by Jorn K. Bramann and Thomas A. Mappes

The purpose of this paper is to shed light on what commonly is called

"logical necessity," i.e., the necessity that characterizes logical and

mathematical operations. We hope to do this by appeal to Ludwig Wittgen-

stein, particularly to his book Remarks on the Foundations of Mathematics.(1)


Ludwig Wittgenstein (1889-1951), although better known for his work in

other areas of philosophy, has thought about philosophical problems of

mathematics throughout his life. Wittgenstein was born in Vienna, Austria,

but he lived most of his life in England, where he moved in 1908 to study

engineering at the University of Manchester. Wliile working on the construc-

tion of a propeller for a jet-reaction engine, he became interested in pro-

blems of mathematics, and it was as a result of this interest that he decided

to study mathematical logic and to take up the study of philosophy under the

guidance of Bertrand Russell. Although his two major works, the Tractatus

(1922) and the Philosophical Investigations (1953) do

not exclusively treat the philosophy of mathematics, reference to the founda-

tions of mathematics plays a major role in both works. We hope to show in

this paper that his reflections on mathematics can be used to dissolve some

of the mystery that surrounds the notion of "logical necessity."

1. Necessary vs. Contingent Truth

It is customary in philosophy to distinguish between statements (judg-

ments; propositions) which are factually true and those which are "necessar-

ily" true (or, what comes to the same, between statements that could conceiv-

ably be false, and those which could not conceivably be false). An example

of the former would be the statement "All of my classmates are unmarried,"

while an example of the latter would be the statement "All bachelors are

unmarried." Whereas it is conceivable to think that there are some of

my classmates who are married, it is not conceivable that one could find a

bachelor who is married. The inconceivability of finding a married bache-

lor is taken to warrant our saying, that "All bachelors are unmarried" is

necessarily true.

Such necessity is also attributed by philosophers to logical inferences. From the two


"All my classmates are unmarried" and

"Angelo is a classmate of mine''

it necessarily follows that "Angelo is unmarried." That is to say, if

somebody admits the two premises, he cannot, without falling into logical

contradiction, refuse the conclusion. In a syllogism of the above kind

the conclusion follows "with logical necessity" (as opposed to cases where

conclusions follow only conditionally, as in the statement "Where a horse

drowned there must have been water'').

Still the same kind of necessity exemplified in the above cases is thought to attend

mathematical statements and calculations. That twelve is the sum of seven plus five, or

that the sequence 2, 4, 6, 8, 10, ... is the result of applying the rule "start with two and

add two to each preceding element," is said to follow with the same necessity that attends

a bachelor's being unmarried or a valid logical inference. Now, the apparently

unassailable necessity displayed in the above examples has given rise to

great philosophical puzzles, puzzles which are by no means resolved even

today. What philosophers are mainly concerned about is the source or the

foundation of the necesssity involved in logic, mathematics, or such state-

ments as "All bachelors are unmarried."

2.. Two Competing Accounts of Mathematical (and Logical) Necessity

Several attempted accounts by philosophers are available, however, and

one such account is classical Platonism. According to classical Platonism

mathematical statements are necessarily true because they correspond to

ideal mathematical objects which stand in certain eternal relations to each

other — independently of actual human construction and calculations. In

the Republic Plato has Socrates say:

In this respect, then, no one who has even a slight acquaintancewith geometry will deny that the nature of this science is in flat contradiction with the absurd language used by mathematicians, for want of better terms. They constantly talk of 'operations' like 'squaring', 'applying', 'adding', and so on, as if the object were to do something, whereas the true purpose of the whole subject is knowledge — knowledge, moreover, of what eternally exists, not of anything that comes to be this or that at some time and ceases to be.(2)

Plato contrasts here geometry (and, by implication, mathematics in general) as a process

of invention with geometry as a process of discovery. Plato wants to argue that

mathematicians have to discover mathematical truths, i.e., truths that exist prior to what

mathematicians do. Mathematical statements are necessarily true, then, because they

correspond to some ideal relations, which they copy.

The opposite account of the nature of mathematical truths, which Plato attacks in the

above quotation, is often called "Conventionalism." The main point of Conventionalism is

the denial that there is only one way of calculating correctly in mathematics, namely

the one way which supposedly corresponds to some ideal mathematical rela-

tions. Conventionalism maintains that the axioms and rules of inference

which govern our mathematics constitute only one possibility among many,

and that alternative systems (with different axioms and different rules of

inference) could be constructed which would be as valid as our actual mathe-

matics. Conventionalists, in other words, insist on just the "doing" of

mathematics which Plato so abhors. They insist that in principle one is free

to choose any system, and that only after the axioms and basic rules are

agreed upon are we then compelled to follow them. The necessity involved

in true mathematical statements, then, is not a result of their correspon-

dence with some ideal relations, but is to be located somewhere in the

axioms and the rules which we adopt when doing mathematics. If certain

axioms and rules are adopted, then certain theorems, series, calculations,

etc. follow necessarily. Whatever follows from certain axioms and rules

is "implied" in the latter; stating what follows consists in nothing more

than mapping out what somehow is already there.

This point is often put by saying that the necessity in mathematical

statements results from the fact that mathematical equations state the

equality of meaning of two expressions, that "7 + 5" simply means "12".

All analytic statements (i.e., all statements that are necessarily true),

thus, are understood as statements about the meanings of certain symbols.

Alfred Jules Ayer in his famous Language, Truth and Logic writes:

We see, then, that there is nothing mysterious about the apodeictic certainty of logic and mathematics. Our knowledge that no observation can ever refute the proposition "7 + 5 = 12" depends simply on the fact that the symbolic expression "7 + 5" is synonymous with "12", just as our knowledge that every oculist is an eye-doctor depends on the fact that the symbol "eye-doctor" is synonymous with "oculist". And the same explanation holds good for every other a priori truth.(3)

It is clear then that classical Platonism, on one hand, and Convention-

alism, on the other, offer contrasting accounts of the "necessity" that

characterizes the realm of mathematics (and logic). Whereas classical

Platonism grounds the necessary truth of mathematical statements in their

correspondence to "real" mathematical relations, Conventionalism grounds

the necessary truth of mathematical statements in linguistic convention,

in the "meaning" of symbols. Though we believe that each of these views has its own

peculiar difficulties, it is our intention to bring to light a difficulty that militates equally

against both views.

The difficulty common to both classical Platonism and Conventiona-

lism can initially be revealed by asking how either the alleged ideal

mathematical relations or the "meaning" of symbols can compel anybody to

proceed in a certain way
(in continuing a mathematical sequence), to

come up with a certain sum (in calculations), or to draw a certain con-

clusion (in a logical inference). There is no mystery in that people do

certain things when they continue a sequence,calculate in certain ways, or

come up with certain conclusions, while rejecting others as incorrect.

But what is mysterious is the "necessity" with which one step in a series

follows another, a certain number results in a calculation, or a certain

conclusion follows. What is puzzling is the logical and mathematical must,

the strange inexorability of logic and mathematics. We say that a certain

step, a certain result or a certain conclusion follows necessarily -- but

why "necessarily"? What is the force behind this peculiar compulsion?

When philosophers say that the results are already "implied" in axioms or

premises, they sound as if one only had to follow what in a mysterious way

is already there, as if one had, as it were, to follow invisible tracks which

are already laid out. Wittgenstein's suggestion is that philosophers, in

thinking about mathematics and logic, are caught by a misleading picture

when they talk about future steps, results, and conclusions which somehow

are already there before they are actually taken or derived.
This picture

is the picture of rails which are laid out before any movement takes place.

And it easily forces itself upon us when we think about mathematical and

logical necessity — but the trouble is that it is nothing more than a

picture, that it is not clear what there is in reality corresponding to

the rails that compels our steps. Thus the nature of mathematical and

logical compulsion remains a mystery.

3. The Incorrigible Tortoise and the Obstinate Student

To bring the above mystery further into focus it is helpful to imagine

what one could say to somebody who does not feel compelled by what we feel

compelled when taking the right steps in a sequence, deriving a correct

result in a calculation, or drawing a logically correct conclusion. One

case of somebody who does not feel compelled (.or who pretends not to feel so)

has been worked out by Lewis Carroll in his puzzle "What the Tortoise Said

to Achilles."
(4) This puzzle deals with the nature of logical necessity, i.e.,

with that which supposedly forcas us to draw a certain conclusion from cer-

tain premises. A standard example of a logical inference is the following


All dogs are mortal
Phido is a dog

Phido is mortal, or



According to standard logical procedure one has. to accept Z, if one accepts

A and B (i.e., Z must be true, if A and B are true). This, however, Lewis

Carroll's Tortoise refuses to do, thus putting Achilles in the position of

having to prove that the Tortoise cannot possibly refuse the conclusion Z.

Achilles (as indeed all of us) believes the following statement (C) to be

true: "If A and B are true, then Z must be true," and with this the Tortoise

disagrees. Finally the Tortoise agrees to accept C, provided Achilles adds

it as a further premise to premises A and B. Thus:



Achilles now thinks that by admitting C the Tortoise has to accept Z. The

Tortoise, however, says that if he did so, he would be subscribing to one

further premise, namely (D). He asks Achilles to write it down as still

another premise, which Achilles does. Achilles then says: "Now that you

accept A and B and C and D, of course you accept Z." The Tortoise, pre-

dictably, does no such thing, but asks instead: "Suppose I still refuse

Z?" To this Achilles replies: "Then logic would take you by the throat,

and force, you to do it: Logic would tell you 'You can't help yourself.

Now that you've accepted A and B and C and D, you must accept Z. So

you've no choice, you see."

The Tortoise does not necessarily believe what "Logic" tells him,

namely (E): "If A and B and C and D are true, Z must be true" (who, after

all, is "logic" to tell him?) and the dialogue continues ad infinitum.

What is so frustrating for Achilles is the fact that the Tortoise simply

does what he is not supposed to do, and that there seems to be nothing, no

further agency or reason, to which Achilles could appeal to make the Tor-

toise conform to the demands of Logic. The Tortoise, no doubt, is un-

logical, i.e., violates the laws of logic, or even "the Laws of Thought",

but there seems to be nothing that would compel him not to do so. No

"Logic" comes to take him by his throat; he seems to get away with breaking

our most sacred, our most fundamental laws. And notice that the Tortoise

is not simply insane, that he is rather calculating and methodical, not

somebody who is gibbering out of ignorance. All of which seems to point

to the fact that there is something mystical about "logical necessity" or the strange

inexorability of logical and mathematical necessity. Ordinarily,

to be sure, we treat correct mathematical calculations and logical inferences

as unassailable, as absolutely true. But once challenged, there does not seem

to be very much that could be offered in the defense of the absolute claims of

logic and mathematics.

Consider now a second and somewhat parallel case, this one suggested by

Wittgenstein himself. The Tortoise case centered around the logical acti-

vity of inferring. This second case centers around the mathematical activity

of "continuing a sequence." Here the role of the incorrigible Tortoise is

played by a student whom all mathematics instructors will be inclined to

label "obstinate, most obstinate.". We are asked by Wittgenstein to visualize

the following interplay between instructor and student.

The instructor is engaged in teaching the student how to continue a

mathematical sequence. He says to the student, "2, 4, 6, 8 — now you con-

tinue in the same way". Ordinarily, if the student were to take up and

say, "10, 12, 14, 16," the instructor would break in, satisfied that the

student had "got the point." But Wittgenstein asks us to visualize the

student continuing the sequence in the expected manner until reaching 1000

and then the student says, "1004, 1008, 1012 ...". The instructor cries out,

"But why are you going on now in a different way?" "But I'm not going on in

a different way, I'm going on in the same way," replies the student. "But

can't you see that you must now say '1002' and not '1004'?" The student

refuses to acknowledge what the instructor claims he must acknowledge, and

the instructor seems to be faced with the same kind of obstinacy as that

which confronted Achilles. The teacher insists that the student was asked

to continue "in the same way" as he did before coming to 1000, and the stu-

dent replies by challenging the teacher's interpretation of what "in the

same way" means. For the student it means to add 2 at every step until 1000,and then to

continue by adding 4 at every step. And that is not what

"continuing in the same way" means to the teacher. What the student's

obstinacy brings out is the question: Whose interpretation of the expres-

sion ''continuing in the sane way" is correct? And Wittgenstein's com-

mentaries in the Remarks on the Foundations of Mathematics suggest tlie

answer: "That depends." In this way Wittgenstein challenges the idea

that the teacher's interpretation is right as a matter of course. He

challenges the idea that there is no conceivable alternative to the answers

the teacher considers correct.

4. Two Levels of Analysis

To be sure, both the student and the Tortoise qualify as "stubborn," but

can they legitimately be called "irrational" ? Yes and no. Here it is crucial to distinguish the

perspective from which the answer is given. Clearly, at the common sense level, both tlie

tortoise and the student are legitimately termed 'irrational.'' The tortoise, within the context

or "institution" of deductive reasoning, refuses to tollow a rule of reasoning,

a rule which effectively serves as a criterion for "rationality" within that

given context. (Think here of the way in which mathematical IQ tests would take the

answer 1002 rather than 1004 as the criterion of mathematical intelligence.) At

the common sense level, then, if we look at the refusal of both the Tortoise

and the student to follow certain rules, we say simply, "Yes, that is wliat

we call 'irrational."

But at the foundational, level, as distinct from the common sense level,

we argue that neither the tortoise nor tlie student can legitimately be called

"irrational." Wittgenstein writes, and we will try to elucidate this none too transparent claim,

"Somebody (such as the incorrigible tortoise or the

obstinate student) may reply like a rational person and yet not be playing

our game." Achilles claims that the tortoise must admit such and such.

The instructor claims that the student must conclude such and such. But,

at the foundational level, the "must" can be sustained (or made sense of)

only if there are available no genuine alternatives. Thus, in order to

show that neither the tortoise nor the student need to be judged "irrational"

at the foundational level, we only have to show that in each case there are

available intelligible alternatives.

We begin with the case of "continuing a sequence", as this seems to

offer fewer difficulties than the case of "logically inferring." Following

a line of thought advanced by Charles S. Chihara, the task is seen to con-

sist in specifying a background framework, against which we may be able to

see that answering "1004" is truly an intelligible alternative. Here it is

important to think that the "obstinate" student operates within a different

background framework than the instructor (who operates within our familiar

common sense framework). And what we need is a specification of this alter-

native background framework.

Consider then the student in question as having been trained in a

hitherto unknown society where it was "natural", i.e. ingrained via training

into one's mathematical intuition, to continue a numerical sequence "2, 4,

6, 8, .... 996, 998, 1000, 1004. 1008, ..., 1992, 1996, 2000, 2008. 2016..."

We need not say why this sort of expansion was taken as primary or natural,

only that it was. (This would rival, as Wittgenstein points out, an imagined

society in which everyone reacted "naturally" to pointing by looking in

the direction opposite to that pointed to.) Thus, against this background

framework, it becomes intelligible how our no longer so obstinate student

would be able to justifiably claim that saying '1004' is to answer correctly, and to say

'1002' is to answer incorrectly. To say '1004' is in a

natural and obvious manner, for him, "to go on in the same way." To say '1002'

is "to go on in a different way." In no way then is the student falling into

logical contradiction, given his background framework. Thus we see that "some-

body may reply like a rational person and yet not be playing our game."

Accordingly, in the case of continuing a sequence, there is no ultimate war-

rant for the must at the foundational level. Here we must acknowledge that

the primacy of practice is paramount. We say, "996, 998, 1000, 1002. ...,

that is the way we do it". The student says, "996, 998, 1000, 1004 .... that is the way we

do it".

5. The Tortoise Beating Achilles On The Foundational Level

We turn now to the more difficult and fundamental case, that of "log-

ically inferring." Here Chihara's strategy, so successful in regard to the

case of continuing a sequence, must be judged inadequate. In the case of

continuing a sequence it was possible to show that a given alternative was

relatively easily imaginable — imaginable within the framework of our own

accepted "Laws of Thought." That is to say, we have shown that the student,

operating within an alternative background framework, could say ''1004, 1008,

etc." without falling into logical contradiction. With the incorrigi-

bility of the tortoise, however, we are faced with a direct challenge of our

"Laws of Thought", for it is a violation of the "Law of Contradiction" to

indulge in saying that all dogs are mortal, that Phido is a dog, and to re-

fuse to admit that Phido is mortal. (For the dog Phido cannot be both mor-

tal and non-mortal at the same time.) And since the violation of a 'Law of

Thought" inescapably generates logical contradiction, it is clear that an

alternative "Law of Thought" cannot be exhibited as intelligible within the

framework of our accepted "Laws of Thought". Yet it may be possible to

conceive of a context in which the Tortoise could refuse, justifiably, to

draw the conclusion that Phido is mortal, even though he has admitted that all dogs are

mortal and Phido is a dog. In other words, it may be possible to

imagine a society where it is natural that people refuse to draw this sort

of conclusion, without being labelled "irrational", or without being repri-

manded for what people (and students:) in our own society are reprimanded for.

Now, there does not seem to be any difficulty in imagining a society

where the "Laws of Thought" are simply ignored. It is, after all, not un-

common in our own society that people indulge in contradictions, if that

should be to their advantage. One often encounters cases where people admit,

e.g., that

All dogs are a nuisance,

His own pet is a dog,

and where they refuse to draw the conclusion

His own pet is a nuisance.

But it is clear that in such cases we do not say that the person refusing

the conclusion does not recognize the "Laws of Thought", that he acts in the

name, as it were, of logical freedom, but rather that an excessive degree of

self-love overrides the proper conclusion. The philosophically interesting

question is whether we can imagine a society in which the "Laws of Thought",

as it were, do not exist, i.e., in which people do not only ignore them on

special occasions (while recognizing them in principle), but in which they

do not play any role at all. We would have to imagine such a society in order

to make plausible Wittgenstein's numerous suggestions in the Remarks, that the

"Laws of Thought" are nothing more than a convention which might exist in some

societies, while lacking in others. (Cf., e.g., the following passage:

What we call "logical inference" is transformation of our expression. For example, the translation of one measure into another. One edge of a ruler is marked in inches, theother in centimeters. I measure the table in inches and go over to centimeters on the ruler.— And of course there'is such a thing as right and wrong in passing from one measure to the other; but what is the reality that 'right' accords with here? Presumably a convention, or a use, and perhaps our practical requirements. (8))

To allow that radically different cultures could refuse to infer in the way we do, that they

could refuse to follow our "Laws of Thought", must strike us as a rather problematic notion.

What are we to make, e.g., of a tribe where every member admits that all dogs are mortal

and that Phido is a dog, but

where nobody ever admits that therefore Phido is mortal? There could, of course, be

the difficulty that this tribe has a different way of deciding what is and what

is not included in "all" dogs (perhaps their practice allows certain exceptions,

so that Phido would not count among "all dogs"). In this case there would not

be any disagreement with our "Laws of Thought", for in that case we too would

admit that "Phido is mortal" does not follow. But given that they agree that

"all dogs" includes all creatures of a certain kind (i.e., with certain speci-

fied features), and given that Phido is such a creature, then we feel absolutely

nobody could possibly refuse the conclusion that Phido is mortal. For nobody

can say that a creature is mortal, and at the same time that he is not.

But what if this tribe does so, anyway?

Then, again, we can say that they contradict themselves, that they are

inconsistent, that they are "irrational", and that they should not do what

they are doing. But the fact is that we are simply repeating ourselves, that

we have nothing further to say, that we are simply stating that they deviate

from our own practice. (For to say that a whole social practice, as opposed

to individual performance within any social practice, is "irrational" would

presuppose something like an ideal that exists independently of all social

practices, a Platonic eidos of a social practice, and it is this which

Wittgenstein wants to expose as a philosophical fiction.) Undoubtedly, they

practice what we call "contradicting one-self", but so what? They do it,

and we do not, and that is all there is to it. Our feeling is, of course,

that we somehow do the "right" thing, and that they in some subtle sense

ought to be reprimanded, perhaps for not living up to the standards of humanity as a

community of rational animals. But who set these standards? Without doubt, these

standards were set by our social practice, and this again testifies to the fact that ultimately

social practices are all we have to fall back upon.

6. Tentative Conclusion

One way of summarizing the foregoing reflections is to say that the pro-

blem of "logical necessity" is solved by recognizing that there is no such

"logical" necessity, that the coercion or inexorability attributed to logic

and mathematics is rather a social phenomenon. That is, of course, misleading,

because it suggests that logical necessity is something people can take con-

trol of in the way they can take control of industrial production, or that a

parliament can legislate about (cf. "Committee for the Revision of the Laws of

Thought"). Our way of thinking is far too intertwined with our way of life

as to be changeable in any way without unheard of changes in the way people

survive. Wittgenstein, himself, constantly reminds the reader of the practical

implications of inferring, calculating, or continuing sequences in a certain

way. Thus lie says :

For what we call 'counting' is an important part of our life's activities. Counting and calculating are not — e.g. — simply a pastime. Counting (and that means: counting like this) is a technique that is employed daily in the most various operations of our lives. And that is why we learn to count as we do: with endless practice, with merciless exactitude; ...(9)

To say that "logical necessity" is social in nature, however, is correct in

that it indicates that necessity, construed as embodied in a Platonic eidos

or in the "meaning" of symbols is a fetish, i.e., revered as somehow existing

over and above social practices, but thoroughly mystical in nature. To re-

lieve us from our preoccupation with this fetish, to show that "logical

necessity" is a way in which we proceed, that seems to be the intent of

Wittgenstein's remarks on logical and mathematical necessity.

We would like to conclude with a short remark about Wittgenstein's notion

of learning. Many readers of Wittgenstein are offended by his apparent identi-

fication of learning with a very mechanical way of training. When he talks

about teaching a student how to continue a sequence,(10) he talks about it in the

same way in which one would teach an animal in a circus to perform a certain

act. (The German word Wittgenstein uses is "abrichten", and this word is

ordinarily used in connection with animals only.) The fact that Wittgenstein

repeatedly uses such cases to illustrate what one means by "learning something'

seems to indicate that Wittgenstein in general had an extremely one-sided

view of learning, that his conception of learning was very much like that of

a highly authoritarian drill-master whose purpose is not to make people

think, but to make them perform mechanically in a certain way. What we can

see now is that learning qua training is more involved at what we called the

"foundational level," i.e., in connection with learning the natural numbers,

drawing simple conclusions, etc., while learning qua understanding is primarily

involved in operations in which the basic things are already learned. That

Wittgenstein did not simply identify all kinds of learning with "abrichten"

is confirmed, e.g., by reports about his teaching as a grade school teacher

in various villages in Lower Austria. William W. Bartley in his book Wittgenstein writes:

Here in grammar, as in mathematics, Wittgenstein argued that the child should learn the principle of a thing through an interesting, though possibly difficult, specific case; even if other standard examples were easier to learn. There was no point in cluttering up the mind of the child with them, unless he understood and could apply the principle behind them.(11)

Thus, it would surely be wrong to accuse Wittgenstein of an overly narrow view of learning.



(1) Ed. G. H. von Wright, R. Rhees, and G. E. M. Auscombe (Oxford: Basil Blackwell, 1964).

(2) Trans. Francis MacUonald Cornford, 527A.

(3) (New York: Dover Publications, 1952), p. 85.

(4) The Works of Lewis Carroll, ed. Roger L. Green (London: Spring Books, 1965), pp. 1049-1051.

(5) Philosophical Investigations, #185.

(6) Remarks on the Foundations of Mathematics, I, 115.

(7) "Wittgenstein and Logical Compulsion," in: Wittgenstein: The Philosophical Investigations: A Collection of Critical Essays, ed. George Pitcher, pp. 469-476.

(8) Remarks on the Foundations of Mathematics, op. cit., I, 9.

(9) Ibid. ,1,4.

(10) Philosophical Investigations, # 185.

(11) (Philadelphia: Lippincott & Co., 1973), p. 114.

Reprinted from

Back to PapersEtc.