Philosophy of Mathematics & Frege - Michael Dummett (1994)

people occasionally express puzzlement

that there being such a thing as

philosophy of mathematics such

puzzlement

arises from a failure to understand that

philosophy like history

is characterized not by its subject

matter but

by its style of thought by the kind of

questions it asks

and the way in which it goes about

trying to answer them

when this is understood phrases

beginning the philosophy of

will cause no more surprise than once

beginning the history of

philosophers attempt to answer questions

we are prompted to ask

by a quite special kind of puzzlement

this puzzlement arises from our

imperfect

mastery of the concepts we employ

these questions occur to everyone but

people are often content to brush them

aside

with answers that will not withstand

scrutiny

scrutiny not accorded them by those

lacking in philosophical curiosity

a philosopher is afflicted by an urge to

subject any proposed answer

to just such scrutiny and to arrive at

an answer that will stand up to scrutiny

because until he does he's conscious

that he does not understand

and what he wants above all is not so

much to know

as to understand characteristic

philosophical question is why can we not

affect the past

although we can affect the future

someone without philosophical curiosity

may answer impatiently

because the past has already happened or

because the previous event has either

occurred or not occurred

and will experience only irritation when

it's pointed out that the first answer

merely repeats the problem without

solving it and the second

may be counted by observing that a

subsequent event

either will or will not occur

the philosopher is irked by the

inadequacies

the inadequacy of these answers that

first come to mind

he's driven to seek one that will

satisfy him

well why because when we are faced with

this question

a question that asks why we cannot do

something that seems on the face of it

nonsensical

and find that we cannot clearly explain

what makes it nonsensical

we become aware that we do not really

know what past and future

are we have no

firm grasp upon the concepts of past and

future

now these are concepts we constantly

employ in the current of everyday life

and everyday conversation we recall what

happened a week ago or ten years ago

we avow what we intend to do tomorrow or

speculate

on what will happen six months from now

we use these concepts all the time of

course we understand them

and yet by our inability to answer the

question

why we cannot affect the past we show

that we have only a superficial

a superficial grasp of them we are like

soldiers in a battle

who know enough to be able to do what

they are meant to do

but have no conception of what is

happening on a larger scale

we can operate with our concepts in the

situations in which we find ourselves in

everyday life in the laboratory

on the stock exchange in the operating

theater but in wittgenstein's phrase

we do not command a clear view of them

or attain a general understanding of

them but can grasp

only how they function in particular

familiar

con contexts this does not feel good

only if the concepts we all employ in

the life of every day

it applies equally in highly technical

regions

quantum mechanics supplies a familiar

example

it's commonplace to remark that it's a

highly successful theory

physicists know how to use it to predict

observations and measurements and yet

frequent conferences

are held to discuss its interpretation

it's well understood how the theory is

to be used

it's not understood what it means that

is what it tells us

about the character of reality

mathematics generates a number of

questions causing just

this kind of puzzlement and has

fascinated and perplexed philosophers

from plato onwards for two reasons in

particular

first it's difficult to say what its

subject matter is

what it's about it's reasonably clear

what physics or geology or biology

investigates but what exactly is it that

mathematics investigates

a standard answer corresponding to the

old-fashioned division of mathematics

into arithmetic and geometry

used to be that it investigated quantity

and space but this unhelpful answer

will no longer suffice since there's so

much mathematics

that will not fit comfortably under

either head

the problem is aggravated by the second

puzzling feature of mathematics

the manner in which the mathematician

sets about

attempting to solve his problems he uses

no telescope or microscope he does not

observe

anything at all rather he reasons

he carries out complex deductive

inferences

the first question must be answered in a

way that accords with this

whatever it is that mathematics is about

must be something that can be found out

just by reasoning philosophy

resembles mathematics in this respect

the philosopher makes no observations

and requires no instruments insofar as

these disciplines can be said to involve

the making of experiments

these are thought experiments to imagine

the experiment made

is quite as good as actually to make it

the two subjects thus appear to be our

priori

their results do not require us to

observe how the world happens to be

but can be arrived at by thought alone

they are thus independent of how the

world happens to be

but would hold good whatever it was like

they are therefore not merely true but

necessarily true

and yet philosophy and mathematics

differ in all other respects

it's a puzzle how there can be even one

subject

that can be investigated our priori that

there should be two

so different from one another appears

baffling

necessary truth and our priori knowledge

are topics

that engage the attention of all

philosophers

it seems straightforward to understand

how there can be contingent truths

things that are so but might have been

different we can get to know contingent

truth

only through our experiences of the

world by observing

that they're so or deducing from what we

observe that they must be so

but how does it come about that there

should also be

necessary truths truth that we can know

independently of our experience of the

world

how is it that if we knew all contingent

truths

there would be some truths left left

over

this is easy enough to understand

concerning trivial necessary truths

such as that there are seven days in a

week or that

every widow was once married to

recognize statements of this sort as

true

we need know nothing other than the

meanings of the words

more we couldn't claim to know the

meanings of the words if we fail to

perceive

that the statements are true but

mathematical theorems are seldom

trivial in this sense we may not need to

start with any initial knowledge

other than the meanings of the words if

we are to come to recognize them as true

but the converse certainly doesn't

appear to hold

we may surely know the meanings of the

words without realizing

that the theorem so good mathematics

presents itself

as by far the most capacious repository

of non-trivial necessary truths and the

knowledge of it

is by far the most extensive body of our

priori knowledge

and this fact alone suffice is to make

it of intense interest

to philosophers admittedly certain

philosophers

of whom john stuart mill is the best

known example

have challenged the our priori character

of mathematics

claiming that mathematical theories rest

upon certain

highly general contingent facts

recognizable

by gross observation but this challenge

does not alter the situation greatly for

the most

that such empiricists can argue is that

the starting point

of this or that mathematical theory the

axioms of the theory

is a collection of readily observable

contingent facts

they cannot explain why mathematicians

fail to set about gathering by devising

experiments by closer observation or

proved observational techniques other

facts of the same kind

as other scientists do why instead they

content themselves with the meager

supply of contingent facts

with which allegedly they begin and

proceed to draw out their consequences

by means of ever lengthening chains of

deductive argument

mathematics is still a science quite

unlike

any other the only upshot

of the empiricist's contention is what

is sometimes called

if if-then-ism which restricts the

necessary truths

discovered by mathematicians to those

expressed by statements of the form

if the axioms of the theory hold good

then such and such a theorem

holds also and this leaves the problem

of necessary truth

untouched

that there should be and such an a

priori subject as philosophy

is comparatively intelligible because

the philosopher's task

consists principally of disentangling

our concepts

it does not aim so much as arriving so

much at arriving at new truths

as it coming to understand better those

which we've already arrived

disentanglement sometimes plays a

critical role in mathematics

as indeed it does in every subject to

attain the right definition of

continuous or of dimension

was a step of the highest importance

nevertheless

hitting on the correct definition of a

concept though often an essential

contribution to progress

remains a preliminary to the discovery

of mathematical truths not a means of

discovering them

it is not the characteristic activity of

ma

of the mathematician to explain the

existence of mathematics

is a greater challenge to the

philosopher

than to explain that of philosophy

itself

the capacity for wonder is a

prerequisite for the activity of

philosophizing

and anyone who retains this capacity

must marvel at the vastness of the body

of our priori knowledge

amassed by mathematicians by pure

deductive reasoning

now having tried to convey an impression

of what the philosophy of mathematics is

about

i will now tell you something of god rob

frager

superficially it might be said that he

lived an uneventful life

born in 1848 his entire professional

career from 1874 to 1918

was spent teaching at the university of

guinea and he died in retirement in

1925.

although he published three interesting

articles towards the end of his life

almost all his work was completed by

1906

but his history during his lifetime and

that his reputation

up to the present are both extraordinary

while he lived very few among them

russell and wittgenstein

paid any attention to his work a state

of affairs that continued

until after the second world war

now he's universally recognized as the

grandfather if not the founder

of analytical philosophy and wherever

that school of philosophy flourishes

his works as indispensable reading for

any student of the subject

yet he was not a professional

philosopher

he was a professor of mathematics still

largely as as neglected by the

mathematicians

as he once was by the philosophers

this neglect was in part due to his

originality

and in part to his having chosen to

devote his life to an enterprise

lying on the borderline between

philosophy and mathematics

with very few exceptions mostly in the

early stages of his career

everything that he wrote was directed

towards

or ancillary to this enterprise

he determined to rectify a situation

that greatly distressed him

the inability of either mathematicians

or philosophers

to explain the basis of our acceptance

of mathematical theories

he wanted to show what justified our

belief in these theories

with what right we assumed the theories

to be correct

and their theorems to be true his

attempt to do this

made him the first modern philosopher of

mathematics

now this description of the task fraga

set himself to accomplish is somewhat

too wide

he had the traditional view of

mathematics

as subdivided into arithmetic and

geometry

and he believed that these two parts of

mathematics

demanded different accounts of the

grounds of our belief in them

his general remarks about geometry were

almost all for the purpose of

contrasting it

in this regard with arithmetic his

positive work

related entirely to the latter by

arithmetic

fragment number theory and analysis

the theory of natural numbers and the

theory of real numbers

he did not view set theory as a separate

mathematical theory but rather as a part

of logic

of which he believed that he could also

give a satisfactory account

and indeed if if the problem can be

solved for number theory

analysis and set theory there should be

little fear

that any other branch of mathematics

will give rise to further difficulties

fragra thought that the inability of

mathematicians to provide

a justification for our acceptance

even of number theory was a scandal to

the science

they could not even give an intelligible

account

of what the natural numbers are and

hence they could not so much as say what

number theory is about

let alone explain how we know that it is

true

his first step in carrying out his

self-appointed task

was to invent modern mathematical logic

i said earlier that fraga was the first

modern philosopher of mathematics

the title might be claimed by some for

but fragra differed from kant and from

all his predecessors

is in his determination to treat of the

mathematical

theories with which he concerned himself

in detail

rather than contenting himself with

observations which even if sound

had not been demonstrated to hold good

for all propositions of those theories

can't have maintained the a priori

character of mathematics

but to distinguish two varieties of our

priori truths

the analytic and the synthetic analytic

truths were those guaranteed by logic

alone

and where on can't view all trivial

they must be recognized immediately by

anyone who understood the words in which

they were expressed

and hence they could not extend our

knowledge

the truths of mathematics by contrast

were synthetic

and hence substantial the recognition of

geometrical truths

depended on our our intuition of space

and the various medical truths on our

intuition of time

our mathematical knowledge was

nevertheless our priori

because can't help that we possess

conceptions of space and time

independently of any particular

experience of

our priori intuitions freyja

accepted consternation of our our priori

intuition of space and with it

his view of geometry he thought that

while non-euclidean geometries are

logically consistent and tense

intelligible

we know our priori that the the geometry

of physical space

is euclidean but he utterly opposed the

kantian view of arithmetic

believing that neither temporal nor

spatial intuition

played any essential role in our

recognition of its truths

in this he was following the footsteps

of the

great czech mathematician and

philosopher boltzano

who died in the year of fragrance birth

bolsano had initiated the process of

rigorizing analysis

explicitly arguing the propositions

concerning real numbers

such as the mean value theorem ought not

to be accepted

in virtue of their apparent obviousness

to geometrical intuition

but could and therefore should be proved

in a purely arithmetical manner

as both sauna had striven to expel

appeals to intuition from analysis

fraga wished to expel them even from

number theory

since proof is the principal instrument

for establishing mathematical truths

he considered the process of

mathematical proof

must be subjected to scrutiny

do we know that all principles of proof

that we use in number theory

or that we need in order to establish

the basic number theoretic propositions

that we frequently take for granted

are of a purely logical character

as long as we reason without paying

conscious attention to the steps we take

in reasoning

we do not we may then mistake for

logical transitions

ones which in fact rely upon intuition

conversely we may suppose an appeal to

intuition to be demanded

by what in fact can be accomplished by

purely logical deduction

it was therefore necessary in fragrance

eyes to attain

an explicit systematization of the

process of mathematical

proof this he did in his first book the

gristrift of 1879

a completely original work that did not

build

on the recent advances in logic by bull

and his successors but adopted a wholly

new approach

not only have they extended the scope of

logic by only a small degree

beyond what aristotle had achieved but

they had provided no more than a means

of

encoding any given argument that fell

within the scope of their theories fraga

desired something different a language

in which mathematical theorems could be

expressed and their proofs carried out

in accordance with strictly formal

principles of inference

it is with hindsight astonishing that

despite an

intense study of logic over many

centuries indeed millennia

it failed to register the glaring fact

that logical theory was incapable of

analyzing

even the simplest piece of mathematical

reasoning

fraga's little book completely rectified

this

using a notation quite different from

but isomorphic with

that used nowadays or by any of his

successes

parts one and two of the book presented

a complete formalization

of what we now call first order logic

but part three

explored the realm of second order logic

also that is the logic

governing statements that generalize not

only over

objects for example the natural numbers

but over properties

of and relations between these objects

and here fragrance scored his first

success

in the war against intuition a logical

analysis

of the notion of a sequence rather

naturally it was common to think of

sequences in temporal terms

but frag observed that the notion was a

far

greater generality a sequence can be

generated by any relation

the term generated by i've just used

itself relies on a way of picturing

sequences

as proceeding in time and i should have

said something like

characterized by reference to

the relation r say r is that which any

term in the sequence

has to the next fragra defined the

expression

b follows a object b follows the object

a in the

r sequence to mean b has every property

f possessed by every object which either

a

or an object having the property f

stands in the relation

r um

this definition and the theorems

concerning sequences that fragra

proved by appeal to it were not

important to him

merely as samples of how one could

dispense

means with what was often thought to

rely upon intuition

it was important because the notion of a

sequence

is fundamental to the theory of numbers

since the natural numbers form precisely

such a sequence

given the number naught and the relation

that holds between any natural number n

and its successor n plus one the natural

numbers could be defined

to be those things that followed naught

in the successor sequence

together with naught itself moreover

such a definition

would render the principle of induction

its most immediate consequence

the principle allows something to be

proved to hold good of all natural

numbers

by showing the totals of naught and of

the successor of any number of which it

holds

and had frequently been said to be a

form of inference

peculiar to number theory but fragrance

definition of natural number

reduced it to pure logic

plainly investigation of what justifies

our accepting

mathematical theories other than

geometry must begin with the most basic

theory

number theory and the first step might

be expected to be

to axiomatize it it's odd that despite

the praise conventionally lavished on

euclid for

his axiomatization of geometry no one

attempted to do this for any

branch of arithmetic until the 19th

century

we can extract an axiomatization from

frager's work

more precisely an abstract

characterization

about which he proved any structuring

exemplifying it

to be isomorphic to the natural numbers

but he did not proceed in this way after

what he had achieved in his first book

it's unsurprising that he came to

believe that the whole of number theory

could be derived from logic alone in

1884 he published the foundations of

arithmetic

in which having subjected all existing

accounts to deadly criticism

he sketched his own demonstration of the

logical character of number theory

without using his logical notation

in 1893 and 1903 he published the first

and second volumes

of his basic laws of arithmetic which

set out

fully formalized proofs in his logical

system

both for number theory and for analysis

given fragra's view that all

arithmetical notions can be defined

in logical terms and all our

arithmetical propositions prove

from logical first principles the task

of

justifying arithmetic necessarily

involved for him

much detailed mathematical work of which

his books are full

largely neglected if only because very

few

were prepared to learn to read his

symbolism

but the task involved philosophical

argument also

if the mathematical proofs were to be

shown to establish what was claimed

and in the process of carrying it out

fraga turned himself into a philosopher

and one of genius as well as a

mathematician

although his philosophical work was

restricted in scope

he plainly became interested in the

topics he tackled

for their own sake so that his

discussions

particularly in the ancillary articles

that he wrote

extend well beyond what was strictly

necessary

for his primary objective it's this that

has made him have

made them of such vivid interest the

modern philosophers

not especially concerned with the

philosophy of mathematics

but that's not our pr our present

concern

what to my mind makes him of primary

importance to present day philosophy of

mathematics

is the clarity with which he presented

its problems

even when he did not attain more than

partial

solutions to them for these are often

problems

which later writers have scarcely

attempted to tackle

given his conclusion that analytic

truths cannot extend our knowledge

can't could could not regard

mathematical truths as analytic

since on the face of it mathematics

massively extends our knowledge

freyja having decided that arithmetical

truths are logical in character and

therefore analytic

was faced with explaining how it is that

analytic truth

can extend our love knowledge and this

is the same problem

as how deductive reasoning can lead to

new knowledge

for such reasoning to be valid a single

inferential step

must be both recognizable and compelling

anyone who understands the statements

figuring as premises and conclusion

must thereby grasp that acknowledging

the former is true

requires acceptance of the latter but if

this

is so how can a sequence of such steps

take you any distance from where you

started

the problem has been addressed by few

philosophers other than fragra notably

by mill

and none has offered a satisfactory

solution

if fragrance solution is not correct in

detail

i'm not going to give the detail it must

i believe be correct in outline

for fragrance not merely to discover a

deductive argument but even to

follow it is to engage in a creative act

that is because it involves more than

merely

understanding the statements involved

to understand the statement to grasp the

thought it expresses

in his terminology it's necessary to

apprehend its structure how it's put

together

out of its paths a sentence is not just

a string of words

but to recognize the validity of an

argument in which it figures

we need to do more than this sentences

exhibit patterns to use a term fragrant

did not

imply that we do not need to discern

in order merely to know what they say

in it's on the relationship between

these patterns

that the validity of an inference

depends

a pattern is not imposed but is there to

be discerned

but discerning it is an intellectual act

and not

mere passive reception

fraga is sometimes supposed to have

thought that arithmetic

describes the plutonic realm of abstract

objects

having no con connection with contingent

reality

such a view would provide no room for

the application of arithmetic

though in fact even plato's ideas bear a

direct relation

to the empirical world but the

misrepresentation of freyja

is nearly total the real fragra

gave a more central role to application

than almost any other philosopher of

mathematics

the applicability of mathematics is

nowadays often described as miraculous

fragra would have denied not merely that

it was a miracle

but that it was even a surprise it is

applicability alone he wrote that raises

arithmetic from the rank of the game

to that of a science his view of

application was a subtle one

arithmetic must remain pure any

importation of notions relating to

particular empirical applications would

sully that purity

and he criticized both mill and

helmholtz

on this ground but the general principle

exemplified by every application of a

fundamental

mathematical notion and in particular by

the notions of the natural numbers and

of the real numbers

must be discerned formulated and made

central to its definition not tagged on

as an appendage

as dedicated did in his treatise only in

his treaties

on the natural numbers rather

the general principle in accordance with

which all applications are made

must be incorporated in the definition

that's why fraga held that since the

fundamental use

of the natural numbers is to specify in

finite cases

how many objects there are that satisfy

some given condition

they must be defined as cardinal numbers

not indeed objects of any specific kind

but of objects of any kind whatever

since all objects can be counted

and it's also why frago regarded the

definitions of the real numbers

given by kanto and dedekind as

unsatisfactory

though their definitions were different

both assumed the rationals

as already given whereas on fragra's

view

the term number is used in just the same

sentence in rational number as in real

number

so that the rationals ought not to be

defined separately

in advance of the real numbers but

simply as a particular type of real

number

by contrast the term number has a

different sense

in natural number making it necessary

to define the natural numbers separately

the natural numbers serve to answer the

question

how many so they together with the

trans-finite cardinals

are numbers in one sense of the word the

real numbers

including the rational ones serve to

give the ratio of a quantity

a mass a temporal duration an electric

charge etc

to a unit quantity of the same kind and

are therefore numbers in a different

sense they must therefore be defined as

such ratios not between quantities of

any specific kind

but between any two quantities of the

same kind

that of course requires a prior

analysis of the notion of a quantity or

a range of quantities i've been

describing a number of near-misses

fragrance account of the fruitfulness of

deductive reasoning

must be close to the truth but i don't

think

it was the precise truth his emphasis

on the applicability of mathematics was

surely correct

but his demand on the way mathematical

concepts are defined

is difficult to satisfy he was clearly

right

to maintain the necessity of

arithmetical truth

and to deny that this would derive from

intuition in can't sense

but from features of concepts sharing

with logical ones their applicability to

any domain of discourse

even if he was wrong to claim them to be

strictly logical in character

the most interesting of all the

components of

his philosophy of arithmetic is however

the most contentious it attempts to

resolve a problem

that hardly any of his successes has

faced

it's not a single thesis but a complex

of ideas

that can with difficulty be disentangled

from one another

and it's quite certainly wrong as a

whole

because it led fraga into the

catastrophe that caused him to

acknowledge

that his life's work had been a failure

someone who knew nothing about fragra

but learned that he believed

arithmetical theorems to be expressible

in purely logical terms and provable

from purely logical principles

would expect on reflection that he did

not take statements of arithmetic

at face value the statement

pi is transcendental is on the face of

it of the same

form as clinton is unsuccessful

and says that a particular object of a

certain kind

has a specific property well there's a

prime number between 44 and 52 appears

to be of the same form as

there's a small country between france

and spain

and says that some object of a certain

kind has a particular property

and stands in a specific relation to two

named objects of the same kind

but statements of forms such as these

are hardly candidates for being

propositions of logic

for the principles of logic must hold

independently of which

particular object or even of how many

objects

there may happen to be so it appears to

follow

that fraga did not think that

arithmetical statements are truly

of the form they appear to be he must

rather have construed them as disguised

versions of statements of very different

forms

for instance reinterpreting any

proposition referring

to a natural number n as involving the

statement

that there are endings of some kind

well this conclusion is highly

reasonable but it is

wrong fraga did understand arithmetical

statements at face value

numbers were in his few objects

it followed that logic must after all

guarantee the existence of certain

abstract objects

indeed of infinitely many of them

numbers

are what he called logical objects

now this didn't to any significant

extent

hamper his current account of the

application of arithmetic

mathematical theorems were for him

encapsulated results of complex chains

of

deductive reasoning enabling us by

specializing them to less general cases

to pass from contingent premises to

contingent

conclusions without having to carry out

the reasoning process afresh

the existence of the objects

constituting the elements of the

structure which each particular theory

number theory or analysis treated was

guaranteed by logic

and in this sense the purity of

arithmetic was secured

but they were not pure in the sense in

which set theorists

nowadays speak of pure sense a pure set

in this sense is one whose

transitive closure contains nothing but

sex

that's to say every element of the set

is a set and every element of an element

of the set is a set

every element an element of an element

of the set is a set and so on

on the contrary for fragrant natural

numbers are cardinal

and the cardinal number n is the class

of all classes

with just n members since that includes

classes whose

members are empirical objects there's no

problem

how theorems about natural numbers can

be applied

to empirical circumstances likewise real

numbers where for him

ratios between quantities and such

quantities include

physical magnitude such as mass and

length

the theory was designed to be directly

applied and the fact

that the numbers with which it deals are

taken to be objects

in no way hinders its applicability

what is problematic is how the existence

of logical objects can be justified

logical objects form a special class of

abstract objects

called by fragra non-actual objects

because they don't

act on other objects or bring about

effects in them

as physical objects do and are therefore

not perceptible by the senses

his account of what we do when we refer

to non-actual objects

was based on his celebrated context

principle

this principle says that it's only in

the context of a sentence

that we can refer to anything otherwise

expressed

we can't mention anything without saying

save as part of a process of saying

something about it

expressing a thought about it not

necessarily asserting anything about it

this principle is the most profound the

most difficult

and the most contentious ingredient in

all fragrance philosophy

philosophers who deny the existence of

abstract objects are labeled

nominalists they usually characterize

abstract objects precisely

by their not being actual in fragrance

sense that is by their

lack of causal powers and the standard

nominalist argument against their

existence

is that since they cannot affect

anything everything must appear exactly

the same if they do not exist

as if they do and hence that we can have

no reason

to suppose them to exist

well the equator is cited by fraga

as a non-actual object so suppose you're

in an airplane and you remark to your

neighbour

that the plane has just crossed the

equator

to your surprise he doesn't know what

the term equator

he doesn't know the term equator and

asks you what it means

you try to explain to him and he asks

whether you can see the equator

or feel the equator when you tell him

that it's not that sort of thing

he asks what reason you have to suppose

that there's any such object

seeing that everything will be the same

exactly the same if they were not

you can do no more than patiently

explain to him

how sentences containing the term the

equator

are used and in particular how we judge

of their truth and falsity

and according to fraga in order to

justify the use of the term

the equator you need to know more than

that

this is what is meant by saying that

it's only in the context of a sentence

that we can refer to an object or at

least to an abstract object

and we may add what's intended to be

part of the content

of the context principle that when we

know

how sentences mentioning such an object

are used

we thereby know what it is to refer to

that object

well in my view this is a wholly

satisfactory vindication

of the general vindication of the use of

terms

referring to abstract objects and there

can

therefore doubt that the context

principle

is in broad outline sound but we should

not

jump to the conclusion that abstract

objects present no further difficulties

and fraga did not jump to that

conclusion

to justify the use of abstract terms of

any

given kind in the light of the context

principle

we must be able to do what that

principle demands that is

to explain without circularity the use

of sentences containing such terms

and the conditions of their truth and

falsity and this is by no means always a

simple task

why did fraga require that numbers be

recognized as objects

to use the for example the sorry

pi is transcendental should be treated

as being of the same form as

clinton is unsuccessful his primary

reason was to guarantee the existence of

sufficiently many

elements of each theory for all possible

applications of it while preserving the

purity of arithmetic

it's easy to imitate fragrance

constructions at a higher level for

example take

cardinal numbers as properties of

properties of objects so

they're no longer themselves construed

as objects and that's essentially what

russell whitehead did

in principia mathematics matica taking

properties

for this purpose extensionally the

drawback of doing this is that one can't

guarantee

then guarantee that there are infinitely

many natural numbers

the solution adopted by russell

whitehead

was to assume an axiom stating that

there are infinitely many individuals

a proposition certainly not a logical

truth and

dubiously drew true at all and thereby

the entire project

of deriving arithmetic from logic was

abandoned

similar difficulty arose with the real

numbers

it was by taking numbers to be objects

that frag was able to circumvent this

problem

he insisted on the generality of the

notion of cardinal number

objects of all kinds can be counted and

among things that can be counted

are numbers themselves as when we speak

of the number of roots of an equation

or of prime numbers less than or equal

to a given number

so numbers must be objects since a

cardinal number is always the odd number

of

objects satisfying some given condition

and this allowed trigger to prove the

existence

of infinitely many natural numbers

independently

the existence of objects of any other

kind

we can show a number n to exist by

producing a predicate true of just

n objects so the number not exists

since is different from itself is true

of not objects

and so is the number naught is true of

just one object

and hence the number one exists and

is the term of the sequence not one is

therefore true

of just two objects and so the number

two also exists

and in general given the existence of

the numbers from not to n

the number n plus one must exist since

it's the number of

terms in that sequence this is a

informal sketch of the theorem fraga

proved really

rigorously from his assumptions the

assumptions are easily stated

natural numbers were to be treated as

cardinal numbers

and the basic notion for the theory of

cardinality

is that expressed in natural language by

saying that there are just as many

objects of one kind

as of another frago adopted as a

definition of this notion

one that had recently been accepted by

other mathematicians of his day

namely the existence of a relation

mapping

the objects of the one kind one to one

onto those of the other

he tacitly assumed that every term

standing for a cardinal number

could be framed by means of the operator

the number of objects which

so given that by appending

any well-defined predicate to this

operator one would obtain a term

standing for an object and given the

definition of justice many

fragra formulated a fundamental

equivalence

namely the number of objects of one kind

is the

same as the number of objects of another

if and only if there are just as many

objects of the one kind

as of the other this fundamental

equivalence

looks at first sight tautologous but it

isn't it's a principle

governing the introduction of terms

standing for numbers

the right hand side says nothing about

any such objects as numbers

the left hand side says that the numbers

denoted by

two basic terms for them coincide

now against the that background fragra

proved

by means of suitable definitions that

all the basic principles of number

theory

could be derived by means of second

order logic

from this fundamental equivalence

so a justification for the fundamental

equivalence is required

if the introduction of terms for

cardinal numbers is to be defended by

appeal to the context principle

we need to show that we succeeded in

specifying

the condition for the truth of any

sentence containing such

terms and fragra discussed whether the

fundamental equivalence

could itself be regarded as affecting

this

well some present day philosophers and

mathematics

enthusiastically answer yes to this

question or

for some rather similar question but

fraga's own answer

was no so he resorted

to his definition in terms of classes

the number of objects of a given kind

is the class of classes a such that just

as many objects

of that kind as members of a

the only use he made of this definition

was to derive the fundamental

equivalence from it

its purpose was simply to introduce

terms for numbers in a way

he considered unexceptionable

his ground for denying that the

fundamental equivalence

served to do what the context principle

required

was that it failed to determine the

condition for the truth or falsity

of a statement of identity between a

number and an

object denoted by a term not given as a

number not

formed by means the operator the number

of objects

the objection is sound but he overlooked

a far more basic one

namely that the fundamental equivalence

does not even determine

the truth or falsity of every statement

of identity between numbers

it doesn't for instance give us any

means of deciding

whether the number of natural numbers is

or is not the same

as the number of all cardinal numbers or

of all objects whatever

now consider what frag was about he was

trying to justify the introduction of

terms for cardinal numbers

satisfying the fundamental equivalence

the numbers denoted by those terms were

then to be treated as belonging to the

same domain of generality

as all other objects those to be covered

by the quantifiers

for every object x or there is an object

x

and in accordance with this the terms

being introduced

were to include ones for the number of

cardinal numbers of some

given kind as specified by means of the

some predicate numbers it's essential

for the proof of the infinity of the

natural numbers and all this was to be

done

without first specifying of what

objects the domain of generality was to

consist

the procedure has a troubling

circularity

which objects there are depends on which

numbers there are

but which numbers there are depends on

which objects

the domain contains

well what do we say about this situation

should we say as the philosophers i

mentioned believe that the conditions

required for appeal to the context

principle were too stringent

must need not determine the truth value

of every statement

if so just how could they be weakened

without destroying the plausibility of

the principle

or should we say that the whole

procedure is misconceived

the usual conception of how an

interpretation of a formal theory

should be laid down is that one should

begin

by specifying the intended domain over

which the variables that arrange

and then interpret with reference to

that

the expression special to the theory

saying for example of which elements of

the domain some given predicate

is to be true so according to this

conception it's impossible

simultaneously to determine the domain

and the interpretation of the symbols

intended to denote

elements of that domain but if that's

the only way in which one can go about

laying down how a mathematical theory is

to be understood

we can never explain how a fundamental

theory

such as number we can never explain a

fundamental theory

such as number theory or as analysis by

a fundamental

mathematical theory i mean one whose

elements cannot be defined as resulting

from some simple operation on those of

another theory

and this amounts in practice to one with

a greater number of elements

than any prior theory the natural

numbers form the prototype

but innumerable totality one all whose

elements can be

generated as the terms of an infinite

sequence

it's by grasping the conception of the

totality of

natural numbers that we first come by an

understanding of the phrase

infinitely many likewise the

real numbers form the prototype of a

larger infinite totality that is a

non-innumerable one with too many

elements for them to be generated

as the terms of a sequence our problem

is to explain

how we first attain a conception of a

totality

the one or the other size cardinality

so long as these theories are presented

to us before any others with domains of

these sizes

we have no means of specifying their

domains

in advance of expounding the

interpretation of the theory as a whole

and yet we do come to understand

it seems that fraga must have been right

in thinking that for these fundamental

theories

we have to acquire a conception of the

domain

simultaneously with that of the meanings

of the basic notions defined

over it and yet such a process

appears doomed to vicious circularity

it's one of the great merits of

fragrance philosophy of arithmetic

that it faces this difficulty squarely

even if he's attempted to solution up to

it failed

he addressed the problem that's usually

ignored

but cannot be evaded his solution did

indeed fail

catastrophically i've discussed the

point is it applies to the cardinal

numbers taken as introduced by

means of the fundamental equivalence to

make clear that it doesn't relate either

to the consistency

or to the power of the method of

introduction but fraga rejected that

method and gave instead his definition

in terms of classes

but that just shifted the problem to the

justification for introducing classes

this he affected by a method precisely

analogous to the use of the

fundamental equivalence he treated the

operator the class of objects which

as a primitive symbol forming a term

standing for an

object whenever supplemented by a

predicate of objects

lay down an axiom stating that the class

of objects

satisfying any given condition is the

same as the class of those satisfying

some other conditions

just in case every object that satisfies

other condition

also satisfies the other the only

difference

was that in this case he had a

supplementary stipulation

to handle the case he believed to give

rise to the only problem

that of a statement equating a class

with some object

not given as a class and notoriously

this axiom rendered his system

inconsistent

it yielded the celebrated paradoxes of

said theory

and after struggling to escape this

calamity

fraga accepted that his entire life's

work

had failed

in setting out the problem i've tacitly

relied on an

attitude towards mathematical objects

that we naturally have

but seldom remark on fragra took for

granted that the logic

appropriate to mathematical theories is

that we now call

classical one that assumes every

statement

to be determinantly either true or false

and considers the truth or falsity of a

complex statement

to depend only on the truth or falsity

of its constituent sub-statements

or in the case of a universal

generalization or existential statement

of its instances this assumes

that the operations of generalization or

existential quantification

will preserve determinateness of truth

failure

that is we shall always obtain a

statement determinantly true or false

by attaching say every number or there

is a number which

to a predicate definitely true or false

of any specific number

well in just consider the empirical case

what then do we require of a domain of

generality

if this is to be so if that's to be

enough

to guarantee determinateness of truth

value we

normally assume it's sufficient that the

concept by means of which we specify the

domain

should have determinate application and

determinant conditions for identity

to guarantee a definite value true or

false

for every statement about all stars or

all books

and every statement to the effect that

there is a star or a book of a certain

kind

we take two conditions two things to

suffice

that the concept star or book should

have a quite precise application

no borderline cases and that it should

be definite what counts as the same

style

or book we don't need in addition

to lay down what stars or books there

are

reality does reality does that

for us now this assumption may be

challenged i'm not concerned with that

i'm concerned with the contrast to how

we normally think about mathematical

objects

to endow every statement about all real

numbers

or asserting the existence of a real

number of a given kind

with a definite truth value we don't

normally think it

enough to lay down what's the count as a

real number

following dedicant we might do that by

requiring

requiring it to have a determinant

relation of magnitude to every rational

but that would

merely tell us how to recognize a real

number

when presented with one it doesn't tell

us

what real numbers there are and it would

need a very robust realism about

mathematics

to think that that could be left the

mathematical reality

to determine normally we think

that that's something that we have

by some means or other to circumscribe

fraga might be accused of having

believed that it was

her classes or cardinal numbers could in

general be specified

and the conditions under which two such

specifications

determined the same class or number but

that would be unfair

he asserted a need to supplement any

such account by

certifying that it did confer on every

statement of the theory

a definite value true or false but

unhappily his attempted proof of this

for the theory with classes was

fallacious

and because of this he left unresolved

the problem

to which he thought he had found the

solution

my own view is that as it stands it

cannot be solved

fragrant sought a justification of our

arithmetical theories satisfying three

criteria

it must accord to arithmetic the status

of a science that is a body of truths

it must exhibit it as apt for

application to empirical reality

while not itself invoking any empirical

notions

or once derived from spatial or temporal

intuition

and it must leave intact the classical

theories

including the classical canons of

mathematical reasoning

it's probable that no account of what

justifies arithmetic

can satisfy all three criteria

no means exists to circumscribe the

totality of real numbers

without circularity and in so definite a

manner

as to warrant confidence that every

general statement about them

has a definite truth value it wouldn't

even be any use to relinquish the

requirement of the purity of arithmetic

the classical continuum cannot be

derived from physical reality

as we experience it but is rather a

concept

generated within mathematics and imposed

by us

in thought upon physical reality

rather we should see the totalities real

numbers

as an immediately indeterminate one

we can prove some statements about all

real numbers on the basis of the

knowledge

of what must hold good of anything for

it to be a real number

and we can prove some statements that a

real number of a certain kind exists

by finding a way to construct one but we

do not have

so sharp a conception of the totality as

to justify

our assuming every statement of either

kind to be true or false independently

of our being able to prove or refute it

and if that's right we are not entitled

to reason

in accordance with the canons of

classical logic

a weaker logic that we should be

justified in using has long been in

existence

so-called intuitionistic logic used by

constructive mathematicians most

mathematicians are reluctant to restrict

the allowable

methods of mathematical proof in this

way because they value the

power of classical reasoning but there's

no real

merit in presenting mathematical results

in a form more course than a careful

attention to the meanings

that can be legitimately attached to

them would warrant

it may well be that a version of

analysis

purified in this way would prove better

adapted to what fraga prized

as the basis for accounting it with

science

its application to the physical world

thank you michael very much um when i

was um

[Applause]

when i was preparing an introduction for

michael i ran into a puzzle

i wanted to try to convey uh

clearly very briefly totally

non-technically

the kind of work that michael did and i

found that

a lot of the things i was noting down

sounded very much like wittgenstein

so a question what's the

difference between michael and

wickenstein and michael has

i think himself told us

given us the slogan in his own lecture

there are of course

myriad differences in detail but

what's significant beyond that is the

very existence

of the detailed fully articulated

and carefully elaborated account um

michael has told us that what

distinguished

fragra from kant was fregger's

determination

to work through his problems in detail

and that is indeed as i believe this

lecture has given us some sense of

what singles out michael himself and

makes him the truly great philosopher

that he is

and i'm very proud to have him here for

this lecture thank you michael

now we've had a

we've had a bit of a gap uh we usually

try to leave a moment or two for people

who

aren't able to stay a little longer for

a question period but

in a moment if it's all right with you

michael we'll have some we'll have some

questions

uh followed then by the award uh

presentation of the award

uh later

oh no it was this beautiful making clear

for the people who don't know about it

yes i mean it was the

you did the balance i thought wonderful

oh

thank you very much because it's

terribly yes it's

it's terribly complicated and one

doesn't want to say what's not exactly

right

sure

i wondered if at some point i should

bring it over i thought well that's a

little bit

distracting it also looks pointy

on the other hand when i heard your

coughing yes well

i'm sorry it was a bit long no no we

didn't it's they take an hour and it

took an hour

yes yeah right now that's uh

just yesterday can we start please i

mean

if people would be a little quieter and

i think you need all this time to

get yourselves organized well it's

really these people though who are

someone else oh yes david papanowa

you said it wasn't there wasn't just a

problem

no that's the whole point of this

example that

it's certainly perfectly it's even

provably consistent if you

take the fundamental equivalent what i

call fundamental equivalence

uh as an axiom and with that definition

of just as many as a

one-on-one mapping it's certainly

uh in a setting of uh of second order

logic

derivation allows you to get the whole

arithmetic by that way and

the theory is provably equivalent that's

something

not equivalent that big or fun

consistent so the objection is not

consistent

in inconsistency the objection is

what the context principle demands

the introduction of a range of abstract

terms

is supposed to be justified providing

that you can lay down

what the truth values are to be of

um senses containing that type

the truth varies or in the case that

reference the empirical world is

necessary at least the truth conditions

right and i just remark

it the problem

of identity statements which are what

are directly dealt with

by the fundamental equivalent doesn't

just arise for once with

numerical terms on one side and some

non-numerical terms on the other

which is what pregame makes all the fuss

about but

even for cases when there are numerical

terms on both sides

how are we to decide i mean quite

obviously

the unless we know a lot more about what

there is in the domain of the

what objects there are we can't

possibly decide uh

identity statements of the kinds i

studied for example

the number of natural numbers equals the

number of cardinal numbers

uh so this

so there are there's a choice here

either the context principle

only justifies the introduction of these

abstract terms if it really does

settle i'll show you how to settle in

the case of

things involving empirical matters the

truth of

every statement in containing one of

these terms

or we've got to place much

less honoris requirement on what has to

be done

in order to justify the introduction of

abstract terms

by appeal to the context principle i

don't know exactly what that

ought to be or what would make it

compelling so fraga doesn't

acknowledge this fact that's all i

complained of

ability of mathematics to uh to heal

nature

and then we have a a long story about

numbers

but the in my domain where one hears

this

is about much of different kinds of

branches of mathematics like

um group theory isn't it amazing that

uh the elaborate mathematics of group

theory has any bearing whatsoever on

reality or hilbert's faces yeah so i

wonder if uh

is there a connection that you can trace

briefly for us to

how to think about the other branches of

mathematics

that get used in modern physics well i'm

not sure that i can

but i mean

what i gave is the recipe

for fragrance says there is

there is a question about

applicability let's say some theory

ought to explain

how mathematics gets applied

and the only theory that can explain

this is mathematics itself otherwise he

says the question falls

into the void right the philosopher the

physicist

uh not my business and the mathematician

says it's not his

uh in the case of groupthink i can't

answer this very well the uh

when you introduce the notion of groups

you're

normally introduced to it via the notion

of a transformation of some

kind of transformations closed under

obvious operations uh under

composition in inverse

and when that's not enough

the the whole idea of the vrega has

and i said it's very difficult to apply

is

you should look to see what is the most

general characteristic of the theory

that enables

one to apply and then you should define

the notions in those terms

that's not very easy to do i don't even

know whether it's practicable

but uh well you can say whether

whether you think that's

look group the very sophisticated theory

but it starts with this

notion of transformation and that

enables you to get

a grip on something

so i don't know that i can say any more

than that at the

moment

uh that was a question discussed between

frank and piana

themselves

yes

piano derived

a great i mean piano was

concerned somewhat

no that would be a wrong thing but piano

was

concerned to develop

[Music]

a certain kind of foundational

presentation of mathematical theories in

a

systematic way but he he wasn't i think

a lot of the what piano did was actually

derivative even what we call the piano

axiom

were given by dedicate in the first

instance that particular accentization

of numbered view

[Music]

and

well i don't know that i can say a great

deal more than that i think that the

the logical notions that piano used were

probably uh

derived from in in part from drago's own

work i

i may be wrong about that this

historical question as you said and

i'm not dead certain of the answer to it

the historical question wasn't what was

discussed between frank and piano what

was discussed was

questions of rigor and frag objected to

the

kind of definitions that piano

yes almost certainly they create nothing

to fragra

and

yes and they also as i said probably

still credit to piano things that was

due to dedicate

um i'm not trying to detract from

piano's reputation i thought things i

said just sound a bit like that but

i think look if you look at

russell's principles of mathematics he

very openly uses a lot of work done

principally by german mathematicians he

doesn't

attempt to to pretend that

uh that it's all his own work by no

means

but that's a very useful summary

of a lot of foundational work that

existed

right at the beginning of this century i

think that

piano did something of the same kind he

systematized a lot of stuff

but i think that a lot of it wasn't due

to his

own work

um no

do you think that's wrong

can you speak more lovely places i

couldn't catch no

no

[Music]

is

he was introducing

yes well i think he was much more on

russell's side than

on puerto rico so far as i know you

never took any notice of that

the particular dispute between those two

but certainly he was in the same sort of

position as

russell namely he did not think

that induction was a

form of reasoning special to mathematics

on the contrary he made a

very specific point of what russell

later

perhaps independently made a point of

this definition that i cited

allows us to exhibit if we define the

natural numbers

using this definition of a sequence

then induction falls out as immediate

consequence of it

and therefore not as depending on any

special

mathematical principle of reasoning

but as uh

just on purely logical principles given

this

definition quackers

position had to do with

the idea that you have to use induction

in seeing that the system is consistent

or the proofs or proofs from the axioms

again

going to yield true conclusions or

something which is

true in a way but it doesn't seem to me

to invalidate the position

of russell and and fragra

and it's like saying you can't

show a logical system to be sound

because look in your argument you've

used some of the principles

of that logical system well of course

any argument has to use

some principles but

there's a difference between a form of

argument that you

use and one that is the object

of of your reasoning

that is that you're talking about so i

don't believe that that argument of

pancreas is sound

at any rate if from the point of view of

your purely historical

question craig was certainly on

russell's side

and would have been against poincare if

he had read him

yes he said it perfectly explicit

well i can't get he certainly says it in

the foundations of arithmetic

um i can't give you an exact reference

at the moment i haven't got the book

with

me but

i'll send you a i'll send you a

reference

um

correct me if i don't exactly answer

your question i

quite got it but the thing was there was

this long

correspondence between russell

and uh freyja in the course of which

russell kept trying all sorts of uh

possible ways out and uh fraga

uh with great confidence despite this

disaster that has occurred to him with

great confidence kept shooting them

down and saying you can't say that and

and in the middle of it fraga announces

that she's found this

solution which then russell pays no

attention to

and says you're well you're probably

right but um

and actually of course the solution

didn't work

now the thing about that is it's

surprising it took so long

it took fraga till 1906 i think to

to to recognize that it didn't work i

don't think he ever knew

that you can still get a contradiction

in his system i'm not sure

but he certainly must have known as soon

as he

addressed himself to the question that

the proofs he had given

would break down under this weakening it

wasn't

a weakening of action five it wasn't

weakened enough

to avoid contradiction but nevertheless

it was weakened enough to invalidate

the simplest proofs in his theory you

couldn't even prove

using that that naught is not equal to

one that's

fatal for any foundations for

arithmetic obviously um

and i think you can locate the exact

point

at which he realized that uh slightly

indirect

but not very injury and namely in august

1906 and after that

he simply acknowledged that that his

work in the philosophy of mathematics

had

had crashed right and he thought

all that remained was the logic that was

all he had achieved but but not the

theory of classes not the

value ranges and therefore not the

foundations of

of number theory or analysis

i want just to say this i suppose

well no problems i'll wait and say

something more in the

moment

was

um

could you just say um what would you

perceive as

the main difference between the research

programs because in the outset

it seems that they shared a good view

yes this is a contentious matter

people disagree about this quite a lot

so i'll just say what uh

briefly what i think uh it's true that

was a converted

no they're not exactly the same fragra

was not a converted mathematician he

remained a mathematician who also

engaged in philosophy engaged in it

while being

while teaching mathematics and ordinary

mathematics counselors

all the time tiene uh

well jose really was a converting this

was someone who had been

started as a mathematician moved into

being a philosopher

um however that's not the main

difference

jose wrote only one book in fact it was

a book which was supposed to have a

second volume and never did

his first book on the philosophy of

arithmetic

it was written after fraker's book it

contained

criticisms of fraga and

it was a book of which fraga later

on four years afterwards wrote

of savage review

um there'd been some correspondence

between them

and i think it's certainly known or

reasonably guessed that jose was deeply

hurt by this review

as indeed anyone would be who read such

a review

his book

i think that fragrance review was

in part unfair in places unfair

i nevertheless think it by and large

fair

that is

the point at which fraga and jose were

close

was let us say just after

jose had published the prologue to his

logical investigation and

containing this attack on psychologism

now i don't know whether jose was

influenced by frago or not but what i

do myself think is that the book on

philosophy of

arithmetic was a thoroughly

psychologistic work

on a basis quite different from

his work at the time of the logical

investigations on the

basis that he had repudiated and i think

it's

a far inferior work to to

to uh to fragrance and

didn't lead in a fruitful direction i

entirely agree that jose the very

interesting philosopher

i think a lot of his work is very well

worth studying i

don't believe this to be true of his

work

on philosophy of mathematics that's my

own view

other people think quite differently and

some people who say

the philosophy of arithmetic isn't

psychologistic at all

there's very little difference between

that logical investigations

i i think this is a bizarre view i can't

do that

so but i will have to discuss the text

in detail but i just that's my reply to

it

michael i think it's time to turn to uh

more of a celebration

um you don't want the clothes by saying

gregor was more interesting

no i don't much want to close by saying

that but

stupid i should have said at that moment

what

it came to me to say um

and sorry i'm just trying to remember

what it was because it was quite an

interesting

point um but

oh yes just get yes i think what i

wanted to say was this

there are a lot of people who study

fragrance philosophy of

mathematics but actually concentrate

only

on the work of number theory and

sometimes you'll see it

argued that well maybe what he ought to

have

done was simply to take this

the numerical operator a number of

objects which as primitive

and we know that gives us a consistent

theory and that's all he needed for

for his construction of listening and so

on

that overlooks two things

um or let's say it overlooks his

attempt in uh the

basic laws of arithmetic which is very

little studied and deserves study it has

some very interesting mathematical

results

to to do a similar thing for

analysis for theory of real numbers

there's no way in which you could set

about reconstructing that as it stands

without using notion of of

of class and you would have to change it

very radically if you wanted to base it

on a

on a consistent century so

that suggestion simply overlooks half of

what

fraga wanted to do the trouble is people

read the foundations for arithmetic

which hardly discusses real numbers at

all

they think it's all about number theory

instead of being just as much about

analysis

that's one thing i want to say the other

thing i want to say is this

you could say fraca took

the fundamental application of the

natural numbers

to be as cardinal numbers say how many

things you

made this very very plain you could it's

perfectly reasonable to

argue that you ought to have taken their

use as ordinal numbers as when you

number houses in the street

as more fundamental because after all

when you count you can't find a number

of things or even if you enumerate

infinite number of things you impose an

order

it doesn't matter what the order is

particular order is

you're in the interest in the

cardinality but you get it the

cardinality by imposing an order so

that's one reason for saying notion of

ordinal number is more

fundamental another reason is that

uh without the notion of the

trans-financial

you have only one way

of getting bigger cardinal numbers

namely the power set operation start

with

the number of natural numbers number of

sets of natural numbers number of sets

sets natural numbers and so on you

haven't got the way that counts or used

by

considering the totality of the

enumerable ordinal gives you a left one

and then there's the interesting

continuum problem how that relates to

the number of real numbers and so on so

there are good reasons for saying a

notion of ordinal number is more basic

actually

the notion of cardinal number you

couldn't as this wasn't my

observation originally somewhere else

you couldn't

set about it in the same way or if you

did you would run into contradiction

much quicker than fraca did because

while the fundamental equivalence which

i discussed is perfectly consistent

if you try to do that

with the notion of the order type of an

ordering

all the type of r is equal to the old

type of best just in case there's a

similarity map order preserving map

on the field once the fields yeah if you

just run into the burally 40 paradox

straight away

without using classes at all

so that's quite that raises a big

question

how far can you push this

way of introducing operators yielding

abstract terms

um well without running into

inconsistencies to start with uh

well i'll just leave it at that you