in some battles had all of his couriers killed and hence his generals autonomous‘. Far from the prideful image of a singular genius strutting as he pleases across the stage of world history, here Napoleon becomes something altogether more monstrous. Foreshadowing Bataille’s argument a few years later that the apex of sovereignty is precisely an absolute moment of unknowing, he becomes a head that has severed itself from its limbs, falling from its body as it gives way to the sharp and militant positive feedback it has unleashed.

To understand its significance, we must begin by recognising that far from being a story of the triumph of a free capitalism over communism, the reality of Park Chung-hee’s rule and the overtaking of the North by the South is more than a little uncomfortable for a right-libertarian (though not, perhaps, for someone like Peter Thiel). Park was not just a sovereign dictator but an inveterate interventionist, who constructed an entire sequence of bureaucracies to oversee the expansion of the economy according to determinate Five-Year Plans. In private notes, he emphasised the ideology of the February 26 incident in Japan, the militarised attempt to effect a ‘Shōwa Restoration’ that would have united the Japanese race politically and economically behind a totalitarian emperor. In Japan this had failed: in Korea, Park himself could be the president-emperor, declaiming on his ‘sacred military revolution’ of 1961 that had brought together the ‘Korean race’. At the same time, he explicitly imitated the communist North, proclaiming the need for spiritual mobilisation and a ‘path of the leader’ 지도자의길 around which the nation would cohere. The carefully-coordinated mass histrionics after his death in 1979 echoed closely the spectacle with which we are still familiar in North Korea.

Park Chung-hee and Napoleon demonstrate at its extreme the tangled structure of the history of capital. Capitalism’s intensities are geographically and temporally uneven; they spread through loops and spectacular digressions. Human agencies and the mechanisms of the state have their important place within this capitalist megamachine. But things never quite work out the way they plan.

Supervenience

Past decades witness an increasing interest in the concept of supervenience, which has traditionally been used as a relation between sets of properties. A set A of properties (called ‘supervenient properties’) is said to supervene on another set B (called ‘subveinent properties’), just in case if B-properties are indistinguishable, then so are A-properties; in other words, agreement in respect of B-properties implies agreement in respect of A-properties. In slogan form, “there cannot be an A-different without a B-difference”. The core idea of supervenience is that fixing subvenient properties fixes its supervenient ones; or equivalently, subvenient properties determine supervenient properties.

The notion of supervenience dates back at least to G. E. Moore’s classical work, where he described some certain dependency relationship between moral and non-moral properties. However, Moore did not use the term ‘supervenience’ explicitly; it was R. M. Hare that introduced the term into the philosophical literature, to characterize a relationship between moral properties and natural properties. Hare stated

First, let us take that characteristic of ‘good’ which has been called its supervenience. Suppose that we say ‘St. Francis was a good man’. It is logically impossible to say this and to maintain at the same time that there might have been another man placed in precisely the same circumstances as St. Francis, and who behaved in them in exactly the same way, but who differed from St. Francis in this respect only, that he was not a good man.

Thanks to Donald Davidson, the term ‘supervenience’ was first introduced into contemporary philosophy of mind, which opened up a new research direction in this area and other branches of philosophy. Donald Davidson used psychophysical supervenience to defend a position of anomalous monism that although the mental supervenes on the physical, the former cannot be reduced to the latter, as he said:

Although the position I describe denies there are psychophysical laws, it is consistent with the view that mental characteristics are in some sense dependent, or supervenient, on physical characteristics. Such supervenience might be taken to mean that there cannot be two events alike in all physical respects but differing in some mental respect, or that an object cannot alter in some mental respect without altering in some physical respect. Dependence or supervenience of this kind does not entail reducibility through law or definition.

It is alleged that every major figure in the history of western philosophy has been at least implicitly committed to some supervenience thesis. For example, Leibniz used the Latin word ‘supervenire’, to state the thesis that relations are supervenient on properties; G. E. Moore stated that “one of the most important facts about qualitative difference · · · [is that] two things cannot differ in quality without differing in intrinsic nature”; David Lewis used a thesis of Humean supervenience to express that the whole truth about a world like ours supervenes on the spatiotemporal distribution of local qualities.

The notion of supervenience is ubiquitous in our daily life. For instance, the aesthetic properties of a work of art supervene on its physical properties, the price of a commodity supervenes on its supply and demand, effects supervene on causes, and the mental supervenes on the physical. According to the chart of levels of existence, atoms supervene on elementary particles, molecules supervene on atoms, cells supervene on molecules, and so on.

Moreover, a number of interesting doctrines and problems can be formulated in terms of supervenience. A paradigmatic example is physicalism, which may be construed as a thesis that “everything supervenes on the physical”. Mereology may be explained as mereological supervenience, i.e., the whole supervenes on its parts. Determinism can be roughly construed as a thesis that everything to the future supervenes on the present, and perhaps past, facts. All of the distinction between internalism and externalism can be characterized by means of supervenience theses. Mind-body problem may be rephrased as to whether the psychophysical supervenience thesis holds, i.e., are psychological properties supervenient upon physical properties?

There are so many distinct formulations for this concept, e.g., individual supervenience, local supervenience, global supervenience, weak supervenience, strong supervenience, similarity-based supervenience, regional supervenience, local-local supervenience and strong-local-local supervenience, multiple domain supervenience, that David Lewis thought of it as an ‘unlovely proliferation’. No matter how different the formulations are, they all conform to the aforementioned core idea of supervenience – that is, fixing the subvenient properties fixes the supervenient properties.

Supervenience has many applications, among which a central use is so-called ‘argument by a false implied supervenience thesis’. It is well known that the reduction of A to B implies the supervenience of A on B; in short, reduction implies supervenience. Thus for one to argue against a reduction thesis, it suffices to falsify the corresponding supervenience thesis. Other applications include characterizing the distinctions between Internalism and Externalism, characterizing physicalism, characterizing haecceitism, and so on.

Conjuncted: Microlinearity

A Frölicher space X is called microlinear providing that any finite limit diagram D in W yields a limit diagram X ⊗ D in FS, where X ⊗ D is obtained from D by putting X ⊗ to the left of every object and every morphism in D.

The following result should be obvious.

Proposition: Convenient vector spaces are microlinear.
Corollary: C^∞-manifolds are microlinear.
Proposition: If X is a Weil exponentiable and microlinear Frölicher space, then so is X ⊗ W for any Weil algebra W.

Proposition: If X and Y are microlinear Frölicher spaces, then so is X × Y. Proof. This follows simply from the familiar fact that the functor · × · : FS × FS → FS preserves limits.

Proposition: If X is a Weil exponentiable and microlinear Frölicher space, then so is X^Y for any Frölicher space Y .

Theorem: Weil exponentiable and microlinear Frölicher spaces, together with smooth mappings among them, form a cartesian closed subcategory FS_WE,ML of the category FS.

We note in passing that microlinearity is closed under transversal limits.

Theorem: If the limit of a diagram F of microlinear Frölicher spaces is transversal, then it is microlinear.

Proof:

Let X be the limit of the diagram F, i.e., X = Lim F

Let W be the limit of an arbitrarily given finite diagram D of Weil algebras, i.e.,

W = Lim D

We denote by F ⊗ D the diagram obtained from the diagrams F and D by the application of the bifunctor ⊗ : FS × W → FS. By recalling that double limits in a complete category commute, we have

Lim (X ⊗ D)

= Lim ((Lim F) ⊗ D)

= Lim_D Lim_F (F ⊗ D)

[since Lim F is the transversal limit] = Lim_F Lim_D (F ⊗ D)

[since double limits commute] = Lim (F ⊗ (Lim D))

[since every object in F is microlinear] = Lim (F ⊗ W ) 

= (Lim F) ⊗ W

[since Lim F is the transversal limit] = X ⊗ W

Therefore we have the desired result.

Proposition: If a Weil exponentiable Frölicher space X is microlinear, then any finite limit diagram D in W yields a transversal limit diagram X ⊗ D in FS.

Emancipating Microlinearity from within a Well-adapted Model of Synthetic Differential Geometry towards an Adequately Restricted Cartesian Closed Category of Frölicher Spaces. Thought of the Day 15.0

Differential geometry of finite-dimensional smooth manifolds has been generalized by many authors to the infinite-dimensional case by replacing finite-dimensional vector spaces by Hilbert spaces, Banach spaces, Fréchet spaces or, more generally, convenient vector spaces as the local prototype. We know well that the category of smooth manifolds of any kind, whether finite-dimensional or infinite-dimensional, is not cartesian closed, while Frölicher spaces, introduced by Frölicher, do form a cartesian closed category. It seems that Frölicher and his followers do not know what a kind of Frölicher space, besides convenient vector spaces, should become the basic object of research for infinite-dimensional differential geometry. The category of Frölicher spaces and smooth mappings should be restricted adequately to a cartesian closed subcategory.

Synthetic differential geometry is differential geometry with a cornucopia of nilpotent infinitesimals. Roughly speaking, a space of nilpotent infinitesimals of some kind, which exists only within an imaginary world, corresponds to a Weil algebra, which is an entity of the real world. The central object of study in synthetic differential geometry is microlinear spaces. Although the notion of a manifold (=a pasting of copies of a certain linear space) is defined on the local level, the notion of microlinearity is defined absolutely on the genuinely infinitesimal level. What we should do so as to get an adequately restricted cartesian closed category of Frölicher spaces is to emancipate microlinearity from within a well-adapted model of synthetic differential geometry.

Although nilpotent infinitesimals exist only within a well-adapted model of synthetic differential geometry, the notion of Weil functor was formulated for finite-dimensional manifolds and for infinite-dimensional manifolds. This is the first step towards microlinearity for Frölicher spaces. Therein all Frölicher spaces which believe in fantasy that all Weil functors are really exponentiations by some adequate infinitesimal objects in imagination form a cartesian closed category. This is the second step towards microlinearity for Frölicher spaces. Introducing the notion of “transversal limit diagram of Frölicher spaces” after the manner of that of “transversal pullback” is the third and final step towards microlinearity for Frölicher spaces. Just as microlinearity is closed under arbitrary limits within a well-adapted model of synthetic differential geometry, microlinearity for Frölicher spaces is closed under arbitrary transversal limits.

Spirit is Matter on the Seventh Plane; Matter is Spirit – on the Lowest Point of its Cyclic Activity; and Both — are MAYA. Note Quote.

In the 1930s the scientist Sir James Jeans wrote:

the tendency of modem physics is to resolve the whole material universe into waves, and nothing but waves. These waves are of two kinds: bottled-up waves, which we call matter, and unbottled waves, which we call radiation or light. If annihilation of matter occurs, the process is merely that of unbottling imprisoned wave-energy and setting it free to travel through space. These concepts reduce the whole universe to a world of light, potential or existent . . . . — The Mysterious Universe

The idea of matter being crystallized light echoes what H. P. Blavatsky wrote half a century earlier in The Secret Doctrine, where she speaks of “that infinite Ocean of Light, whose one pole is pure Spirit lost in the absoluteness of Non-Being, and the other, the matter in which it condenses, crystallizing into a more and more gross type as it descends into manifestation” (The Secret Doctrine). Material particles, she said, were infinitely divisible centers of force, and matter could therefore exist in infinitely varying degrees of density. Our physical senses have been evolved to perceive only one particular plane of matter, which is interpenetrated by countless other worlds or planes invisible to us because composed of ranges of energy-substance both finer and grosser than our own.

Modern science has analyzed matter down to the point where it vanishes into wisps of energy. Energy is said to be a measure of motion or activity. But motion of what? It is a truism that there can be no motion without something that moves. Scientists in the last century believed that wave-motion took place in a universal medium called the ether. This hypothesis was abandoned because the ether proved to be chemically and physically undetectable, and science was left with the unlikely idea that waves are transmitted through “empty space.”

Modern physicists believe that underlying the material world there is a quantum field, also called the quantum void or vacuum. The quantum field is said to be “a continuous medium which is present everywhere in space” (The Tao of Physics) and matter is said to be constituted by regions of space in which the field is extremely intense. Scientists assert that the quantum field is non-material, but deny that it is mere nothingness. Paul Davies states that the quantum void is not inert and featureless but throbbing with energy and vitality, a seething ferment of “Virtual” particles and “ghost” particles. (Superforce) It therefore seems to be actually a form of ether, which is non-material only in the sense that it is not composed of physical matter. Rather than material particles being “knots of nothingness,” as Davies calls them, they may therefore be seen as vibrations in an etheric medium composed of a subtler, superphysical grade of substance. The same reasoning applies to all the other “non-material” fields and forces postulated by science.

Everything is relative. Physical matter is condensed energy, but what for us is energy would be matter for beings on a higher plane than ours, as is suggested by the fact that energy does not exist in a continuous flow but is composed of discrete units or quanta. Likewise, the energy on the next plane would be matter to an even higher plane. The loftiest form of energy in any particular hierarchy of worlds is what we call spirit or consciousness. As H. P. Blavatsky put it: “Spirit is matter on the seventh plane; matter is Spirit – on the lowest point of its cyclic activity; and both — are MAYA.” (The Secret Doctrine). To say that spirit and matter are “maya” or illusion does not mean that they do not exist, but that we do not understand them as they really are. Any particular plane of energy-substance can be understood only with reference to superior, causal planes. Everything — from atom to human, from star to universe — is the expression of something higher.

Throughout the ages, sages and seers have suggested that hidden within the phenomenal world in which we live there are inner worlds of reality — astral, mental, and spiritual — and that the physical world is but a pale shadow of the spiritual world. These inner worlds cannot be investigated with physical instruments, but only by delving into the depths of our own minds and consciousness, and this requires many lives of self-purification and self-conquest. Scientists using only materialistic methods are in no position to deny point-blank the possibility of such higher planes.

Most scientists, in fact, now believe that some 90% of the matter in the universe exists in a state unknown to them; it is called “dark matter” because it is physically unobservable, and its existence is known of only by its gravitational effects. Such matter is suggestive of the higher subplanes and planes postulated by theosophy, which are composed of matter of increasingly slower rates of vibration and are therefore beyond our range of perception. Given scientists’ confessed ignorance of most of the matter in the universe and their inability to explain satisfactorily the evolution of life and consciousness and the “laws of nature” along materialistic lines, any suggestion that they are on the verge of discovering the innermost secrets of nature or of reducing the mystery of existence to a single equation is premature to say the least!

In theosophical philosophy, the physical universe is regarded as no more than a cross section through infinitude. Universal nature is composed of worlds within worlds within worlds, filled full of conscious, living beings at infinitely varying stages of their evolutionary awakenment. Our finite minds cannot embrace the infinite. As G. de Purucker says in his Fundamentals of the Esoteric Philosophy, we can do no more than to try and form a simple conception of the Boundless All: never-ending life and consciousness in unceasing motion everywhere. The ancients, he says, were never so foolish as to try to fathom infinitude. They recognized the reality of being and let it go at that, knowing that an ever-expanding consciousness and an ever-growing understanding of existence is all that we can ever attain to during our eternal evolutionary journey through the fields of infinitude.