Axiomatic Method and Category Theoryphilomatica.org/wp-content/uploads/2012/12/amcat.pdf ·...

OutlinePast there stages of development of the Axiomatic MethodPresent Formal Axiomatic Method in the the 20th century mathematics

Future the Geometrical TurnConclusion

Axiomatic Method and Category Theory

Andrei Rodin

21 avril 2012

Andrei Rodin Axiomatic Method and Category Theory



Past there stages of development of the Axiomatic MethodEuclidHilbert 1899Hilbert 1927-39

Present Formal Axiomatic Method in the the 20th centurymathematics

Axiomatic Set theoryBoorbaki

Future the Geometrical TurnPrehistory Venn and TarskiPrehistory Schonfinkel and CurryCategorical LogicToposesHomotopy Type theory

Conclusion




EuclidHilbert 1899Hilbert 1927-39

Theorem 15 of Euclidrsquos ELEMENTS

[enunciation ]

For isosceles triangles the angles at the base are equal toone another and if the equal straight lines are producedthen the angles under the base will be equal to oneanother





Theorem 15 (continued)

[exposition]

Let ABC be an isosceles triangle having the side ABequal to the side AC and let the straight lines BD andCE have been produced further in a straight line with ABand AC (respectively) [Post 2]






[specification ]

I say that the angle ABC is equal to ACB and (angle)CBD to BCE






[construction ]

For let a point F be taken somewhere on BD and letAG have been cut off from the greater AE equal to thelesser AF [Prop 13] Also let the straight lines FC GBhave been joined [Post 1]






[proof ]

In fact since AF is equal to AG and AB to AC the two(straight lines) FA AC are equal to the two (straightlines) GA AB respectively They also encompass acommon angle FAG Thus the base FC is equal to thebase GB and the triangle AFC will be equal to thetriangle AGB and the remaining angles subtended by theequal sides will be equal to the corresponding remainingangles [Prop 14] (That is) ACF to ABG and AFC toAGB And since the whole of AF is equal to the whole ofAG within which AB is equal to AC the remainder BFis thus equal to the remainder CG [Ax3] But FC wasalso shown (to be) equal to GB






So the two (straight lines) BF FC are equal to the two(straight lines) CG GB respectively and the angle BFC(is) equal to the angle CGB while the base BC iscommon to them Thus the triangle BFC will be equal tothe triangle CGB and the remaining angles subtended bythe equal sides will be equal to the correspondingremaining angles [Prop 14] Thus FBC is equal to GCBand BCF to CBG Therefore since the whole angle ABGwas shown (to be) equal to the whole angle ACF withinwhich CBG is equal to BCF the remainder ABC is thusequal to the remainder ACB [Ax 3] And they are at thebase of triangle ABC And FBC was also shown (to be)equal to GCB And they are under the base






[conclusion ]

Thus for isosceles triangles the angles at the base areequal to one another and if the equal sides are producedthen the angles under the base will be equal to oneanother (Which is) the very thing it was requiredto show





Problem 11 of Euclidrsquos ELEMENTS

[enunciation ]

To construct an equilateral triangle on a given finitestraight-line





Problem 11 (continued)

[exposition ]

Let AB be the given finite straight-line

[specification ]

So it is required to construct an equilateral triangle onthe straight-line AB






[construction ]

Let the circle BCD with center A and radius AB havebeen drawn [Post 3] and again let the circle ACE withcenter B and radius BA have been drawn [Post 3] Andlet the straight-lines CA and CB have been joined fromthe point C where the circles cut one another to thepoints A and B [Post 1]






[proof ]

And since the point A is the center of the circle CDBAC is equal to AB [Def 115] Again since the point Bis the center of the circle CAE BC is equal to BA [Def115] But CA was also shown (to be) equal to ABThus CA and CB are each equal to AB But thingsequal to the same thing are also equal to one another[Axiom 1] Thus CA is also equal to CB Thus the three(straight-lines) CA AB and BC are equal to oneanother






[conclusion ]

Thus the triangle ABC is equilateral and has beenconstructed on the given finite straight-line AB (Whichis) the very thing it was required to do





3 Kinds of First Principles in Euclidrsquos ELEMENTS

I Definitions play the same role as axioms in the modern sense ex radii ofa circle are equal

I Axioms (Common Notions) play the role similar to that of logical rules restricted tomathematics cf the use of the term by Aristotle

I Postulates non-logical constructive rules





























Common Notions

A1 Things equal to the same thing are also equal to one anotherA2 And if equal things are added to equal things then the wholesare equalA3 And if equal things are subtracted from equal things then theremainders are equalA4 And things coinciding with one another are equal to oneanotherA5 And the whole [is] greater than the part





Common Notions (continued)

Euclidrsquos Common Notions hold both for numbers and magnitudes(hence the title of ldquocommonrdquo) they form the basis of a regionalldquomathematical logicrdquo applicable throughout the mathematicsAristotle transform them into laws of logic applicable throughoutthe episteme which in Aristotlersquos view does not reduce tomathematics but also includes physics





Postulates 1-3

P1 to draw a straight-line from any point to any pointP2 to produce a finite straight-line continuously in astraight-lineP3 to draw a circle with any center and radius





Postulates 1-3 (continued)

Postulates 1-3 are NOT propositions They are not first truthsThey are basic (non-logical) operations





Operational interpretation of Postulates

Postulates input output

P1 two points straight segment

P2 straight segment straight segment

P3 straight segment and its endpoint circle





Key points on Euclid

I Problems et Theorems share a common 6-part structure (enunciationexposition specification construction proofconclusion) which does NOT reduce to the binary structureproposition - proof

I Postulates 1-3 and enunciations of Problems are NOTpropositions but (non-logical) operations

I Euclidrsquos mathematics aims at doing AND showing but notonly at showing and moreover not only to proving certainpropositions





























Hilbert on Euclid

Euclidrsquos axiomatics was intended to be contentual andintuitive and the intuitive meaning of the figures is notignored in it Furthermore its axioms are not inexistential form either Euclid does not presuppose thatpoints or lines constitute any fixed domain of individualsTherefore he does not state any existence axioms eitherbut only construction postulates(HilbertampBernays 1934)





Foundations 1899

Let us consider three distinct systems of things Thethings composing the first system we will call points anddesignate them by the letters A B C those of thesecond we will call straight lines and designate them bythe letters a b c and those of the third system wewill call planes and designate them by the Greek lettersα β γ [] We think of these points straight lines andplanes as having certain mutual relations which weindicate by means of such words as ldquoare situatedrdquoldquobetweenrdquo ldquoparallelrdquo ldquocongruentrdquo ldquocontinuousrdquo etcThe complete and exact description of these relationsfollows as a consequence of the axioms of geometryThese axioms [] express certain related fundamentalfacts of our intuition





On Truth and Existence (circa1900)

[A]s soon as I posited an axiom it will exist and beldquotruerdquo [] If the arbitrarily posited axioms together withall their consequences do not contradict each other thenthey are true and the things defined by these axiomsexist For me this is the criterion of truth and existence





Hintikka on Hilbert

[For Hilbert t]he basic clarified form of mathematicaltheorizing is a purely logical axiom system(Hintikka 1997)





Consistency

How we know if a given system of axioms is consistent Makingmodels provides only proofs of relative consistency Reduction toArithmetic works for Euclidean Geometry but not universally IsArithmetic consistent





Formalization of Logic

[I]t appears necessary to axiomatize logic itself and toprove that number theory and set theory are only parts oflogic This method was prepared long ago (not least byFregersquos profound investigations) it has been mostsuccessfully explained by the acute mathematician andlogician Russell One could regard the completion of thismagnificent Russellian enterprise of the axiomatization oflogic as the crowning achievement of the work ofaxiomatization as a whole (Hilbert 1918 p 1113)






[I]n my theory contentual inference is replaced bymanipulation of signs [ausseres Handeln] according torules in this way the axiomatic method attains thatreliability and perfection that it can and must reach if itis to become the basic instrument of all research(Hilbert 1927)






If formalization means making explicit the logical form of a giventheory how one can formalize logic The answer seems to be thatformalization of logic puts logic into a symbolic form By extensionin the new setting this applies also to mathematical theories






Remark that in Hilbert-Ackermann symbolic logic logical symbolsunlike non-logical ones cannot be re-interpreted Symbolic logic isstill contentual in a sense in which mathematical theories(expressed symbolically) are not





Proof theory as foundation

With this new way of providing a foundation formathematics which we may appropriately call aproof theory I pursue a significant goal for I should liketo eliminate once and for all the questions regarding thefoundations of mathematics in the form in which they arenow posed by turning every mathematical propositioninto a formula that can be concretely exhibited andstrictly derived thus recasting mathematical definitionsand inferences in such a way that they are unshakableand yet provide an adequate picture of the whole scienceI believe that I can attain this goal completely with myproof theory even if a great deal of work must still bedone before it is fully developed (Hilbert 1927)






No more than any other science can mathematics befounded by logic alone rather as a condition for the useof logical inferences and the performance of logicaloperations something must already be given to us in ourfaculty of representation certain extralogical concreteobjects that are intuitively present as immediateexperience prior to all thought(Hilbert 1927)






If logical inference is to be reliable it must be possible tosurvey these objects completely in all their parts and thefact that they occur that they differ from one anotherand that they follow each other or are concatenated isimmediately given intuitively together with the objectsas something that neither can be reduced to anythingelse nor requires reduction This is the basic philosophicalposition that I regard as requisite for mathematics and ingeneral for all scientific thinking understanding andcommunication (Hilbert 1927)






And in mathematics in particular what we consider isthe concrete signs themselves whose shape according tothe conception we have adopted is immediately clear andrecognizable This is the very least that must bepresupposed no scientific thinker can dispense with itand therefore everyone must maintain it consciously ornot (Hilbert 1927)





Real and Ideal Mathematical Objects

Just as for example the negative numbers areindispensable in elementary number theory and just asmodern number theory and algebra become possible onlythrough the Kummer-Dedekind ideals so scientificmathematics becomes possible only through theintroduction of ideal propositions (Hilbert 1927)





Metamathematics

When we now approach the task of such an impossibilityproof we have to be aware of the fact that we cannotagain execute this proof with the method ofaxiomatic-existential inference Rather we may onlyapply modes of inference that are free from idealizingexistence assumptions(HilbertampBernays 1934)





Formal Axiomatic Method

Formal Axiomatic Method splits into genetic part (buildingformulas) and axiomatic-existential part (interpreting formulas asproofs) Doing is Showing Real mathematical objects are onlysymbolic expressions built after the model of alphabetic linguisticexpressions In Metamathematics one needs to show (prove) thingsdifferently





Hilbert Program

failed but the method survived





Axiomatic Set theory

It is a metamathematics of ZFC and the like Ex Continuumhypothesis





Boorbaki on Axiomatic Method

After more or less evident bankruptcy of the differentsystems [] it looked at the beginning of the present[20th] century as if the attempt had just about beenabandoned to conceive of mathematics as a sciencecharacterized by a definitely specified purpose andmethod [] Today we believe however that the internalevolution of mathematical science has in spite ofappearance brought about a closer unity among itsdifferent parts so as to create something like a centralnucleus that is more coherent than it has ever been Theessential aspect of this evolution has been the systematicstudy of the relation existing between differentmathematical theories and which has led to what isgenerally known as the ldquoaxiomatic methodrdquo(Bourbaki 1950) Andrei Rodin Axiomatic Method and Category Theory





[E]very mathematical theory is a concatenation ofpropositions each one derived from the preceding ones inconformity with the rules of a logical system [] It istherefore a meaningless truism to say that thisldquodeductive reasoningrdquo is a unifying principle formathematics [] [I]t is the external form which themathematician gives to his thought the vehicle whichmakes it accessible to others in short the languagesuited to mathematicians this is all no furthersignificance should be attached to it (Bourbaki 1950)





Ex Axiomatic Group theory

It is a model theory of the Axiomatic Group theory in the(axiomatic ) Set theory Axioms of Group theory by themselvesimply very little G1 x (y z) = (x y) z (associativity of )G2 there exists an item 1 (called unit) such that for all xx 1 = 1 x = xG3 for all x there exists xminus1 (called inverse of x) such thatx xminus1 = xminus1 x = 1





Kant on the role of geometrical constructions

Give a philosopher the concept of triangle and let him tryto find out in his way how the sum of its angles might berelated to a right angle He has nothing but the conceptof figure enclosed by three straight lines and in it theconcept of equally many angles Now he may reflect onhis concept as long as he wants yet he will never produceanything new He can analyze and make distinct theconcept of a straight line or of an angle or of thenumber three but he will not come upon any otherproperties that do not already lie in these concepts






But now let the geometer take up this question Hebegins at once to construct a triangle In such a waythrough a chain of inferences that is always guided byintuition he arrives at a fully illuminated and at the sametime general solution of the question (Critique of PureReason)





Mathematics and Metamathematics

ldquoRealrdquo mathematics ( = mathematics as it is practiced) qualifiesas metamathematics (Whether we are talking about formalstudies or the mainstream ldquoinformalrdquo mathematics)




Prehistory Venn and TarskiPrehistory Schonfinkel and CurryCategorical LogicToposesHomotopy Type theory

Venn on Logical Diagrams

The relative cumbousness of such a mode of expression[in a natural language] is obviously the real measure ofour need for a reformed or symbolic language [] Thatsuch a scheme ]of symbolic representation] is completethere can be no doubt But unfortunately owing to thisvery completeness it is apt to prove terribly lengthy []This then is the state of thing which a reformed schemeof diagrammatic notation has to meet It mustcorrespond in all essential respects to that regular systemof class subdivision which has just been referred to underits verbal and its literal or symbolic aspect Theoreticallyas we shall see this is perfectly attainable (Venn 1881)






It has been said above that this question of the Universeonly arises when we apply our formulas Now diagramsare strictly speaking a form of application and thereforesuch considerations at once meet us when we come tomake use of diagrams I draw a circle to represent Xthen what is outside of that circle represents not-X butthe limits of that outside are whatever I choose toconsider them (Venn 1881)





Tarski on Topological Interpretation of PropositionalCalculus

(both Classical and Intuitionistic)

The present discussion seems to me to have a certaininterest not only from the purely formal point of view italso throws an interesting light on the content relationsbetween the two systems of the sentential calculus andthe intuitions underlying these systems [zugrundeliegendeIntuitionen] (Tarski 1956)





Combinatory Logic

The successes that we have encountered thus far on theroad taken encourage us to attempt further progress Weare led to the idea which at first glance certainly appearsextremely bold of attempting to eliminate by suitablereduction the remaining fundamental notions those ofproposition propositional function and variable [] Toexamine this possibility more closely and to pursue itwould be valuable not only from the methodologicalpoint of view that enjoins us to strive for the greatestpossible conceptual uniformity but also from a certainphilosophic or if you wish aesthetic point of view(Schonfinkel 1926)





Combinatory Logic

Combinatory logic is a branch of mathematical logicwhich concerns itself with the ultimate foundations Itspurpose is the analysis of certain notions of such basiccharacter that they are ordinarly taken for granted Theseinclude [(i)] the process of substitution usually indicatedby the use of variables and also [(ii)] the classification ofthe entities constructed by these processes into types orcategories which in many systems has to be doneintuitively before the theory can be applied It has beenobserved that these notions although generallypresupposed are not simple they constitute a prelogicso to speak whose analysis is by no means trivial(CurryampFeysampCraig 1958)





Curry-Howard-Lambek

Isomorphism between typed lambda-calculi and formal deductivesystems and cartesian closed categories





Axiomatic Method and Foundations of Mathematics

Replacement of Set theory by Category theory in Foundations doesnot necessarily requires a modification of the Formal AxiomaticMethod ex ETCS (Lawvere 1964) But





Lawvere on foundations

In my own education I was fortunate to have twoteachers who used the term ldquofoundationsrdquo in acommon-sense way (rather than in the speculative way ofthe Bolzano-Frege-Peano-Russell tradition) [] Theorientation of these works seemed to be ldquoconcentrate theessence of practice and in turn use the result to guidepracticerdquo I propose to apply the tool of categorical logicto further develop that inspirationFoundations is derived from applications by unificationand concentration in other words by the axiomaticmethod Applications are guided by foundations whichhave been learned through education (Lawvere 2003)





Lawvere Categories

The formalism of category theory is itself often presentedin ldquogeometricrdquo terms In fact to give a category is togive a meaning to the word morphism and to thecommutativity of diagrams like

Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime

which involve morphisms in such a way that the obviousassociativity and identity conditions hold





Lawvere Categories

as well as the condition that whenever

Af B B

f C

are commutative then there is just one h such that

Bg

A

f

h C

is commutative





Lawvere Categories

To save printing space one also says that A is thedomain and B the codomain of f when

Af B

is commutative and in particular that h is thecomposition f g if

Bg

A

f

h C

is commutative





Lawvere Categories

We regard objects as co-extensive with identitymorphisms or equivalently with those morphisms whichappear as domains or codomains As usual we call amorphism which has a two-sided inverse an isomorphism(Lawvere 1969)





Lawvere Categories

[In a category w]e identify objects with their identitymaps and we regard a diagram

Af B

as a formula which asserts that A is the (identity map ofthe) domain of f and that B is the (identity map of the)codomain of f Thus for example the following is auniversally valid formula

Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic

The term ldquologicrdquo has always had two meanings - abroader one and a narrower one (1) All the general laws about the movement of humanthinking should ultimately be made explicit so thatthinking can be a reliable instrument but(2) already Aristotle realized that one must start on thatvast program with a more sharply defined subcase





Lawvere on Logic

The achievements of this subprogram include therecognition of the necessity of making explicit(a) a limited universe of discourse as well as(b) the correspondence assigning to each adjective thatis meaningful over a whole universe the part of thatuniverse where the adjective applies This correspondencenecessarily involves(c) an attendant homomorphic relation betweenconnectives (like and and or) that apply to the adjectivesand corresponding operations (like intersection andunion) that apply to the parts ldquonamedrdquo by the adjectives





Lawvere on Logic

When thinking is temporarily limited to only oneuniverse the universe as such need not be mentioned however thinking actually involves relationships betweenseveral universes [] Each suitable passage from oneuniverse of discourse to another induces





Lawvere on Logic

(0) an operation of substitution in the inverse directionapplying to the adjectives meaningful over the seconduniverse and yielding new adjectives meaningful over thefirst universe The same passage also induces twooperations in the forward direction (1) one operation corresponds to the idea of the directimage of a part but is called ldquoexistential quantificationrdquoas it applies to the adjectives that name the parts (2) the other forward operation is called ldquouniversalquantificationrdquo on the adjectives and corresponds to adifferent geometrical operation on the parts of the firstuniverse





Lawvere on Logic

It is the study of the resulting algebra of parts of auniverse of discourse and of these three transformationsof parts between universes that we sometimes call ldquologicin the narrow senserdquo Presentations of algebraic structuresfor the purpose of calculation are always needed but it isa serious mistake to confuse the arbitrary formulations ofsuch presentations with the objective structure itself or toarbitrarily enshrine one choice of presentation as thenotion of logical theory thereby obscuring even theexistence of the invariant mathematical content





Lawvere on Logic

In the long run it is best to try to bring the form of thesubjective presentation paradigm as much as possibleinto harmony with the objective content of the objects tobe presented with the help of the categorical method wewill be able to approach that goal(LawvereampRosebrugh 2003)





Adjunction

An adjoint situation (called also an adjunction) is a pair ofcategories AB with two functors f g going in opposite directions

Af

Bg

oo

provided with natural transformations α Ararr fg and β gf rarr Bsuch that (gα)(βf ) = g and (αf )(gβ) = f This latter conditionis tantamount to saying that the following diagrams commute

ggα gfgβg

oo

fαf fgff β

oo





Quantifiers as Adjoints

Let us for simplicity think of these universes as sets but have inmind that the functorial construction of logical quantifiers does notrequire this assumption Suppose now that we have a one-placepredicate (a property) P which is meaningful on set Y so thatthere is a subset PY of Y (in symbols PY sube Y ) such that for ally isin Y P(y) is true just in case y isin PY Now using these data(together with morphism f as above) we can define a newpredicate R on X as follows we say that for all x isin X R(x) istrue when f (x) isin PY and false otherwise So we get subsetRX sube X such that for all x isin X R(x) is true just in case x isin RX






Let us assume in addition that every subset PY of Y is determinedby some predicate P meaningful on Y Then given morphism(ldquopassagerdquo) f from ldquouniverserdquo X to ldquouniverserdquo Y we get a way toassociate with every subset PY (every part of universe Y ) a subsetRX and correspondingly a way to associate with every predicate Pmeaningful on Y a certain predicate R meaningful on X Sincesubsets of given set Y form Boolean algebra B(Y ) we get a mapbetween Boolean algebras (notice the change of direction )

f lowast B(Y ) B(X )






Since Boolean algebras themselves are categories (where objectsare subsets and maps are their inclusions) f lowast is a functor For everyproposition of form P(y) where y isin Y functor f lowast takes somex isin X such that y = f (x) and produces a new propositionP(f (x)) = R(x) (for a single given y it may produce a set ofdifferent propositions of this form) Since it replaces y in P(y) byf (x) = y it is appropriate to call f lowast substitution functor






The left adjoint to the substitution functor f lowast is functor

existf B(X ) B(Y )

which sends every R isin B(X ) (ie every subset of X ) intoP isin B(Y ) (subset of Y ) consisting of elements y isin Y such thatthere exists some x isin R such that y = f (x) in (some more)symbols

existf (R) = y |existx(y = f (x) and x isin R)






In other words existf sends R into its image P under f Now if (asabove) we think of R as a property R(x) meaningful on X andthink of P as a property P(y) meaningful on Y we can describe existfby saying that it transformes R(x) into P(y) = existf xP prime(x y) andinterpret existf as the usual existential quantifier






The right adjoint to the substitution functor f lowast is functor

forallf B(X ) B(Y )

which sends every subset R of X into subset P of Y defined asfollows

forallf (R) = y |forallx(y = f (x)rArr x isin R)

and thus transforms R(X ) into P(y) = forallf xP prime(x y)






Notice that functors existf and forallf are defined here as adjoints tofunctor f lowast ie quite independently from their interpretation aslogical quantifiers explained above





Grothendieck (circa 1962)

Topos 2-fold generalization

I Topological space rarr Space of sheafs (detach sets from opens)

I Poset of opens rarr (small) category of opens (Grothendiecktopology)





Elementary Topos Geometry and Logic

The unity of opposites in the title [Quantifiers andSheafs] is essentially that between logic and geometryand there are compelling reasons for maintaining thatgeometry is the leading aspect At the same time in thepresent joint work with Myles Tierney there are importantinfluences in the other direction a Grothendieckldquotopologyrdquo appears most naturally as a modal operatorof the nature ldquoit is locally the case thatrdquo the usuallogical operators such as forall exist rArr have natural analogueswhich apply to families of geometrical objects rather thanto propositional functions and an important technique isto lift constructions first understood for ldquotherdquo category Sof abstract sets to an arbitrary topos (Lawvere 1970)





Lawvere

Toposes = continually variable sets





Johnstone Sketches of the Elephant

Three Sketches Space Logical Framework Universe of discourse





Features of Topos Logic

I Geometry comes first (the ldquoleading aspectrdquo)

I (Internal) logic of a given topos reflects its geometricalstructure

I Connectives quantifiers and truth-values are internalized (theldquolocal truthrdquo)

I Identity is not





Fundamental group

Fundamental group G 0T of a topological space T

I a base point P

I loops through P (loops are circular paths l I rarr T )

I composition of the loops (up to homotopy only - see below)

I identification of homotopic loops

I independence of the choice of the base point





Fundamental (1-) groupoid

G 1T

I all points of T (no arbitrary choice)

I paths between the points (embeddings s I rarr T )

I composition of the consecutive paths (up to homotopy only -see below)

I identification of homotopic paths

Since not all paths are consecutive G 1T contains more information

about T than G 0T





Path Homotopy and Higher Homotopies

s I rarr T p I rarr T where I = [0 1] paths in Th I times I rarr T homotopy of paths s t if h(0times I ) = s h(1times I ) = thn I times I nminus1 rarr T n-homotopy of n minus 1-homotopies hnminus1

0 hnminus11 if

hn(0times I nminus1) = hnminus10 hn(1times I nminus1) = hnminus1

1 Remark Paths are zero-homotopies





Higher Groupoids and Omega-Groupoids


I paths between the points

I homotopies of paths

I homotopies of homotopies (2-homotopies)

I higher homotopies up to n-homotopies

I higher homotopies ad infinitum

GnT contains more information about T than Gnminus1

T





Composition of Paths

Concatenation of paths produces a map of the form 2I rarr T butnot of the form I rarr T ie not a path We have the whole spaceof paths I rarr 2I to play with But all those paths are homotopicalSimilarly for higher homotopies (but beware that n-homotopies arecomposed in n different ways )On each level when we say that aoplus b = c the sign = hides aninfinite-dimensional topological structure





Grothendick Conjecture

GωT contains all relevant information about T an omega-groupoid

is a complete algebraic presentation of a topological space





Martin-Lof

Constructive Intensional Type theory (with dependent types) Is itreally a ldquologicrdquo





Voevodsky Univalent Foundtions

(circa 2008)Higher (homotopical) groupoids model higher identity types inMartin-Lofrsquos Intensional Constructive Type theory Identities (ie identity types of all orders) get internalizedDisclaimer 1-groupoids model Martin-Lofrsquos theory too but theykill higher identity types (ldquoextensionality one level uprdquo)Only omega-groupoids provide ldquofully instensionalrdquo models Voevodskyrsquos Univalence Axiom rules out all groupoid modelsexcept the omega-groupoid model





Lurie Higher Topos Theory

(2011)Unification of Topos theory and Higher-categorical Homotopytheory




Kant-Friedman on Euclid

Euclidean geometry [] is not to be compared withHilbertrsquos axiomatization [of Euclidean geometry] say butrather with Fregersquos Begriffsschrift It is not a substantivedoctrine but a form of rational representation a form ofrational argument and inference[] There remains aserious question about Euclidrsquos axioms of course whenpressed Kant would most likely claim that they representthe most general conditions under which alone a conceptof extended magnitude - and therefore a rigorousconception of an external world - is possible (seeA163B204) And of course we now know that Kant isfundamentally mistaken here (Friedman 1992)




Quod Erat Faciendum

I Doing and Showing go closely together again

I Foundations are no longer purely logical Operations withsymbols represent operations with mathematical (geometrical)objects but not just operations with linguistic expressions

I No straightforward distinction between real and idealmathematical objects

I Geometrical intuition plays a role in foundations




Back to Euclid

I No uniqueness but a broad unification anyway

I A new mathematical language for the Book of Nature

Logic is rather a feature of a physical system or region ofspacetime than the universal ambient environmentunchanged whatsoever the scales of energies areconsidered (Krol circa 2010)




movie

x = r(t minus sint)y = r(1minus cost)


cyloidGenerationswf
Media File (applicationx-shockwave-flash)



THE END


Outline

Past there stages of development of the Axiomatic Method

Euclid

Hilbert 1899

Hilbert 1927-39

Present Formal Axiomatic Method in the the 20th century mathematics


Boorbaki

Future the Geometrical Turn

Prehistory Venn and Tarski

Prehistory Schoumlnfinkel and Curry

Categorical Logic

Toposes

Homotopy Type theory

Conclusion



Past there stages of development of the Axiomatic MethodEuclidHilbert 1899Hilbert 1927-39

Present Formal Axiomatic Method in the the 20th centurymathematics


Future the Geometrical TurnPrehistory Venn and TarskiPrehistory Schonfinkel and CurryCategorical LogicToposesHomotopy Type theory

Conclusion






[enunciation ]







[exposition]







[specification ]







[construction ]







[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[enunciation ]







[exposition]







[specification ]







[construction ]







[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[exposition]







[specification ]







[construction ]







[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[specification ]







[construction ]







[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[construction ]







[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[proof ]













[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion











[conclusion ]







[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[enunciation ]







[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[exposition ]


[specification ]







[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[construction ]







[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[proof ]







[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





[conclusion ]






































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




































Common Notions












Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion










Postulates 1-3





















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion



















































Hilbert on Euclid






Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Foundations 1899












Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion










Hintikka on Hilbert






Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Consistency




























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion


























































Metamathematics












Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion










Hilbert Program


































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion
































































Combinatory Logic






Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Combinatory Logic






Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Curry-Howard-Lambek


















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion
















Lawvere Categories


Af B B

g

A

f

h C

Af

a

B

b

Aprime g B prime






Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere Categories


Af B B

f C


Bg

A

f

h C

is commutative





Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere Categories


Af B


Bg

A

f

h C

is commutative





Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere Categories






Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere Categories


Af B


Af B rArr A

A A and Af B and B

B B andAf = f = fB





Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Lawvere on Logic






Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Adjunction


Af

Bg

oo


ggα gfgβg

oo

fαf fgff β

oo


























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion

























existf B(X ) B(Y )















forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion












forallf B(X ) B(Y )




























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion
























Lawvere
















I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion














I Identity is not





Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




Fundamental group


I a base point P










G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion





G 1T






about T than G 0T







0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion






0 hnminus11 if















T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion












T


















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion

















Martin-Lof






















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion




















Quod Erat Faciendum








Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion



Back to Euclid







movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion



movie



cyloidGenerationswf



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion



THE END


Outline


Euclid

Hilbert 1899

Hilbert 1927-39



Boorbaki




Categorical Logic

Toposes


Conclusion

Axiomatic Method and Category Theoryphilomatica.org/wp-content/uploads/2012/12/amcat.pdf ·...

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Transcript of Axiomatic Method and Category Theoryphilomatica.org/wp-content/uploads/2012/12/amcat.pdf ·...