Morphed Ideologies. Thought of the Day 105.0

 

edited political spectrum

The sense of living in a post-fascist world is not shared by Marxists, of course, who ever since the first appearance of Mussolini’s virulently anti-communist squadrismo have instinctively assumed fascism to be be endemic to capitalism. No matter how much it may appear to be an autonomous force, it is for them inextricably bound up with the defensive reaction of bourgeoisie elites or big business to the attempts by revolutionary socialists to bring about the fundamental changes needed to assure social justice through a radical redistribution of wealth and power. According to which school or current of Marxism is carrying out the analysis, the precise sector or agency within capitalism that is the protagonist or backer of fascism’s elaborate pseudo-revolutionary pre-emptive strike, its degree of independence from the bourgeoisie elements who benefit from it, and the amount of genuine support it can win within the working class varies appreciably. But for all concerned, fascism is a copious taxonomic pot into which is thrown without too much intellectual agonizing over definitional or taxonomic niceties. For them, Brecht’s warning at the end of Arturo Ui has lost none of its topicality: “The womb that produced him is still fertile”.

The fact that two such conflicting perspectives can exist on the same subject can be explained as a consequence of the particular nature of all generic concepts within the human sciences. To go further into this phenomenon means entering a field of studies where philosophy of the social sciences has again proliferated conflicting positions, this time concerning the complex and largely subliminal processes involved in conceptualization and modeling in the pursuit of definite, if not definitive, knowledge. According to Max Weber, terms such as capitalism and socialism are ideal types, heuristic devices created by an act of idealizing abstraction. This cognitive process, which in good social scientific practice is carried out as consciously and scrupulously as possible, extracts a small group of salient features perceived as common to a particular generic phenomenon and assembles them into a definitional minimum which is at bottom a utopia.

The result of idealizing abstraction is a conceptually pure, artificially tidy model which does not correspond exactly to any concrete manifestation of the generic phenomenon being investigated, since in reality these are always inextricably mixed up with features, attributes, and surface details which are not considered definitional or as unique to that example of it. The dominant paradigm of the social sciences at any one time, the hegemonic political values and academic tradition prevailing in a particular geography, the political and moral values of the individual researcher all contribute to determining what common features are regarded as salient or definitional. There is no objective reality or objective definition of any aspect of it, and no simple correspondence between a word and what it means, the signifier and the signified, since it is axiomatic to Weber’s world-view that the human mind attaches significance to an essentially absurd universe and thus literally creates value and meaning, even when attempting to understand the world objectively. The basic question to be asked about any definition of fascism therefore, is not whether it is true, but whether it is heuristically useful: what can be seen or understood about concrete human phenomenon when it is applied that could not otherwise be seen, and what is obscured by it.

In his theory of ideological morphology, the British political scientist Michael Freeden has elaborated a nominalist and hence anti-essentialist approach to the definition of generic ideological terms that is deeply compatible with Weberian heuristics. He distinguishes between the ineliminable attributes or properties with which conventional usage endows them and those adjacent and peripheral to them which vary according to specific national, cultural or historical context. To cite the example he gives, liberalism can be argued to contain axiomatically, and hence at its definitional core, the idea of individual, rationally defensible liberty. however, the precise relationship of such liberty to laissez-faire capitalism, nationalism, the sanctuary, or the right of the state to override individual human rights in the defense of collective liberty or the welfare of the majority is infinitely negotiable and contestable. So are the ideal political institutions and policies that a state should adopt in order to guarantee liberty, which explains why democratic politics can never be fully consensual across a range of issues without there being something seriously wrong. It is the fact that each ideology is a cluster of concepts comprising ineliminable with eliminable ones that accounts for the way ideologies are able to evolve over time while still remaining recognizably the same and why so many variants of the same ideology can arise in different societies and historical contexts. It also explains why every concrete permutation of an ideology is simultaneously unique and the manifestation of the generic “ism”, which may assume radical morphological transformations in its outward appearance without losing its definitional ideological core.

 

The Left Needs the Stupid to Survive…

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Social pathologies, or the social pathologist undoubtedly. Orwell developed his Newspeak dictionary in order to explain the cognitive phenomenon he observed about him with regard to those committed to the left. Thats not to say that the cognitive phenomenon cannot be on the right, since many mass movement type ideologies are logically contradictory and to sustain themselves their adherents must engage themselves in mental gyrations to upkeep their belief. Orwell needed the Newspeak as part of the apparatus of totalitarian control, something forced on to an unwitting and unwilling public. It never occurred to Orwell that the masses would never care as long as their animal desires were being provided for. The party, much like the Juvenal before them, recognized that the public would not much care about the higher concepts such as truth or freedom as ling as their bread and circuses, in the form of the cynical statement Prolefeed were supplied. In fact, trying to pry them away from such materialities or ‘truth’ would likely cause them the to support the existing regime. This means that a capitalist totalitarianism, with its superior ability to provide for material goods would be harder to dislodge than a socialist one.

Take for example the notion of Doublethink, the idea of keeping two mutually opposing ideas in one’s head without noticing the difference. Orwell saw this mode as an aberration with regard to normal thought but never realized the fact that this was in the common man a mode of cognition. Or the concept of Bellyfeel, which Orwell states,

Consider, for example, a typical sentence from a Times leading article as “Oldthinkers unbellyfeel Ingsoc”. the shortest rendering one could make of this in Oldspeak would be: “Those whose ideas formed before the revolution cannot have a full understanding of the principle of English socialism.” But, this is not an adequate translation…only a person thoroughly grounded in Ingsoc could appreciate the full force of the word bellyful, which implied a blind, enthusiastic and casual acceptance difficult to imagine today.

“Gut-Instinct”, more than reason, is mass man’s mechanism of political orientation. This is why Fascism and Socialism is better understood as appeals to the gut-brain rather than logically and empirically justified modes of political thought. Totalitarian regimes cannot solely rely on oppression for their survival, they also need to rely on some of cooperation  amongst the population, and they bring this about by exploiting the cognitive miserliness of the average man. Orwell, just like many other left-wing intellectuals never really appreciated the mindset of just outside the proletariat that he was. His fundamental misunderstanding of Newspeak lay in the assumption of rationalist fallacy, which assumes that the average man is rational when it counts, but the problem lies in the fact that for the average man cognitive miserliness is the norm. the problem is that a lot of mainstream conservative thought is based on this premise, which in turn undermines its own survival and helps feed the leftist beast. Any conservatives that believes in the right of the conservative miser to choose is a dead man walking. This criticism of the prole-mind is not based on any snobbery, rather it is of functional basis. Competency, not class should be the eligibility for decision-making, and thus no wonder left needs the stupid to survive.

Homotopies. Thought of the Day 35.0

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One of the major innovations of Homotopy Type Theory is the alternative interpretation of types and tokens it provides using ideas from homotopy theory. Homotopies can be thought of as continuous distortions between functions, or between the images of functions. Facts about homotopy theory are therefore only given ‘up to continuous distortions’, and only facts that are preserved by all such distortions are well-defined. Homotopy is usually presented by starting with topological spaces. Given two such spaces X and Y , we say that continuous maps f, g : X → Y are homotopic, written ‘f ∼ g’, just if there is a continuous map h : [0, 1] × X → Y with h(0, x) = f(x) and h(1, x) = g(x) ∀ x ∈ X. Such a map is a homotopy between f and g. For example, any two curves between the same pair of points in the Euclidean plane are homotopic to one another, because they can be continuously deformed into one another. However, in a space with a hole in it (such as an annulus) there can be paths between two points that are not homotopic, since a path going one way around the hole cannot be continuously deformed into a path going the other way around the hole.

Two spaces X and Y are homotopy equivalent if there are maps f : X → Y and f′ : Y → X such that f′◦ f ∼ idX and f ◦ f′ ∼ idY. This is an equivalence relation between topological spaces, so we can define the equivalence class [X] of all topological spaces homotopy equivalent to X, called the homotopy type of X. Homotopy theory does not distinguish between spaces that are homotopy equivalent, and thus homotopy types, rather than the topological spaces themselves, are the basic objects of study in homotopy theory.

In the homotopy interpretation of the basic language of HoTT we interpret types as homotopy types or ‘spaces’. It is then natural to interpret tokens of a type as ‘points’ in a space. The points of topological space have what we might call absolute identity, being elements of the underlying set. But a homotopy equivalence will in general map a given point x ∈ X to some other x′ ∈ X, and so when we work with homotopy types the absolute identity of the points is lost. Rather, we must say that a token belonging to a type is interpreted as a function from a one-point space into the space.

Given two points a and b in a space X, a path between them is a function γ : [0, 1] → X with γ(0) = a and γ(1) = b. However, given any such path, X can be smoothly distorted by retracting the path along its length toward a. Thus a space containing two distinct points and a path between them is homotopic to a space in which both points coincide (and the path is just a constant path at this point). We may therefore interpret a path between points as an identification of those points. Thus the identity type IdX(a,b) corresponds to the path space of paths from a to b. This also gives a straightforward justification for the principle of path induction: since any path is homotopic to a constant path (which corresponds to a trivial self-identification), any property (that respects homotopy) that holds of all trivial self-identifications must hold of all identifications.

Activism and Militancy: Empire of the Sands. Note Quote.

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Negri writes:

In the post-modern era, as the figure of the people dissolves, the militant is the one who best expresses the life of the multitude: the agent of biopolitical production and resistance against Empire […] When we speak of the militant, we are not thinking of anything like the sad, ascetic agent of the Third International […] We are thinking of nothing like that and of no one who acts on the basis of duty and discipline, who pretends his or her actions are deduced from an ideal plan […] Today the militant cannot even pretend to be a representative, even of the fundamental human needs of the exploited. Revolutionary political militancy today, on the contrary, must rediscover what has always been its proper form: not representational but constituent activity.[…] Militants resist imperial command in a creative way. In other words, resistance is linked immediately with a constitutive investment in the biopolitical realm and to the formation of co-operative apparatuses of production and community.[…] There is an ancient legend that might serve to illuminate the future life of communist militancy: that of Saint Francis of Assisi. Consider his work. To denounce the poverty of the multitude he adopted that common condition and discovered there the ontological power of a new society. The communist militant does the same, identifying in the common condition of the multitude its enormous wealth. Francis in opposition to nascent capitalism refused every instrumental discipline, and in opposition to the mortification of the flesh (in poverty and in the constituted order) he posed a joyous life, including all of being and nature […] Once again in postmodernity we find ourselves in Francis’s situation, posing against the misery of power the joy of being. This is a revolution that no power will control – because biopower and communism, co-operation and revolution remain together, in love, simplicity, and also innocence. This is the irrepressible lightness and joy of being communist.

Once again it is particularly difficult to find any ideas that bear any relation to classical Marxism in the extract above. For Negri, the militant [activist] becomes an individualist who confronts the capitalist system in a “creative” way and who draws his own revolutionary strength from his or her own very uniqueness and his or her capacity to identify with the conditions of the masses. On top of this, the hero of this type of militancy is St. Francis of Assisi! In reality, genuine Marxist activists are able to place themselves at the vanguard of the working class, not only because they have won the trust and respect of workers through their ideas but also because they are able to connect with the political consciousness of the working class at a particular given moment and raise it towards the accomplishment of the socialist transformation of society. These types of activists never act on the basis of their own individuality, but know how to use it by linking it up with the individualities of other activists and put it at the service of the revolution. The political activist is in no way some sort of dour killjoy, but is the driving force of a whole class, the proletariat.

For the activist, being part of the proletariat also means not being afraid to represent it. On the contrary, each day of the activist’s life is dedicated to advancing the working class in its quest for the final victory. The Marxist activist’s revolutionary duty is to organise and lead, without ever becoming separated from his or her own class. Lenin, in a critique of Rosa Luxemburg’s conception of party organisation – which he saw as a vanguard based on revolutionary discipline – says the following in “Left-wing communism, an infantile disorder” about how the discipline of the proletariat’s revolutionary party can be maintained.

First, by the class-consciousness of the proletarian vanguard and by its devotion to the revolution, by its tenacity, self-sacrifice and heroism. Second, by its ability to link up, maintain the closest contact, and – if you wish – merge, in certain measure, with the broadest masses of the working people – primarily with the proletariat, but also with the non-proletarian masses of working people. Third, by the correctness of the political leadership exercised by this vanguard, by the correctness of its political strategy and tactics, provided the broad masses have seen, from their own experience, that they are correct.

All this has little to do with the ideal kind of activist described in the pages of Empire. In conclusion, we have a good suggestion for bringing Negri’s theory face to face with stark reality. What would happen if Negri’s “activist” went to a factory gate, or any other workplace at the beginning of the day’s shift, and invited the workers to “have fun” and “disobey”, in order to subvert the established order? We do not claim to know the conditions of every single workplace or factory, but we are certain that in those places that we know and where we often go to give out leaflets and organise campaigns, the level of alienation and fatigue caused by waged labour under the control of the capitalists is very high. Activists going to workers and proposing to them the type of activity that Negri suggests would be lucky to get away with less than a scratch! Again, once petit-bourgeois theories are confronted with the reality of the situation, they show their completely bankrupt nature.

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).

Badiou’s Materiality as Incorporeal Ontology. Note Quote.

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Badiou criticises the proper form of intuition associated with multiplicities such as space and time. However, his own ’intuitions’ are constrained by set theory. His intuition is therefore as ‘transitory’ as is the ontology in terms of which it is expressed. Following this constrained line of reasoning, however, let me now discuss how Badiou encounters the question of ‘atoms’ and materiality: in terms of the so called ‘atomic’ T-sets.

If topos theory designates the subobject-classifier Ω relationally, the external, set-theoretic T-form reduces the classificatory question again into the incorporeal framework. There is a set-theoretical, explicit order-structure (T,<) contra the more abstract relation 1 → Ω pertinent to categorical topos theory. Atoms then appear in terms of this operator <: the ‘transcendental grading’ that provides the ‘unity through which all the manifold given in an intuition is united in a concept of the object’.

Formally, in terms of an external Heyting algebra this comes down to an entity (A,Id) where A is a set and Id : A → T is a function satisfying specific conditions.

Equaliser: First, there is an ‘equaliser’ to which Badiou refers as the ‘identity’ Id : A × A → T satisfies two conditions:

1) symmetry: Id(x, y) = Id(y, x) and
2) transitivity: Id(x, y) ∧ Id(y, z) ≤ Id(x, z).

They guarantee that the resulting ‘quasi-object’ is objective in the sense of being distinguished from the gaze of the ‘subject’: ‘the differences in degree of appearance are not prescribed by the exteriority of the gaze’.

This analogous ‘identity’-function actually relates to the structural equalization-procedure as appears in category theory. Identities can be structurally understood as equivalence-relations. Given two arrows X ⇒ Y , an equaliser (which always exists in a topos, given the existence of the subobject classifier Ω) is an object Z → X such that both induced maps Z → Y are the same. Given a topos-theoretic object X and U, pairs of elements of X over U can be compared or ‘equivalized’ by a morphism XU × XUeq ΩU structurally ‘internalising’ the synthetic notion of ‘equality’ between two U-elements. Now it is possible to formulate the cumbersome notion of the ‘atom of appearing’.

An atom is a function a : A → T defined on a T -set (A, Id) so that
(A1) a(x) ∧ Id(x, y) ≤ a(y) and
(A2) a(x) ∧ a(y) ≤ Id(x, y).
As expressed in Badiou’s own vocabulary, an atom can be defined as an ‘object-component which, intuitively, has at most one element in the following sense: if there is an element of A about which it can be said that it belongs absolutely to the component, then there is only one. This means that every other element that belongs to the component absolutely is identical, within appearing, to the first’. These two properties in the definition of an atom is highly motivated by the theory of T-sets (or Ω-sets in the standard terminology of topological logic). A map A → T satisfying the first inequality is usually thought as a ‘subobject’ of A, or formally a T-subset of A. The idea is that, given a T-subset B ⊂ A, we can consider the function
IdB(x) := a(x) = Σ{Id(x,y) | y ∈ B}
and it is easy to verify that the first condition is satisfied. In the opposite direction, for a map a satisfying the first condition, the subset
B = {x | a(x) = Ex := Id(x, x)}
is clearly a T-subset of A.
The second condition states that the subobject a : A → T is a singleton. This concept stems from the topos-theoretic internalization of the singleton-function {·} : a → {a} which determines a particular class of T-subsets of A that correspond to the atomic T-subsets. For example, in the case of an ordinary set S and an element s ∈ S the singleton {s} ⊂ S is a particular, atomic type of subset of S.
The question of ‘elements’ incorporated by an object can thus be expressed externally in Badiou’s local theory but ‘internally’ in any elementary topos. For the same reason, there are two ways for an element to be ‘atomic’: in the first sense an ‘element depends solely on the pure (mathematical) thinking of the multiple’, whereas the second sense relates it ‘to its transcendental indexing’. In topos theory, the distinction is slightly more cumbersome. Badiou still requires a further definition in order to state the ‘postulate of materialism’.
An atom a : A → T is real if if there exists an element x ∈ T so that a(y) = Id(x,y) ∀ y ∈ A.
This definition gives rise to the postulate inherent to Badiou’s understanding of ‘democratic materialism’.
Postulate of Materialism: In a T-set (A,Id), every atom of appearance is real.
What the postulate designates is that there really needs to exist s ∈ A for every suitable subset that structurally (read categorically) appears to serve same relations as the singleton {s}. In other words, what ‘appears’ materially, according to the postulate, has to ‘be’ in the set-theoretic, incorporeal sense of ‘ontology’. Topos theoretically this formulation relates to the so called axiom of support generators (SG), which states that the terminal object 1 of the underlying topos is a generator. This means that the so called global elements, elements of the form 1 → X, are enough to determine any particular object X.
Thus, it is this specific condition (support generators) that is assumed by Badiou’s notion of the ‘unity’ or ‘constitution’ of ‘objects’. In particular this makes him cross the line – the one that Kant drew when he asked Quid juris? or ’Haven’t you crossed the limit?’ as Badiou translates.
But even without assuming the postulate itself, that is, when considering a weaker category of T-sets not required to fulfill the postulate of atomism, the category of quasi-T -sets has a functor taking any quasi-T-set A into the corresponding quasi-T-set of singletons SA by x → {x}, where SA ⊂ PA and PA is the quasi-T-set of all quasi-T-subsets, that is, all maps T → A satisfying the first one of the two conditions of an atom designated by Badiou. It can then be shown that, in fact, SA itself is a sheaf whose all atoms are ‘real’ and which then is a proper T-set satisfying the ‘postulate of materialism’. In fact, the category of T-Sets is equivalent to the category of T-sheaves Shvs(T, J). In the language of T-sets, the ‘postulate of materialism’ thus comes down to designating an equality between A and its completed set of singletons SA.

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Diffeomorphism Diffeology via Leaves of Lie Foliation

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The notion of diffeological space is due to Jean-Marie Souriau.

Let M be a set. Any set map α: U ⊂ Rn → M defined on an open set U of some Rn, n ≥ 0, will be called a plot on M. The name plot is chosen instead of chart to avoid some confusion with the usual notion of chart in a manifold. When possible, a plot α with domain U will be simply denoted by αU.

A diffeology of class C on the set M is any collection P of plots α: Uα ⊂ Rnα → M, nα ≥ 0, verifying the following axioms:

  1. (1)  Any constant map c: Rn → M, n ≥ 0, belongs to P;
  2. (2)  Let α ∈ P be defined on U ⊂ Rn and let h: V ⊂ Rm → U ⊂ Rn beany C map; then α ◦ h ∈ P;
  3. (3)  Let α: U ⊂ Rn → M be a plot. If any t ∈ U has a neighbourhood Ut such that α|Ut belongs to P then α ∈ P.

Usually, a diffeology P on the set M is defined by means of a generating set, that is by giving any set G of plots (which is implicitly supposed to contain all constant maps) and taking the least diffeology containing it. Explicitly, the diffeology ⟨G⟩ generated by G is the set of plots α: U → M such that any point t ∈ U has a neighbourhood Ut where α can be written as γ ◦ h for some C map h and some γ ∈ G.

A finite dimensional manifold M is endowed with the diffeology generated by the charts U ⊂ Rn → M, n = dimM, of any atlas.

Basic constructions. A map F : (M, P) → (N, Q) between diffeological spaces is differentiable if F ◦ α ∈ Q for all α ∈ P. A diffeomorphism is a differentiable map with a differentiable inverse.

Let (M,P) be a diffeological space and F : M → N a map of sets. The final diffeology FP on N is that generated by the plots F ◦ α, α ∈ P. A particular case is the quotient diffeology associated to an equivalence relation on M.

Analogously, let (N, Q) be a diffeological space and F : M → N a map of sets. The initial diffeology FQ on M is that generated by the plots α in M such that F ◦ α ∈ Q. A particular case is the induced diffeology on any subset M ⊂ N.

Finally, let D(M,N) be the space of differentiable maps between two diffeological spaces (M,P) and (N,Q). We define the functional diffeology on it by taking as a generating set all plots α: U → D(M,N) such that the associated map α~ : U × M → N given by α~ (t, x) = α(t)(x) is differentiable.

Diffeological groups.

Definition 2.3. A diffeological group is a diffeological space (G,P) endowed with a group structure such that the division map δ : G × G → G, δ(x, y) = xy−1, is differentiable.

A typical example of diffeological group is the diffeomorphism group of a finite dimensional manifold M, endowed with the diffeology induced by D(M,M). It is proven that the diffeomorphism group of the space of leaves of a Lie foliation is a diffeological group too.

Both constructions verify the usual universal properties.

Let (M, P), (N, Q) be two diffeological spaces. We can endow the cartesian product M × N with the product diffeology P × Q generated by the plots α × β, α ∈ P, β ∈ Q.

To Err or Not? Neo-Kantianism’s Logical Flaw. Note Quote.

According to Bertrand Russell, the sense in which every object is ‘one’ is a very shadowy sense because it is applicable to everything alike. However, Russell argues, the sense in which a class may be said to have one member is quite precise. “A class u has one member when u is not null, and ‘x and y are us’ implies ‘x is identical with y’.” In this case the one-ness is a property of a class and Russell calls this class a unit-class. Thus, Russell claims further, the number ‘one’ is not to be asserted of terms but of classes having one member in the above-defined sense. The same distinction between the different uses of ‘one’ was also made by Frege and Couturat. Frege says that the sense in which every object is ‘one’ is very imprecise, that is, every single object possesses this property. However, Frege argues that when one speaks of ‘the number one’, one indicates by means of the definite article a definite and unique object of scientific study. In his reply to Poincaré’s critique of the logicist programme, Couturat says that the confusion which exists in Poincaré’s mind arises from the double meaning of the word for ‘one’, that is, it is used both as a name of a number and as an indefinite article:

To sum up, it is not enough to conceive any one object to conceive the number one, nor to think of two objects together to have by that alone the idea of the number two.

According to Couturat, from the fact that the proposition “x and y are the elements of the class u” contains the symbols x and y one should not conclude that the number two is implied in this proposition. As a result, from the viewpoint of Russell, Couturat and Frege, the neo-Kantians are making here an elementary logical mistake. This awakens an interesting question. Why the neo-Kantians did not notice the mistake they had made? The answer is not that they would not have been aware of the opinion of the logicists. Both Cohn and Cassirer discuss the above-mentioned passage in Russell’s Principles. However, although Cohn and Cassirer were familiar with the distinction presented by Russell, it did not convince them. In Cohn’s view, Russell’s unit-class does not define ‘one’ but ‘only one’. As Cohn sees it, ‘only one’ means the limitation of a class to one object. Thus Russell’s ‘unit-class’ already presupposes that an object is seen as a unit. As a result, Russell’s definition of ‘one’ is unsuccessful since it already presupposes the number ‘one’. Cassirer, too, refers to Russell’s explanation, according to which it is naturally incontestable that every member of a class is in some sense one, but, Cassirer says, it does not follow from this that the concept of ‘one’ is presupposed. Cassirer mentions also Russell’s explanation according to which the meaning of the assertion that a class u possesses ‘one’ member is determined by the fact that this class is not null and that if x and y are u, then x is identical with y. According to Cassirer, the logical function of number is here not so much deduced as rather described by a technical circumlocution. Cassirer argues that in order to comprehend Russell’s explanation it is necessary that the term x is understood as identical with itself, and at the same time it is related to another term y and the former is judged as agreeing with or differing from the latter. In Cassirer’s view, if this process of positing and differentiation is accepted, then all that has been done will be to presuppose the number in the sense of the theory of ordinal number.

The neo-Kantian critique cannot be explained away as a mere logical error. The real reason why they did not accept the distinction is that to accept it would be to accept at least part of the logicist programme. As Warren Goldfarb has pointed out, Poincaré’s argument will be logically in error only if one simultaneously accepts the analysis of notions ‘in no case’ and ‘a class with one object’ that was first made available through modern mathematical logic. In other words, the logicists claim that the appearance of circularity is eliminated when one distinguishes uses of numerical expressions that can be replaced by purely quantificational devices from the full-blooded uses of such expressions that the formal definition is meant to underwrite. Hence the notions ‘in no case’ and ‘a class with one object’ do not presuppose any number theory if one simultaneously accepts the analysis which first-order quantificational logic provides for them. Poincaré does not accept this analysis, and, as result, he can bring the charge of petitio principii.

Like Poincaré, the neo-Kantians were not ready to accept Russell’s analysis of the expression ‘a class with one object’. As they see it, although the notion ‘a class with one object’ does not presuppose the number ‘one’ if one accepts the logicist definition of number, it will presuppose it if one advocates a neo-Kantian theory of number. According to Cassirer, the concept of number is the first and truest expression of rational method in general. Later Cassirer added that number is not merely a product of pure thought but its very prototype and source. It not only originates from the pure regularities of thought but designates the primary and original act to which these regularities ultimately go back. In Natorp’s view, number is the purest and simplest product of thought. Natorp claims that the first precondition for the logical understanding of number is the insight that number has nothing to do with the existing things but that number is only concerned with the pure regularities of thought. Natorp connects number to the fundamental logical function of quantity. In his view, the quantitative function of thought is produced when multiplicity is singled out from the fundamental relation between unity and multiplicity. Moreover, multiplicity is a plurality of distinguishable elements. Plurality, in turn, is necessarily a plurality of unities. Thus unity in the sense of numerical oneness is the unavoidable starting-point, the indispensable foundation of every quantitative positing of pure thought. According to Natorp, the quantitative positing of thought proceeds in three steps. First, pure thought posits something as one. What is posited as one is not important (it can be the world, an atom, and so on). It is only something to which the thought attaches the character of oneness. Second, the positing of the one can be repeated in the sense that while the one remains posited, we can posit always another in comparison with it. This is the way in which we attain plurality. Third and last, we collect the individual positings into a whole, that is, to a new unity in the sense of a unity of several. In this way we attain a definite plurality, that is, “so much” as distinguished from an indefinite set. In other words, one and one and one, and so forth, are here joined to new mental unities (duality, triplicity, and so forth).

According to Cohn, the natural numbers are the most abstract objects possible. Everything thinkable can be an object, and every object has two elements: the thinking-form and the objectivity. The thinking-form belongs to every object, and Cohn calls it “positing”. It can be described by saying that every object is identical with itself. This formal definiteness of an object has nothing to do with the determination of an object with regard to content. Since the thinking-form belongs to every object in the same way, it alone is not enough to form any specific object. The particularity of any individual object, or as Cohn puts it, the objectivity of any individual object, is something new and foreign when compared to the thinking-form of the object. In other words, Cohn argues that the necessary elements of every object are the thinking-form, and the objectivity. As a result, natural numbers are objects which have the thinking-form of identity and the minimum of objectivity, that is, the form of identity must be thought to be filled with something in some way or other. Moreover, Cohn says that his theory of natural numbers presupposes the possibility of arbitrary object-formation, that is, the possibility to construct arbitrarily many objects. On the basis of these two logical presuppositions, Cohn says that we are able to form arbitrarily many objects which are all equal with each other. According to Cohn, all of these objects can be described by the same symbol 1, and after this operation the fundamental equation 1 = 1 can be presented. Cohn says that the two separate symbols 1 in the equation signify different unities and the sign of equality means only that in any arithmetical relation any arbitrary unity can be replaced with any other unity. Moreover, Cohn says that we can collect an arbitrary number of objects into an aggregate, that is, into a new object. This is expressed by the repeated use of the word ‘and’. In arithmetic the combination of unities into a new unity has the form: 1 + 1 + 1 and so on (when ‘and’ is replaced by ‘+’). The most simple combination (1 + 1) can be described as 2, the following one (1 + 1 + 1) as 3, and so on. Thus a new number can always be attained by adding a new unity.