Hochschild Cohomology Tethers to Closed String Algebra by way of Cyclicity.

When we have an open and closed Topological Field Theory (TFT) each element ξ of the closed algebra C defines an endomorphism ξ_a = i_a(ξ) ∈ O_aa of each object a of B, and η ◦ ξ_a = ξ_b ◦ η for each morphism η ∈ O_ba from a to b. The family {ξ_a} thus constitutes a natural transformation from the identity functor 1_B : B → B to itself.

For any C-linear category B we can consider the ring E of natural transformations of 1_B. It is automatically commutative, for if {ξ_a}, {η_a} ∈ E then ξ_a ◦ η_a = η_a ◦ ξ_a by the definition of naturality. (A natural transformation from 1_B to 1_B is a collection of elements {ξ_a ∈ O_aa} such that ξ_a ◦ f = f ◦ ξ_b for each morphism f ∈ O_ab from b to a. But we can take a = b and f = η_a.) If B is a Frobenius category then there is a map π_a^b : O_bb → O_aa for each pair of objects a, b, and we can define j^b : O_bb → E by j^b(η)_a = π_a^b(η) for η ∈ O_bb. In other words, j^b is defined so that the Cardy condition ι_a ◦ j^b = π_a^b holds. But the question arises whether we can define a trace θ : E → C to make E into a Frobenius algebra, and with the property that

θ_a(ι_a(ξ)η) = θ(ξj^a(η)) —– (1)

∀ ξ ∈ E and η ∈ O_aa. This is certainly true if B is a semisimple Frobenius category with finitely many simple objects, for then E is just the ring of complex-valued functions on the set of classes of these simple elements, and we can readily define θ : E → C by θ(ε_a) = θ_a(1_a)², where a is an irreducible object, and ε_a ∈ E is the characteristic function of the point a in the spectrum of E. Nevertheless, a Frobenius category need not be semisimple, and we cannot, unfortunately, take E as the closed string algebra in the general case. If, for example, B has just one object a, and O_aa is a commutative local ring of dimension greater than 1, then E = O_aa, and so ι_a : E → O_aa is an isomorphism, and its adjoint map j^a ought to be an isomorphism too. But that contradicts the Cardy condition, as π_a^a is multiplication by ∑ψ_iψⁱ, which must be nilpotent.

The commutative algebra E of natural endomorphisms of the identity functor of a linear category B is called the Hochschild cohomology HH⁰(B) of B in degree 0. The groups HH^p(B) for p > 0, vanish if B is semisimple, but in the general case they appear to be relevant to the construction of a closed string algebra from B. For any Frobenius category B there is a natural homomorphism K(B) → HH⁰(B) from the Grothendieck group of B, which assigns to an object a the transformation whose value on b is π_b^a(1_a) ∈ O_bb. In the semisimple case this homomorphism induces an isomorphism K(B) ⊗ C → HH⁰(B).

For any additive category B the Hochschild cohomology is defined as the cohomology of the cochain complex in which a k-cochain F is a rule that to each composable k-tuple of morphisms

Y₀ →^φ₁ Y₁ →^φ₂ ··· →^φ_k Y_k —– (2)

assigns F(φ₁,…,φ_k) ∈ Hom(Y₀,Y_k). The differential in the complex is defined by

(dF)(φ₁,…,φ_k+1) = F(φ₂,…,φ_k+1) ◦ φ₁ + ∑_i=1^k(−1)ⁱ F(φ₁,…,φ_i+1 ◦ φ_i,…,φ_k+1) + (−1)^k+1φ_k+1 ◦ F(φ₁,…,φ_k) —– (3)

(Notice, in particular, that a 0-cochain assigns an endomorphism F_Y to each object Y, and is a cocycle if the endomorphisms form a natural transformation. Similarly, a 2-cochain F gives a possible infinitesimal deformation F(φ₁, φ₂) of the composition law (φ₁, φ₂) ↦ φ₂ ◦ φ₁ of the category, and the deformation preserves the associativity of composition iff F is a cocycle.)

In the case of a category B with a single object whose algebra of endomorphisms is O the cohomology just described is usually called the Hochschild cohomology of the algebra O with coefficients in O regarded as a O-bimodule. This must be carefully distinguished from the Hochschild cohomology with coefficients in the dual O-bimodule O^∗. But if O is a Frobenius algebra it is isomorphic as a bimodule to O^∗, and the two notions of Hochschild cohomology need not be distinguished. The same applies to a Frobenius category B: because Hom(Y_k, Y₀) is the dual space of Hom(Y₀, Y_k) we can think of a k-cochain as a rule which associates to each composable k-tuple of morphisms a linear function of an element φ₀ ∈ Hom(Y_k, Y₀). In other words, a k-cochain is a rule which to each “circle” of k + 1 morphisms

···→^φ₀ Y₀ →^φ₁ Y1 →^φ₂···→^φ_k Y_k →^φ₀··· —– (4)

assigns a complex number F(φ₀,φ₁,…,φ_k).

If in this description we restrict ourselves to cochains which are cyclically invariant under rotating the circle of morphisms (φ₀,φ₁,…,φ_k) then we obtain a sub-cochain complex of the Hochschild complex whose cohomology is called the cyclic cohomology HC^∗(B) of the category B. The cyclic cohomology, which evidently maps to the Hochschild cohomology is a more natural candidate for the closed string algebra associated to B than is the Hochschild cohomology. A very natural Frobenius category on which to test these ideas is the category of holomorphic vector bundles on a compact Calabi-Yau manifold.

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Affine Schemes

Let us associate to any commutative ring A its spectrum, that is the topological space Spec A. As a set, Spec A consists of all the prime ideals in A. For each subset S A we define as closed sets in Spec A:

V(S) := {p ∈ Spec A | S ⊂ p} ⊂ Spec A

If X is an affine variety, defined over an algebraically closed field, and O(X) is its coordinate ring, we have that the points of the topological space underlying X are in one-to-one correspondence with the maximal ideals in O(X).

We also define the basic open sets in Spec A as

U_ƒ := Spec A \ V(ƒ) = Spec A_ƒ with ƒ ∈ A,

where A_ƒ = A[ƒ^-1] is the localization of A obtained by inverting the element ƒ. The collection of the basic open sets U_ƒ, ∀ ƒ ∈ A forms a base for Zariski topology. Next, we define the structure sheaf O_A on the topological space Spec A. In order to do this, it is enough to give an assignment

U ↦ O_A(U) for each basic open set U = U_ƒ in Spec A.

The assignment

U_ƒ ↦ A_ƒ

defines a B-sheaf on the topological space Spec A and it extends uniquely to a sheaf of commutative rings on Spec A, called the structure sheaf and denoted by O_A. Moreover, the stalk at a point p ∈ Spec A, O_A,p is the localization A_p of the ring at the prime p. While the differentiable manifolds are locally modeled, as ringed spaces, by (Rⁿ, C^∞_Rⁿ), the schemes are geometric objects modeled by the spectrum of commutative rings.

Affine scheme is a locally ringed space isomorphic to Spec A for some commutative ring A. We say that X is a scheme if X = (|X|, O_X) is a locally ringed space, which is locally isomorphic to affine schemes. In other words, for each x ∈ |X|, ∃ an open set U_x ⊂ |X| such that (U_x, O_X|Ux) is an affine scheme. A morphism of schemes is just a morphism of locally ringed spaces.

There is an equivalence of categories between the category of affine schemes (aschemes) and the category of commutative rings (rings). This equivalence is defined on the objects by

(rings)^op → (aschemes), A ↦ Spec A

In particular a morphism of commutative rings A → B contravariantly to a morphism Spec B → Spec A of the corresponding affine superschemes.

Since any affine variety X is completely described by the knowledge of its coordinate ring O(X), we can associate uniquely to an affine variety X, the affine scheme Spec O(X). A morphism between algebraic varieties determines uniquely a morphism between the corresponding schemes. In the language of categories, we say we have a fully faithful functor from the category of algebraic varieties to the category of schemes.

Tantric Initiation. Thought of the Day 131.0

Man, universe, gods and ritual are not considered separate entities but rather different manifestations of the same Śakti. Therefore, during a particular ritual every element of it is symbolic of something else. The flowers are representative of something else, the incense is representative of something else and so on. This viewpoint is based upon the crucial teaching that “worldly and spiritual” are the two faces of a same coin. One often thinks that “spirituality” is associated with something which is “within”, while “worldliness” is associated with something which is “without”. So, if you see a light “within”, that is a “spiritual” experience, while if you see a light “without”, that is a “worldly” experience. Besides, the worldliness is based on “day-to-day experiences”. It is approximately so. Tantricism considers all to be the manifestation of Śakti, the Divine Mother. So, an external light is as spiritual as an internal one and vice versa. In fact, there is neither spirituality nor worldliness because only one Supreme Consciousness is permeating everything and everyone.

Śakti or the Divine Mother is the core of all tantric practices. She is known as Kuṇḍalinī when residing in a living being. She is the bestower of the Supreme Bliss for all those followers that worship Her according to the sacred rituals and meditations contained in the Tantra-s. Her importance has been emphasized in Niruttaratantra:

बहूनां जन्मनामन्ते शक्तिज्ञानं प्रजायते।
शक्तिज्ञानं विना देवि निर्वाणं नैव जायते॥

Bahūnaṁ janmanāmante śaktijñānaṁ prajāyate|
Śaktijñānaṁ vinā devi nirvāṇaṁ naiva jāyate||

After (ante) many (bahūnām) births (janmanām), the knowledge (jñānam) of Śakti (śakti) is born (in oneself) (prajāyate). Oh goddess (devi)!, without (vinā) the knowledge (jñānam) of Śakti (śakti), Nirvāṇa — final Liberation — (nirvāṇam) does not (na eva) spring up (jāyate).

However, Tantricism should not be “strictly” equated to Shaktism, because there are groups of Śākta-s (followers of Śakti) which are not “tantric” at all. In turn, there are tantric groups that worship Śiva, Viṣṇu, etc. as well as Śakti.

Consequently, one may use a set of elements as representative of other realities. For example: a man represents Śiva and a woman represents Śakti. Then, their union is representative of that of Śiva and Śakti. Microcosm and macrocosm are closely allied to each other, because the two are the manifestation of only one Power. The following fragment extracted from the ancient Tantra-s clearly shows the aforesaid correlation between man, universe, gods and ritual. The sādhaka or practitioner is meditating on the Divine Mother (Śakti) in his heart lotus. He forms a mental image of Śakti there, and begins worshipping Her this way:

हृत्पद्मासनं दद्यात् सहस्रारच्युतामृतैः।
पाद्यं चरणयोर्दद्यान्मनसार्घ्यं निवेदयेत्॥

तेनामृतेनाचमनं स्नानीयमपि कल्पयेत्।
आकाशतत्त्वं वसनं गन्धं तु गन्धतत्त्वकम्॥

चित्तं प्रकल्पयेत् पुष्पं धूपं प्राणान् प्रकल्पयेत्।
तेजस्तत्त्वं च दीपार्थे नैवेद्यं च सुधाम्बुधिम्॥

अनाहतध्वनिं घण्टां वायुतत्त्वं च चामरम्।
नृत्यमिन्द्रियकर्माणि चाञ्चल्यं मनसस्तथा॥

पुष्पं नानाविधं दद्यादात्मनो भावसिद्धये।
अमायामनहङ्कारमरागममदं तथा॥

अमोहकमदम्भं च अद्वेषाक्षोभके तथा।
अमात्सर्यमलोभं च दशपुष्पं प्रकीर्तितम्॥

अहिंसा परमं पुष्पं पुष्पमिन्द्रियनिग्रहम्।
दयाक्षमाज्ञानपुष्पं पञ्चपुष्पं ततः परम्॥

इति पञ्चदशैर्पुष्पैर्भावपुष्पैः प्रपूजयेत्॥

Hṛtpadmāsanaṁ dadyāt sahasrāracyutāmṛtaiḥ|
Pādyaṁ caraṇayordadyānmanasārghyaṁ nivedayet||

Tenāmṛtenācamanaṁ snānīyamapi kalpayet|
Ākāśatattvaṁ vasanaṁ gandhaṁ tu gandhatattvakam||

Cittaṁ prakalpayet puṣpaṁ dhūpaṁ prāṇān prakalpayet|
Tejastattvaṁ ca dīpārthe naivedyaṁ ca sudhāmbudhim||

Anāhatadhvaniṁ ghaṇṭāṁ vāyutattvaṁ ca cāmaram|
Nṛtyamindriyakarmāṇi cāñcalyaṁ manasastathā||

Puṣpaṁ nānāvidhaṁ dadyādātmano bhāvasiddhaye|
Amāyāmanahaṅkāramarāgamamadaṁ tathā||

Amohakamadambham ca adveṣākṣobhake tathā|
Amātsaryamalobhaṁ ca daśapuṣpaṁ prakīrtitam||

Ahiṁsā paramaṁ puṣpamindriyanigraham|
Dayākṣamājñānapuṣpaṁ pañcapuṣpaṁ tataḥ param||

Iti pañcadaśairpuṣpairbhāvapuṣpaiḥ prapūjayet||

He gives (dadyāt… dadyāt) (his) heart (hṛt) lotus (padma) as the seat (āsanam), and the water for washing (pādyam) the feet (caraṇayoḥ) in the form of the nectars (amṛtaiḥ) flowing (cyuta) from Sahasrāra — the supreme Cakra placed at the crown of the head– (sahasrāra). He presents (nivedayet) the offering — lit. water offered to a guest — (arghyam) in the form of (his) mind (manasā).

He also (api) prepares (kalpayet) the water to be sipped from the palm of the hand — a purificatory ceremony that is performed before any ritual or meal — (ācamanam) (as well as) the water to be used in ablutions (snānīyam) by means of that very (tena) nectar (amṛtena). (He gives) the principle (tattvam) of Ākāśa — ether or space– (ākāśa) as the dress (vasanam), and the power of smelling (gandhatattvakam) as the odor (gandham).

He prepares (prakalpayet) (his) mind (manas) as the flower (vai) (and) arranges (prakalpayet) (his) vital energies (prāṇān) as incense (dhūpam). (He) also (ca) (arranges) the principle (tattvam) of Tejas — fire — (tejas) for it to act as (arthe) the lamp (dīpa), and (ca) the ocean (ambudhim) of nectar (sudhā) as the offering of food (naivedyam).

(He prepares) the Anāhata (anāhata) sound — which keeps sounding constantly in the heart lotus — (dhvanim) as the bell (ghaṇṭām), and (ca) the principle (tattvam) of Vāyu –air– (vāyu) as the fly-whisk made of tail of Yak (cāmaram). (He offers) the actions (karmāṇi) of the senses (indriya) as well as (tathā) the unsteadiness (cāñcalyam) of mind (manasaḥ) as dance (nṛtyam).

For realizing (siddhaye) the state (bhāva) of the Self (ātmanaḥ), he gives (dadyāt) flower(s) (puṣpam) of various sorts (nānāvidham): absence of delusion (amāyām), nonegotism (anahaṅkāram), dispassion and detachment (arāgam) as well as (tathā) absence of arrogance (amadam);…… absence of both bewilderment (amohakam) and (ca) deceit (adambham), as well as (tathā) nonmalevolence (adveṣa) and freedom from agitation (akṣobhake); absence of envy (amātsaryam) and (ca) liberty from covetousness (alobham)” — (these virtues) are named (prakīrtitam) the ten (daśa) flower(s) (puṣpam) –.

The supreme (paramam) flower(s) (puṣpam) (known as) Áhiṁsā — nonviolence and harmlessness — (ahiṁsā) and subjugation (nigraham) of the senses (indriya) (along with) the flower(s) (puṣpam) (known as) compassion (dayā), patience (kṣamā) and knowledge (jñāna), (are) therefore (tatas) the highest (param) five (pañca) flowers (puspam). Thus (iti), through (these) fifteen (pañcadaśaiḥ) flowers (puṣpaiḥ), (which are actually fifteen) flowers (puṣpaiḥ) formed from feelings (bhāva), he performs the worship (prapūjayet).

The sādhaka or practitioner uses every object in the ritual as representative of a virtue, state and so on. Therefore, one “must” be initiated in order to understand the Truth according to the Tantra-s, since only then the well-known vedic spirit of renunciation could be replaced for “a reintegration of the worldly life to the purposes of Enlightenment”. The “desire” and all values associated with it are then employed to achieve final Liberation. The tantric practitioner is both a master in spiritual matters and a master in worldly matters, because, in fact, there is no difference between “spiritual” and “worldly”. They are the two aspects in which the Divine Mother (Śakti) is manifested. So, a freed person is one who has transcended all pains and Saṁsāra (transmigration of the souls, that is, to be born and then to die, and to die and then to be born), and one who has acquired astonishing skills to lead a mundane life which is full of fulfillments.

मद्यपानेन मनुजो यदि सिद्धिं लभेत वै।
मद्यपानरताः सर्वे सिद्धिं गच्छन्तु पामराः॥११७॥

मांसभक्षणमात्रेण यदि पुण्यगतिर्भवेत्।
लोके मांसाशिनः सर्वे पुण्यभाजो भवन्त्विह॥११८॥

स्त्रीसम्भोगेन देवेशि यदि मोक्षं व्रजन्ति वै।
सर्वेऽपि जन्तवो लोके मुक्ताः स्युः स्त्रीनिषेवणात्॥११९॥

Madyapānena manujo yadi siddhiṁ labheta vai|
Madyapānaratāḥ sarve siddhiṁ gacchantu pāmarāḥ||117||

Māṁsabhakṣaṇamātreṇa yadi puṇyagatirbhavet|
Loke māṁsāśinaḥ sarve puṇyabhājo bhavantviha||118||

Strīsambhogena deveśi yadi mokṣaṁ vrajanti vai|
Sarve’pi jantavo loke muktāḥ syuḥ strīniṣevaṇāt||119||

If (yadi) a man (manujaḥ) really (vai) could attain (labheta) to Perfection (siddhim) by drinking (pānena) wine (madya), (then) may all (sarve) (those) vile (pāmarāḥ) people who are addicted to drinking (pānaratāḥ) wine (madya) achieve (gacchantu) Perfection (siddhim)!||117||

If (yadi) the achievement (gatiḥ) of Virtue (puṇya) would result (bhavet) from merely (mātreṇa) eating (bhakṣaṇa) meat (māṁsa), (then) may all (sarve) carnivorous beings (māṁsāśinaḥ) in this world (loke… iha) be (bhavantu) virtuous (puṇyabhājaḥ)!||118||

Oh goddess (deveśi)!, if (yadi) (the beings) indeed (vai) attain (vrajanti) to Liberation (mokṣam) through the enjoyment (sambhogena) of women (strī), (then) all (sarve) creatures (jantavaḥ) in this world (loke) would become (syuḥ) liberated (muktāḥ) by frequenting (niṣevaṇāt) women (strī)||119||

Underlying the Non-Perturbative Quantum Geometry of the Quartic Gauge Couplings in 8D.

A lot can be learned by simply focussing on the leading singularities in the moduli space of the effective theory. However, for the sake of performing really non-trivial quantitative tests of the heterotic/F-theory duality, we should try harder in order to reproduce the exact functional form of the couplings ∆_eff(T) from K3 geometry. The hope is, of course, to learn something new about how to do exact non-perturbative computations in D-brane physics.

More specifically, the issue is to eventually determine the extra contributions to the geometric Green’s functions. Having a priori no good clue from first principles how to do this, the results of the previous section, together with experience with four dimensional compactifications with N = 2 supersymmetry, suggest that somehow mirror symmetry should be a useful tool.

The starting point is the observation that threshold couplings of similar structure appear also in four dimensional, N = 2 supersymmetric compactifications of type II strings on Calabi-Yau threefolds. More precisely, these coupling functions multiply operators of the form TrF_G² (in contrast to quartic operators in d = 8), and can be written in the form

∆^(4d)_eff ∼ ln[λ^α₁(1-λ)^α₂(λ’)³] + γ(λ) —– (1)

which is similar to Green’s function

∆eff (λ) = ∆∑N-1prime form (λ) + δ(λ)

It is to be noted that a Green’s function is in general ambiguous up to the addition of a finite piece, and it is this ambiguous piece to which we can formally attribute those extra non-singular, non-perturbative corrections.

The term δ(λ) contributes to dilation flat coordinate. The dilation S is a period associated with the CY threefold moduli space, and like all period integrals, it satisfies a system of linear differential equations. This differential equation may then be translated back into geometry, and this then would hopefully give us a clue about what the relevant quantum geometry is that underlies those quartic gauge couplings in eight dimensions.

The starting point is the families of singular K3 surfaces associated with which are the period integrals that evaluate to the hypergeometric functions. Generally, period integrals satisfy the Picard-Fuchs linear differential equations.

The four-dimensional theories are obtained by compactifying the type II strings on CY threefolds of special type, namely they are fibrations of the K3 surfaces over P^l. The size of the P¹ yields then an additional modulus, whose associated fiat coordinate is precisely the dilaton S (in the dual, heterotic language; from the type II point of view, it is simply another geometric modulus). The K3-fibered threefolds are then associated with enlarged PF systems of the form:

L_N(z, y) = θ_z(θ_z – 2θ_y) – z(θ_z + 1/2N)(θ_z + 1/2 – 1/2N)

L₂(y) = θ_y² – 2y(2θ_y +1)θ_y —– (2)

For perturbative, one-loop contributions on the heterotic side (which capture the full story in d = 8, in contrast to d = 4), we need to consider only the weak coupling limit, which corresponds to the limit of large base space: y ∼ e^-S → 0. Though we might now be tempted to drop all the θ_y ≡ y∂y terms in the PF system, we better note that the θ_y term in L_N(z, y) can a non-vanishing contribution, namely in particular when it hits the logarithmic piece of the dilaton period, S = -In[y] + γ. As a result one finds that the piece , that we want to compute satisfies in the limit y → 0 the following inhomogenous differential equation

L_N . (γϖ₀)(z) = ϖ₀(z) —– (3)

We now apply the inverse of this strategy to our eight dimensional problem. Since we know from the perturbative heterotic calculation what the exact answer for δ must be, we can work backwards and see what inhomogenous differential equation the extra contribution δ(λ) obeys. It satisfies

L_N^⊗2 . (δϖ₀)(z) = ϖ₀²(z) —– (4)

whose homogenous operator

L_N^⊗2(z) = θ_z³ – z(θ_z + 1 – 1/N)(θ_z + 1/2)(θ_z + 1/N) —– (5)

is the symmetric square of the K3 Picard-Fuchs operator. This means that its solution space is given by the symmetric square of the solution space of L_N(z), i.e.,

L_N^⊗2 . (ϖ₀², ϖ₀ϖ₁, ϖ₁²) = 0 —– (6)

Even though the inhomogenous PF equation (4) concisely captures the extra corrections in the eight-dimensional threshold terms, the considerations leading to this equation have been rather formal and it would be clearly desirable to get a better understanding of what it mathematically and physically means.

Note that in the four dimensional situation, the PF operator L_N(z), which figures as homogenous piece in (3), is by construction associated with the K3 fiber of the threefold. By analogy, the homogenous piece of equation (4) should then tell us something about the geometry that is relevant in the eight dimensional situation. Observing that the solution space (6) is given by products of the K3 periods, it is clear what the natural geometrical object is: it must be the symmetric square Sym²(K3) = (K3 x K3)/Ζ₂. Being a hyperkähler manifold, its periods (not subject to world-sheet instanton corrections) indeed enjoy the factorization property exhibited by (6).

Formal similarity of the four and eight-dimensional string compactifications: the underlying quantum geometry that underlies the quadratic or quartic gauge couplings appears to be given by three- or five-folds, which are fibrations of K3 or its symmetric square, respectively. The perturbative computations on the heterotic side are supposdly reproduced by the mirror maps on these manifolds in the limit where the base P^l‘s are large.

The occurrence of such symmetric products is familiar in D-brane physics. The geometrical structure that is relevant to us is however not just the symmetric square of K3, but rather a fibration of it, in the limit of large base space – this is precisely what the content of the inhomogenous PF equation (4) is. It is however not at all obvious to us why this particular structure of a hyperkähler-fibered five-fold would underlie the non-perturbative quantum geometry of the quartic gauge couplings in eight dimensions.

Quantifier – Ontological Commitment: The Case for an Agnostic. Note Quote.

What about the mathematical objects that, according to the platonist, exist independently of any description one may offer of them in terms of comprehension principles? Do these objects exist on the fictionalist view? Now, the fictionalist is not committed to the existence of such mathematical objects, although this doesn’t mean that the fictionalist is committed to the non-existence of these objects. The fictionalist is ultimately agnostic about the issue. Here is why.

There are two types of commitment: quantifier commitment and ontological commitment. We incur quantifier commitment to the objects that are in the range of our quantifiers. We incur ontological commitment when we are committed to the existence of certain objects. However, despite Quine’s view, quantifier commitment doesn’t entail ontological commitment. Fictional discourse (e.g. in literature) and mathematical discourse illustrate that. Suppose that there’s no way of making sense of our practice with fiction but to quantify over fictional objects. Still, people would strongly resist the claim that they are therefore committed to the existence of these objects. The same point applies to mathematical objects.

This move can also be made by invoking a distinction between partial quantifiers and the existence predicate. The idea here is to resist reading the existential quantifier as carrying any ontological commitment. Rather, the existential quantifier only indicates that the objects that fall under a concept (or have certain properties) are less than the whole domain of discourse. To indicate that the whole domain is invoked (e.g. that every object in the domain have a certain property), we use a universal quantifier. So, two different functions are clumped together in the traditional, Quinean reading of the existential quantifier: (i) to assert the existence of something, on the one hand, and (ii) to indicate that not the whole domain of quantification is considered, on the other. These functions are best kept apart. We should use a partial quantifier (that is, an existential quantifier free of ontological commitment) to convey that only some of the objects in the domain are referred to, and introduce an existence predicate in the language in order to express existence claims.

By distinguishing these two roles of the quantifier, we also gain expressive resources. Consider, for instance, the sentence:

(∗) Some fictional detectives don’t exist.

Can this expression be translated in the usual formalism of classical first-order logic with the Quinean interpretation of the existential quantifier? Prima facie, that doesn’t seem to be possible. The sentence would be contradictory! It would state that ∃ fictional detectives who don’t exist. The obvious consistent translation here would be: ¬∃x Fx, where F is the predicate is a fictional detective. But this states that fictional detectives don’t exist. Clearly, this is a different claim from the one expressed in (∗). By declaring that some fictional detectives don’t exist, (∗) is still compatible with the existence of some fictional detectives. The regimented sentence denies this possibility.

However, it’s perfectly straightforward to express (∗) using the resources of partial quantification and the existence predicate. Suppose that “∃” stands for the partial quantifier and “E” stands for the existence predicate. In this case, we have: ∃x (Fx ∧¬Ex), which expresses precisely what we need to state.

Now, under what conditions is the fictionalist entitled to conclude that certain objects exist? In order to avoid begging the question against the platonist, the fictionalist cannot insist that only objects that we can causally interact with exist. So, the fictionalist only offers sufficient conditions for us to be entitled to conclude that certain objects exist. Conditions such as the following seem to be uncontroversial. Suppose we have access to certain objects that is such that (i) it’s robust (e.g. we blink, we move away, and the objects are still there); (ii) the access to these objects can be refined (e.g. we can get closer for a better look); (iii) the access allows us to track the objects in space and time; and (iv) the access is such that if the objects weren’t there, we wouldn’t believe that they were. In this case, having this form of access to these objects gives us good grounds to claim that these objects exist. In fact, it’s in virtue of conditions of this sort that we believe that tables, chairs, and so many observable entities exist.

But recall that these are only sufficient, and not necessary, conditions. Thus, the resulting view turns out to be agnostic about the existence of the mathematical entities the platonist takes to exist – independently of any description. The fact that mathematical objects fail to satisfy some of these conditions doesn’t entail that these objects don’t exist. Perhaps these entities do exist after all; perhaps they don’t. What matters for the fictionalist is that it’s possible to make sense of significant features of mathematics without settling this issue.

Now what would happen if the agnostic fictionalist used the partial quantifier in the context of comprehension principles? Suppose that a vector space is introduced via suitable principles, and that we establish that there are vectors satisfying certain conditions. Would this entail that we are now committed to the existence of these vectors? It would if the vectors in question satisfied the existence predicate. Otherwise, the issue would remain open, given that the existence predicate only provides sufficient, but not necessary, conditions for us to believe that the vectors in question exist. As a result, the fictionalist would then remain agnostic about the existence of even the objects introduced via comprehension principles!

Homotopically Truncated Spaces.

The Eckmann–Hilton dual of the Postnikov decomposition of a space is the homology decomposition (or Moore space decomposition) of a space.

A Postnikov decomposition for a simply connected CW-complex X is a commutative diagram

such that p_n∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. Let F_n be the homotopy fiber of q_n. Then the exact sequence

π_r+1(P_nX) →^q_n∗ π_r+1(P_n−1X) → π_r(F_n) → π_r(P_nX) →^q_n∗ π_r(P_n−1X)

shows that F_n is an Eilenberg–MacLane space K(π_nX, n). Constructing P_n+1(X) inductively from P_n(X) requires knowing the nth k-invariant, which is a map of the form k_n : P_n(X) → Y_n. The space P_n+1(X) is then the homotopy fiber of k_n. Thus there is a homotopy fibration sequence

K(π_n+1X, n+1) → P_n+1(X) → P_n(X) → Y_n

This means that K(π_n+1X, n+1) is homotopy equivalent to the loop space ΩY_n. Consequently,

π_r(Y_n) ≅ π_r−1(ΩY_n) ≅ π_r−1(K(π_n+1X, n+1) = π_n+1X, r = n+2,

= 0, otherwise.

and we see that Y_n is a K(π_n+1X, n+2). Thus the nth k-invariant is a map k_n : P_n(X) → K(π_n+1X, n+2)

Note that it induces the zero map on all homotopy groups, but is not necessarily homotopic to the constant map. The original space X is weakly homotopy equivalent to the inverse limit of the P_n(X).

Applying the paradigm of Eckmann–Hilton duality, we arrive at the homology decomposition principle from the Postnikov decomposition principle by changing:

• • the direction of all arrows
  • π_∗ to H_∗
  • loops Ω to suspensions S
  • fibrations to cofibrations and fibers to cofibers
  • Eilenberg–MacLane spaces K(G, n) to Moore spaces M(G, n)
  • inverse limits to direct limits

A homology decomposition (or Moore space decomposition) for a simply connected CW-complex X is a commutative diagram

such that j_n∗ : H_r(X_≤n) → H_r(X) is an isomorphism for r ≤ n and H_r(X_≤n) = 0 for

r > n. Let C_n be the homotopy cofiber of i_n. Then the exact sequence

H_r(X_≤n−1) →^i_n∗ H_r(X_≤n) → H_r(C_n) →^i_n∗ H_r−1(X_≤n−1) → H_r−1(X_≤n)

shows that C_n is a Moore space M(H_nX, n). Constructing X_≤n+1 inductively from X_≤n requires knowing the nth k-invariant, which is a map of the form k_n : Y_n → X_≤n.

The space X_≤n+1 is then the homotopy cofiber of k_n. Thus there is a homotopy cofibration sequence

Y_n →^k_n X_≤n →^i_n+1 X_≤n+1 → M(H_n+1X, n+1)

This means that M(H_n+1X, n+1) is homotopy equivalent to the suspension SY_n. Consequently,

H^˜_r(Y_n) ≅ H^∼_r+1(SY_n) ≅ H_r+1(M(H_n+1X, n+1)) = H_n+1X, r = n,

= 0, otherwise

and we see that Y_n is an M(H_n+1X, n). Thus the nth k-invariant is a map k_n : M(H_n+1X, n) → X_≤n

It induces the zero map on all reduced homology groups, which is a nontrivial statement to make in degree n:

k_n∗ : H_n(M(H_n+1X, n)) ∼= H_n+1(X) → H_n(X) ∼= H_n(X_≤n)

The original space X is homotopy equivalent to the direct limit of the X_≤n. The Eckmann–Hilton duality paradigm, while being a very valuable organizational principle, does have its natural limitations. Postnikov approximations possess rather good functorial properties: Let p_n(X) : X → P_n(X) be a stage-n Postnikov approximation for X, that is, p_n(X)_∗ : π_r(X) → π_r(P_n(X)) is an isomorphism for r ≤ n and π_r(P_n(X)) = 0 for r > n. If Z is a space with π_r(Z) = 0 for r > n, then any map g : X → Z factors up to homotopy uniquely through P_n(X). In particular, if f : X → Y is any map and p_n(Y) : Y → P_n(Y) is a stage-n Postnikov approximation for Y, then, taking Z = P_n(Y) and g = p_n(Y) ◦ f, there exists, uniquely up to homotopy, a map p_n(f) : P_n(X) → P_n(Y) such that

homotopy commutes. Let X = S² ∪₂ e³ be a Moore space M(Z/2,2) and let Y = X ∨ S³. If X_≤2 and Y_≤2 denote stage-2 Moore approximations for X and Y, respectively, then X_≤2 = X and Y_≤2 = X. We claim that whatever maps i : X_≤2 → X and j : Y_≤2 → Y such that i_∗ : H_r(X_≤2) → H_r(X) and j_∗ : H_r(Y_≤2) → H_r(Y) are isomorphisms for r ≤ 2 one takes, there is always a map f : X → Y that cannot be compressed into the stage-2 Moore approximations, i.e. there is no map f_≤2 : X_≤2 → Y_≤2 such that

commutes up to homotopy. We shall employ the universal coefficient exact sequence for homotopy groups with coefficients. If G is an abelian group and M(G, n) a Moore space, then there is a short exact sequence

0 → Ext(G, π_n+1Y) →^ι [M(G, n), Y] →^η Hom(G, π_nY) → 0,

where Y is any space and [−,−] denotes pointed homotopy classes of maps. The map η is given by taking the induced homomorphism on π_n and using the Hurewicz isomorphism. This universal coefficient sequence is natural in both variables. Hence, the following diagram commutes:

Here we will briefly write E₂(−) = Ext(Z/2,−) so that E₂(G) = G/2G, and E^Y (−) = Ext(−, π₃Y). By the Hurewicz theorem, π₂(X) ∼= H₂(X) ∼= Z/2, π₂(Y) ∼= H₂(Y) ∼= Z/2, and π₂(i) : π₂(X_≤2) → π₂(X), as well as π₂(j) : π₂(Y_≤2) → π₂(Y), are isomorphisms, hence the identity. If a homomorphism φ : A → B of abelian groups is onto, then E₂(φ) : E₂(A) = A/2A → B/2B = E₂(B) remains onto. By the Hurewicz theorem, Hur : π₃(Y) → H₃(Y) = Z is onto. Consequently, the induced map E₂(Hur) : E₂(π₃Y) → E₂(H₃Y) = E₂(Z) = Z/2 is onto. Let ξ ∈ E₂(H₃Y) be the generator. Choose a preimage x ∈ E₂(π₃Y), E₂(Hur)(x) = ξ and set [f] = ι(x) ∈ [X,Y]. Suppose there existed a homotopy class [f_≤2] ∈ [X_≤2, Y_≤2] such that

j_∗[f_≤2] = i_∗[f].

Then

η_≤2[f_≤2] = π₂(j)_∗η_≤2[f_≤2] = ηj_∗[f_≤2] = ηi^∗[f] = π₂(i)^∗η[f] = π₂(i)^∗ηι(x) = 0.

Thus there is an element ε ∈ E₂(π₃Y_≤2) such that ι_≤2(ε) = [f_≤2]. From ιE₂π₃(j)(ε) = j_∗ι_≤2(ε) = j_∗[f_≤2] = i^∗[f] = i^∗ι(x) = ιE^Y π₂(i)(x)

we conclude that E₂π₃(j)(ε) = x since ι is injective. By naturality of the Hurewicz map, the square

commutes and induces a commutative diagram upon application of E₂(−):

It follows that

ξ = E₂(Hur)(x) = E₂(Hur)E₂π₃(j)(ε) = E₂H₃(j)E₂(Hur)(ε) = 0,

a contradiction. Therefore, no compression [f_≤2] of [f] exists.

Given a cellular map, it is not always possible to adjust the extra structure on the source and on the target of the map so that the map preserves the structures. Thus the category theoretic setup automatically, and in a natural way, singles out those continuous maps that can be compressed into homologically truncated spaces.

Kant and Non-Euclidean Geometries. Thought of the Day 94.0

The argument that non-Euclidean geometries contradict Kant’s doctrine on the nature of space apparently goes back to Hermann Helmholtz and was retaken by several philosophers of science such as Hans Reichenbach (The Philosophy of Space and Time) who devoted much work to this subject. In a essay written in 1870, Helmholtz argued that the axioms of geometry are not a priori synthetic judgments (in the sense given by Kant), since they can be subjected to experiments. Given that Euclidian geometry is not the only possible geometry, as was believed in Kant’s time, it should be possible to determine by means of measurements whether, for instance, the sum of the three angles of a triangle is 180 degrees or whether two straight parallel lines always keep the same distance among them. If it were not the case, then it would have been demonstrated experimentally that space is not Euclidean. Thus the possibility of verifying the axioms of geometry would prove that they are empirical and not given a priori.

Helmholtz developed his own version of a non-Euclidean geometry on the basis of what he believed to be the fundamental condition for all geometries: “the possibility of figures moving without change of form or size”; without this possibility, it would be impossible to define what a measurement is. According to Helmholtz:

the axioms of geometry are not concerned with space-relations only but also at the same time with the mechanical deportment of solidest bodies in motion.

Nevertheless, he was aware that a strict Kantian might argue that the rigidity of bodies is an a priori property, but

then we should have to maintain that the axioms of geometry are not synthetic propositions… they would merely define what qualities and deportment a body must have to be recognized as rigid.

At this point, it is worth noticing that Helmholtz’s formulation of geometry is a rudimentary version of what was later developed as the theory of Lie groups. As for the transport of rigid bodies, it is well known that rigid motion cannot be defined in the framework of the theory of relativity: since there is no absolute simultaneity of events, it is impossible to move all parts of a material body in a coordinated and simultaneous way. What is defined as the length of a body depends on the reference frame from where it is observed. Thus, it is meaningless to invoke the rigidity of bodies as the basis of a geometry that pretend to describe the real world; it is only in the mathematical realm that the rigid displacement of a figure can be defined in terms of what mathematicians call a congruence.

Arguments similar to those of Helmholtz were given by Reichenbach in his intent to refute Kant’s doctrine on the nature of space and time. Essentially, the argument boils down to the following: Kant assumed that the axioms of geometry are given a priori and he only had classical geometry in mind, Einstein demonstrated that space is not Euclidean and that this could be verified empirically, ergo Kant was wrong. However, Kant did not state that space must be Euclidean; instead, he argued that it is a pure form of intuition. As such, space has no physical reality of its own, and therefore it is meaningless to ascribe physical properties to it. Actually, Kant never mentioned Euclid directly in his work, but he did refer many times to the physics of Newton, which is based on classical geometry. Kant had in mind the axioms of this geometry which is a most powerful tool of Newtonian mechanics. Actually, he did not even exclude the possibility of other geometries, as can be seen in his early speculations on the dimensionality of space.

The important point missed by Reichenbach is that Riemannian geometry is necessarily based on Euclidean geometry. More precisely, a Riemannian space must be considered as locally Euclidean in order to be able to define basic concepts such as distance and parallel transport; this is achieved by defining a flat tangent space at every point, and then extending all properties of this flat space to the globally curved space (Luther Pfahler Eisenhart Riemannian Geometry). To begin with, the structure of a Riemannian space is given by its metric tensor g_μν from which the (differential) length is defined as ds² = g_μν dx^μ dx^ν; but this is nothing less than a generalization of the usual Pythagoras theorem in Euclidean space. As for the fundamental concept of parallel transport, it is taken directly from its analogue in Euclidean space: it refers to the transport of abstract (not material, as Helmholtz believed) figures in such a space. Thus Riemann’s geometry cannot be free of synthetic a priori propositions because it is entirely based upon concepts such as length and congruence taken form Euclid. We may conclude that Euclids geometry is the condition of possibility for a more general geometry, such as Riemann’s, simply because it is the natural geometry adapted to our understanding; Kant would say that it is our form of grasping space intuitively. The possibility of constructing abstract spaces does not refute Kant’s thesis; on the contrary, it reinforces it.

How are Topological Equivalences of Structures Homeomorphic?

Given a first-order vocabulary 𝜏, 𝐿_𝜔𝜔(𝜏) is the set of first-order sentences of type 𝜏. The elementary topology on the class 𝑆𝑡_𝜏 of first-order structures type 𝜏 is obtained by taking the family of elementary classes

𝑀𝑜𝑑(𝜑) = {𝑀:𝑀 |= 𝜑}, 𝜑 ∈ 𝐿_𝜔𝜔(𝜏)

as an open basis. Due to the presence of classical negation, this family is also a closed basis and thus the closed classes of 𝑆𝑡_𝜏 are the first-order axiomatizable classes 𝑀𝑜𝑑(𝑇), 𝑇 ⊆ 𝐿_𝜔𝜔(𝜏). Possible foundational problems due to the fact that the topology is a class of classes may be settled observing that it is indexed by a set, namely the set of theories of type 𝜏.

The main facts of model theory are reflected by the topological properties of these spaces. Thus, the downward Löwenheim-Skolem theorem for sentences amounts to topological density of the subclass of countable structures. Łoś theorem on ultraproducts grants that U-limits exist for any ultrafilter 𝑈, condition well known to be equivalent to topological compactness, and to model theoretic compactness in this case.

These spaces are not Hausdorff or T₁, but having a clopen basis they are regular; that is, closed classes and exterior points may be separated by disjoint open classes. All properties or regular compact spaces are then available: normality, complete regularity, uniformizability, the Baire property, etc.

Many model theoretic properties are related to the continuity of natural operations between classes of structures, where operations are seen to be continuous and play an important role in abstract model theory.

A topological space is regular if closed sets and exterior points may be separated by open sets. It is normal if disjoint closed sets may be separated by disjoint open sets. Thus, normality does not imply regularity here. However, a regular compact space is normal. Actually, a regular Lindelöf space is already normal

Consider the following equivalence relation in a space 𝑋: 𝑥 ≡ 𝑦 ⇔ 𝑐𝑙{𝑥} = 𝑐𝑙{𝑦}

where 𝑐𝑙 denotes topological adherence. Clearly, 𝑥 ≡ 𝑦 iff 𝑥 and 𝑦 belong to the same closed (open) subsets (of a given basis). Let 𝑋_/≡ be the quotient space and 𝜂 : 𝑋 → 𝑋_/≡ the natural projection. Then 𝑋_/≡ is T₀ by construction but not necessarily Hausdorff. The following claims thus follow:

a) 𝜂 : 𝑋 → 𝑋_/≡ induces an isomorphism between the respective lattices of Borel subsets of 𝑋 and 𝑋_/≡. In particular, it is open and closed, preserves disjointedness, preserves and reflects compactness and normality.

b) The assignment 𝑋 → 𝑋_/≡ is functorial, because ≡ is preserved by continuous functions and thus any continuous map 𝑓 : 𝑋 → 𝑌 induces a continuous assignment 𝑓_/≡ : 𝑋_/≡ → 𝑌_/≡ which commutes with composition.

c) 𝑋 → 𝑋_/≡ preserves products; that is, (𝛱_𝑖𝑋_𝑖)_/≡ is canonically homeomorphic to 𝛱_𝑖(𝑋_𝑖/≡) with the product topology (monomorphisms are not preserved).

d) If 𝑋 is regular, the equivalence class of 𝑥 is 𝑐𝑙{𝑥} (this may fail in the non-regular case).

e) If 𝑋 is regular, 𝑋_/≡ is Hausdorff : if 𝑥 ≢ 𝑦 then 𝑥 ∉ 𝑐𝑙{𝑦} by (d); thus there are disjoint open sets 𝑈, 𝑉 in 𝑋 such that 𝑥 ∈ 𝑈, 𝑐𝑙{𝑦} ⊆ 𝑉, and their images under 𝜂 provide an open separation of 𝜂𝑥 and 𝜂𝑦 in 𝑋_/≡ by (a).

f) If 𝐾₁ and 𝐾₂ are disjoint compact subsets of a regular topological space 𝑋 that cannot be separated by open sets there exist 𝑥_𝑖 ∈ 𝐾_𝑖, 𝑖 = 1, 2, such that 𝑥₁ ≡ 𝑥₂. Indeed, 𝜂𝐾₁ and 𝜂𝐾₂ are compact in 𝑋_/≡ by continuity and thus closed because 𝑋_/≡ is Hausdorff by (e). They can not be disjoint; otherwise, they would be separated by open sets whose inverse images would separate 𝐾₁ and 𝐾₂. Pick 𝜂𝑥 = 𝜂𝑦 ∈ 𝜂𝐾₁ ∩ 𝜂𝐾₂ with 𝑥 ∈ 𝐾₁, 𝑦 ∈ 𝐾₂.

Clearly then, for the elementary topology on 𝑆𝑡_𝜏, the relation ≡ coincides with elementary equivalence of structures and 𝑆𝑡_𝜏/≡ is homeomorphic to the Stone space of complete theories.

Intuition

During his attempt to axiomatize the category of all categories, Lawvere says

Our intuition tells us that whenever two categories exist in our world, then so does the corresponding category of all natural transformations between the functors from the first category to the second (The Category of Categories as a Foundation).

However, if one tries to reduce categorial constructions to set theory, one faces some serious problems in the case of a category of functors. Lawvere (who, according to his aim of axiomatization, is not concerned by such a reduction) relies here on “intuition” to stress that those working with categorial concepts despite these problems have the feeling that the envisaged construction is clear, meaningful and legitimate. Not the reducibility to set theory, but an “intuition” to be specified answers for clarity, meaningfulness and legitimacy of a construction emerging in a mathematical working situation. In particular, Lawvere relies on a collective intuition, a common sense – for he explicitly says “our intuition”. Further, one obviously has to deal here with common sense on a technical level, for the “we” can only extend to a community used to the work with the concepts concerned.

In the tradition of philosophy, “intuition” means immediate, i.e., not conceptually mediated cognition. The use of the term in the context of validity (immediate insight in the truth of a proposition) is to be thoroughly distinguished from its use in the sensual context (the German Anschauung). Now, language is a manner of representation, too, but contrary to language, in the context of images the concept of validity is meaningless.

Obviously, the aspect of cognition guiding is touched on here. Especially the sensual intuition can take the guiding (or heuristic) function. There have been many working situations in history of mathematics in which making the objects of investigation accessible to a sensual intuition (by providing a Veranschaulichung) yielded considerable progress in the development of the knowledge concerning these objects. As an example, take the following account by Emil Artin of Emmy Noether’s contribution to the theory of algebras:

Emmy Noether introduced the concept of representation space – a vector space upon which the elements of the algebra operate as linear transformations, the composition of the linear transformation reflecting the multiplication in the algebra. By doing so she enables us to use our geometric intuition.

Similarly, Fréchet thinks to have really “powered” research in the theory of functions and functionals by the introduction of a “geometrical” terminology:

One can [ …] consider the numbers of the sequence [of coefficients of a Taylor series] as coordinates of a point in a space [ …] of infinitely many dimensions. There are several advantages to proceeding thus, for instance the advantage which is always present when geometrical language is employed, since this language is so appropriate to intuition due to the analogies it gives birth to.

Mathematical terminology often stems from a current language usage whose (intuitive, sensual) connotation is welcomed and serves to give the user an “intuition” of what is intended. While Category Theory is often classified as a highly abstract matter quite remote from intuition, in reality it yields, together with its applications, a multitude of examples for the role of current language in mathematical conceptualization.

This notwithstanding, there is naturally also a tendency in contemporary mathematics to eliminate as much as possible commitments to (sensual) intuition in the erection of a theory. It seems that algebraic geometry fulfills only in the language of schemes that essential requirement of all contemporary mathematics: to state its definitions and theorems in their natural abstract and formal setting in which they can be considered independent of geometric intuition (Mumford D., Fogarty J. Geometric Invariant Theory).

In the pragmatist approach, intuition is seen as a relation. This means: one uses a piece of language in an intuitive manner (or not); intuitive use depends on the situation of utterance, and it can be learned and transformed. The reason for this relational point of view, consists in the pragmatist conviction that each cognition of an object depends on the means of cognition employed – this means that for pragmatism there is no intuitive (in the sense of “immediate”) cognition; the term “intuitive” has to be given a new meaning.

What does it mean to use something intuitively? Heinzmann makes the following proposal: one uses language intuitively if one does not even have the idea to question validity. Hence, the term intuition in the Heinzmannian reading of pragmatism takes a different meaning, no longer signifies an immediate grasp. However, it is yet to be explained what it means for objects in general (and not only for propositions) to “question the validity of a use”. One uses an object intuitively, if one is not concerned with how the rules of constitution of the object have been arrived at, if one does not focus the materialization of these rules but only the benefits of an application of the object in the present context. “In principle”, the cognition of an object is determined by another cognition, and this determination finds its expression in the “rules of constitution”; one uses it intuitively (one does not bother about the being determined of its cognition), if one does not question the rules of constitution (does not focus the cognition which determines it). This is precisely what one does when using an object as a tool – because in doing so, one does not (yet) ask which cognition determines the object. When something is used as a tool, this constitutes an intuitive use, whereas the use of something as an object does not (this defines tool and object). Here, each concept in principle can play both roles; among two concepts, one may happen to be used intuitively before and the other after the progress of insight. Note that with respect to a given cognition, Peirce when saying “the cognition which determines it” always thinks of a previous cognition because he thinks of a determination of a cognition in our thought by previous thoughts. In conceptual history of mathematics, however, one most often introduced an object first as a tool and only after having done so did it come to one’s mind to ask for “the cognition which determines the cognition of this object” (that means, to ask how the use of this object can be legitimized).

The idea that it could depend on the situation whether validity is questioned or not has formerly been overlooked, perhaps because one always looked for a reductionist epistemology where the capacity called intuition is used exclusively at the last level of regression; in a pragmatist epistemology, to the contrary, intuition is used at every level in form of the not thematized tools. In classical systems, intuition was not simply conceived as a capacity; it was actually conceived as a capacity common to all human beings. “But the power of intuitively distinguishing intuitions from other cognitions has not prevented men from disputing very warmly as to which cognitions are intuitive”. Moreover, Peirce criticises strongly cartesian individualism (which has it that the individual has the capacity to find the truth). We could sum up this philosophy thus: we cannot reach definite truth, only provisional; significant progress is not made individually but only collectively; one cannot pretend that the history of thought did not take place and start from scratch, but every cognition is determined by a previous cognition (maybe by other individuals); one cannot uncover the ultimate foundation of our cognitions; rather, the fact that we sometimes reach a new level of insight, “deeper” than those thought of as fundamental before, merely indicates that there is no “deepest” level. The feeling that something is “intuitive” indicates a prejudice which can be philosophically criticised (even if this does not occur to us at the beginning).

In our approach, intuitive use is collectively determined: it depends on the particular usage of the community of users whether validity criteria are or are not questioned in a given situation of language use. However, it is acknowledged that for example scientific communities develop usages making them communities of language users on their own. Hence, situations of language use are not only partitioned into those where it comes to the users’ mind to question validity criteria and those where it does not, but moreover this partition is specific to a particular community (actually, the community of language users is established partly through a peculiar partition; this is a definition of the term “community of language users”). The existence of different communities with different common senses can lead to the following situation: something is used intuitively by one group, not intuitively by another. In this case, discussions inside the discipline occur; one has to cope with competing common senses (which are therefore not really “common”). This constitutes a task for the historian.

Mathematical Reductionism: As Case Via C. S. Peirce’s Hypothetical Realism.

During the 20th century, the following epistemology of mathematics was predominant: a sufficient condition for the possibility of the cognition of objects is that these objects can be reduced to set theory. The conditions for the possibility of the cognition of the objects of set theory (the sets), in turn, can be given in various manners; in any event, the objects reduced to sets do not need an additional epistemological discussion – they “are” sets. Hence, such an epistemology relies ultimately on ontology. Frege conceived the axioms as descriptions of how we actually manipulate extensions of concepts in our thinking (and in this sense as inevitable and intuitive “laws of thought”). Hilbert admitted the use of intuition exclusively in metamathematics where the consistency proof is to be done (by which the appropriateness of the axioms would be established); Bourbaki takes the axioms as mere hypotheses. Hence, Bourbaki’s concept of justification is the weakest of the three: “it works as long as we encounter no contradiction”; nevertheless, it is still epistemology, because from this hypothetical-deductive point of view, one insists that at least a proof of relative consistency (i.e., a proof that the hypotheses are consistent with the frequently tested and approved framework of set theory) should be available.

Doing mathematics, one tries to give proofs for propositions, i.e., to deduce the propositions logically from other propositions (premisses). Now, in the reductionist perspective, a proof of a mathematical proposition yields an insight into the truth of the proposition, if the premisses are already established (if one has already an insight into their truth); this can be done by giving in turn proofs for them (in which new premisses will occur which ask again for an insight into their truth), or by agreeing to put them at the beginning (to consider them as axioms or postulates). The philosopher tries to understand how the decision about what propositions to take as axioms is arrived at, because he or she is dissatisfied with the reductionist claim that it is on these axioms that the insight into the truth of the deduced propositions rests. Actually, this epistemology might contain a short-coming since Poincaré (and Wittgenstein) stressed that to have a proof of a proposition is by no means the same as to have an insight into its truth.

Attempts to disclose the ontology of mathematical objects reveal the following tendency in epistemology of mathematics: Mathematics is seen as suffering from a lack of ontological “determinateness”, namely that this science (contrarily to many others) does not concern material data such that the concept of material truth is not available (especially in the case of the infinite). This tendency is embarrassing since on the other hand mathematical cognition is very often presented as cognition of the “greatest possible certainty” just because it seems not to be bound to material evidence, let alone experimental check.

The technical apparatus developed by the reductionist and set-theoretical approach nowadays serves other purposes, partly for the reason that tacit beliefs about sets were challenged; the explanations of the science which it provides are considered as irrelevant by the practitioners of this science. There is doubt that the above mentioned sufficient condition is also necessary; it is not even accepted throughout as a sufficient one. But what happens if some objects, as in the case of category theory, do not fulfill the condition? It seems that the reductionist approach, so to say, has been undocked from the historical development of the discipline in several respects; an alternative is required.

Anterior to Peirce, epistemology was dominated by the idea of a grasp of objects; since Descartes, intuition was considered throughout as a particular, innate capacity of cognition (even if idealists thought that it concerns the general, and empiricists that it concerns the particular). The task of this particular capacity was the foundation of epistemology; already from Aristotle’s first premisses of syllogism, what was aimed at was to go back to something first. In this traditional approach, it is by the ontology of the objects that one hopes to answer the fundamental question concerning the conditions for the possibility of the cognition of these objects. One hopes that there are simple “basic objects” to which the more complex objects can be reduced and whose cognition is possible by common sense – be this an innate or otherwise distinguished capacity of cognition common to all human beings. Here, epistemology is “wrapped up” in (or rests on) ontology; to do epistemology one has to do ontology first.

Peirce shares Kant’s opinion according to which the object depends on the subject; however, he does not agree that reason is the crucial means of cognition to be criticised. In his paper “Questions concerning certain faculties claimed for man”, he points out the basic assumption of pragmatist philosophy: every cognition is semiotically mediated. He says that there is no immediate cognition (a cognition which “refers immediately to its object”), but that every cognition “has been determined by a previous cognition” of the same object. Correspondingly, Peirce replaces critique of reason by critique of signs. He thinks that Kant’s distinction between the world of things per se (Dinge an sich) and the world of apparition (Erscheinungswelt) is not fruitful; he rather distinguishes the world of the subject and the world of the object, connected by signs; his position consequently is a “hypothetical realism” in which all cognitions are only valid with reservations. This position does not negate (nor assert) that the object per se (with the semiotical mediation stripped off) exists, since such assertions of “pure” existence are seen as necessarily hypothetical (that means, not withstanding philosophical criticism).

By his basic assumption, Peirce was led to reveal a problem concerning the subject matter of epistemology, since this assumption means in particular that there is no intuitive cognition in the classical sense (which is synonymous to “immediate”). Hence, one could no longer consider cognitions as objects; there is no intuitive cognition of an intuitive cognition. Intuition can be no more than a relation. “All the cognitive faculties we know of are relative, and consequently their products are relations”. According to this new point of view, intuition cannot any longer serve to found epistemology, in departure from the former reductionist attitude. A central argument of Peirce against reductionism or, as he puts it,

the reply to the argument that there must be a first is as follows: In retracing our way from our conclusions to premisses, or from determined cognitions to those which determine them, we finally reach, in all cases, a point beyond which the consciousness in the determined cognition is more lively than in the cognition which determines it.

Peirce gives some examples derived from physiological observations about perception, like the fact that the third dimension of space is inferred, and the blind spot of the retina. In this situation, the process of reduction loses its legitimacy since it no longer fulfills the function of cognition justification. At such a place, something happens which I would like to call an “exchange of levels”: the process of reduction is interrupted in that the things exchange the roles performed in the determination of a cognition: what was originally considered as determining is now determined by what was originally considered as asking for determination.

The idea that contents of cognition are necessarily provisional has an effect on the very concept of conditions for the possibility of cognitions. It seems that one can infer from Peirce’s words that what vouches for a cognition is not necessarily the cognition which determines it but the livelyness of our consciousness in the cognition. Here, “to vouch for a cognition” means no longer what it meant before (which was much the same as “to determine a cognition”), but it still means that the cognition is (provisionally) reliable. This conception of the livelyness of our consciousness roughly might be seen as a substitute for the capacity of intuition in Peirce’s epistemology – but only roughly, since it has a different coverage.