Mathematical Necessity

Tel Aviv University Faculty of Humanities

Mathematical Necessity Seminar on Logic and Necessity

Lecturer: Professor H. Putnam Author: Yoav Aviram Date: July 2004

Logic and Necessity

Table of Contents Table of Contents.................................................................................................................2 Abstract................................................................................................................................3 Rethinking Mathematical Necessity – An Exegesis............................................................7 The Unique Status of Logical Truths..............................................................................8 A Suggestion for a New Distinction................................................................................9 A Relative Notion of Truth.............................................................................................11 Unrevisability of Some a Posteriori Statements............................................................12 Giving Sense to Riddles.................................................................................................13 Speaking of Logic..........................................................................................................14 How a Change of Context Affects Statements...............................................................15 Is There Absolute Necessity?.........................................................................................17 The Need for Justification..............................................................................................18 Discussion..........................................................................................................................20 Revising Absolute a Priori.............................................................................................20 Conclusions .......................................................................................................................25 Works Cited........................................................................................................................27

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Abstract A major concern of analytic philosophy in the last century has been whether or not we can say that there are true statements that are necessary, and if so, on what grounds. It has been argued that the notion of such absolute necessity is a dogma, and that statements that we call a priori are more similar to empirical hypothesis. A reason for this can be given by analogy; since historically some statements considered at one time to be necessary were later revised, it is possible that some future experience will motivate us to revise statements that we consider to be necessary today. But can we really make sense of the notion of revising a statement such as "7+5=12"? Hillary Putnam suggests that even though the notion of absolute necessity is compromised by the argumentation offered against it, there are statements we consider to be necessary relative to our "conceptual schema". Einstein's theory of relativity showed that Euclidian geometry dose not correctly describe physical space. But in the 18 th century statements of Euclidian geometry had an epistemic status quite different than that of empirical hypothesis. By analogy, this may be the case with statements within our present conceptual schema, for example - statements of arithmetic. This essay begins with an historical introduction to the discussion of the analytic synthetic distinction. The historical overview in the introduction is brief, and represents one of several possible narratives. The introduction is followed by an exegesis of Putnam's paper titled "Rethinking Mathematical Necessity". The arguments in that paper are explained and major points are discussed in detail. Views, examples and opinions that are not in the original paper or in other referenced works are remarked as my own. The paper concludes with my own suggestion for an alternative view of the matter, and a conclusion to the entire essay.

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Introduction The discussion about the distinction between analytic and synthetic knowledge has its roots deep within the history of Western philosophy. The very justification of a philosophical discussion might be seen as one that depends on there being an analytic, a priori way of knowing facts about the world. This paper deals with a narrow aspect of this discussion - the status of mathematical knowledge. Understanding the issues at hand requires an understanding of the context of the philosophical debate in which the issues were introduced. In this introduction I hope to briefly explain this context, limiting the discussion to the major views expressed in the Twentieth Century. It is important to note that these views are based on earlier views which are also important, such as the philosophy of Kant, which I will only discuss briefly. Kant explained that truths of logic are analytic a priori and that they are trivial. He explains that logical laws govern relationships between concepts. He also explains that logic is the vary structure of thought. After Kant, the next important milestone was the work that began with Gottlob Frege. Frege elaborated on the ideas of Kant by identifying the logical structure of an ideal language with the nature of thought. The status of mathematical truths, in terms of the analytic-synthetic discussion, was up until this point controversial1. In his works, Frege showed that it is possible to base all mathematical concepts on logical ones. This attempt was later improved in the works of Russell and Whitehead and was called logicism. Showing that the truths of mathematics are truths based on logic alone situated mathematics for the first time clearly within the analytic domain, at least according the common view that considered logic to be analytic. Logicism reinstated the analytic-synthetic discussion in the twentieth century. For the first time it seemed possible to classify all truths as either analytic or synthetic. The logical positivist movement attempted such a classification of truths. Rudolf Carnap explains that "it became possible for the first time to combine the basic tenet of empiricism with a satisfactory explanation of the nature of logic and mathematics" (Carnap 1963). The positivist philosophers attempted to reduce all knowledge to a basis 1

Kant argued that there is a third realm of synthetic a priori truths to which mathematics belonged, while

Mill suggested that mathematical knowledge was empirical. Wittgenstein explained in his early writings that all analytic truths are tautologies, but did not include mathematics amongst them.

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of certainty. For this task they chose to adopt a phenomenological language whose basic sentences were about sense impressions. More complex sentences had to be reducible to these simpler ones, and a statement was considered true if there was a viable method of confirming it. By formalizing language in such a way, the hope was to create a basis for sciences to thrive on. Other than the class of empirical sentences and sentences that were reducible to empirical claims, there was also a class of analytic sentences. The positivists explained that analytic statements were true by virtue of convention or stipulation. As a result of the adoption of logicism, the class of analytic sentences now included all of mathematics. This attempt sharpened the analytic-synthetic distinction creating a large class of analytic statements that were also thought to be a priori, and therefore immune from revision. In 1951 Quine set out to criticize the views of Carnap and the positivists in his paper "Two Dogmas of Empiricism" (Quine 1951). The first dogma of which Quine speaks is that the distinction between analytic and synthetic statements really exists. The second dogma is that every meaningful sentence in our language is structured from logical components that refer to immediate experience. The argument against the first dogma is that the idea of analyticity rests on other notions, such as that of synonymy 2, which are just as unclear as the notion of analyticity, and that any attempt at explaining analyticity using these notions inherits this unclarity. What is implicit in Quine's arguments is that not only is there no meaningful distinction, but that all statements considered analytic are, in fact, synthetic3. Since Quine, like the positivists, identifies in his writings the notion of analyticity with the notion of a priori, the conclusion is that no statement is immune from revision. As for the second dogma, Quine explains that only a holistic theory of meaning is an adequate one. He draws a picture of what he calls our "web of believes"; our knowledge about the world is organized so that some statements take a more central role amongst our beliefs. Only statements in the periphery are statements about immediate 2

Quine refers to a linguistic notion of analyticity: A sentence is analytic if it can be obtained from a logical

truth by substituting synonyms for synonyms. A logical truth is one in which only words of formal logic occur essentially. 3

Quine does agrees to one notion of analytic he calls "stimulus analytic". This is a relativised notion of

analyticity, much like Putnam's notion of quasi-necessity.

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experience and all other statements are logically constructed of peripheral ones. Any statement in a language might be revised, even a central statement of the kind we call logical laws. A revision of such a statement is likely to require us to make extensive modification to our web of beliefs. Similarly, it is possible to hold a statement about immediate experience immune from revision by altering neighboring statements or even more central statements4. Quine's account of the distinction triggered numerous responses. Some followed in his footsteps while others objected to Quine's radical views. In a paper titled "In Defense of a Dogma", Grice and Strawson (1956) argued that Quine's claim that we have not yet clarified the distinction in a rigorous manner is also true of many other distinctions that we use in philosophy and are not willing to reject. Hillary Putnam argues in his philosophy that although there is a lot of sense to Quine's theses, its radical form misses out on some important methodological distinctions. The claim that every statement is revisable does not account well enough to the difference between empirical statements and the statements once considered a priori. In this paper I will attempt to follow Putnam's investigation of this matter, and towards the end introduce some views of my own.

4

This can be done, as Quine suggests, by pleading hallucination.

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Rethinking Mathematical Necessity – An Exegesis The problematic nature of discussing the epistemic status of mathematical truths may be described by the following claim: A good justification we can give for the necessity of mathematical truths is that they are true, without exception. Unfortunately it is this vary claim that situates mathematical truths as something to be justified in the light of use and experience. Similarly, the ontological status of mathematical entities as real existing ones seems to be justified well by the claim that mathematics is simply so successful and useful as a prediction tool, that we do not know how to explain its success other than to say that it deals with real entities. Unfortunately this type of justification also situates mathematics on par with empirical science. Putnam begins the discussion of mathematical necessity in his paper "Rethinking Mathematical Necessity" (Putnam 1994) with an overview of Quine's view. Putnam calls our attention to the type of reasoning Quine employs when he points to the successes of mathematics as a reason for the existence of mathematical entities. This type of argument, Putnam explains, unavoidably reduces mathematical statements to the same level of necessity as that of statements of physics, making the difference between the two one of degree and not of kind. Mathematical statements by Quine's account are subject to revision in the light of experience just the same as statements of physics, except mathematical statements are closer to the center of our "web of beliefs" and thus we are far less likely to revise them. Quine's attack on the views of Carnap and the positivists leaves mathematical knowledge indistinguishable from scientific knowledge from the epistemic point of view. Putnam claims that even though there is a lot of sense to Quine's theses, it misses a critical distinction that exists between the necessities of the two kinds of statements. This distinction exists even if we accept that ultimately no statement is immune from change. Putnam intends to introduce a different way of viewing the situation, one in the light of the views of Kant and Frege. I will now attempt to explain his view in relation to the views of Quine and Carnap.

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The Unique Status of Logical Truths Putnam shares with us some thoughts he had about the writings of Wittgenstein on logic which he claims to have found a little puzzling at first glance. In the T ractatus (Wittgenstein 1922) it seems that for Wittgenstein it is clear that logical truths are without sense, and at the same time are not nonsense5. I think that Wittgenstein spends a large amount of time (relative to his succinct style) on this fine distinction. He explains that "A logical picture of facts is a thought" and that "Thought can never be of anything illogical". Propositions for Wittgenstein express thoughts, and thoughts do not actually contain their sense, but only the possibility of expressing it6. This fine distinction between containment and expression is the line Wittgenstein draws between logic that is present in our thoughts, but which we cannot exactly speak about7. We cannot speak about logic because it does not have a sense, but since logic is thought, we cannot say that it is nonsense. Putnam's puzzlement has to do with this ambivalence in Wittgenstein's attitude towards logic. While trying to clarify Wittgenstein's notion of logic, Putnam is reminded of what Kant has to say about it. In the Critique of Pure Reason (Kant 1787) Kant identifies logic with the structure of rational thought. He introduces the notion that the ways we view the world have something in common that we can investigate. These common features of thought are the only type of objectivity to which Kant is willing to commit 8. Since logic is a feature of human thought, it is incorrect to say that it provides us with any positive descriptive knowledge of the world. This notion of logic differs greatly from the Platonist metaphysical claim that the laws of logic are the most general laws there are. Even though it is less metaphysically ambitious, there is still quite a lot that we can say about Kant's view of logic and thought. No thought is allowed to violate the laws of logic and any thought that does so is considered to be irrational. For Kant any sort of talk, event talk of "noumena", must conform to the laws of logic. This must be so because talk is a 5

This line of thought is still noticeable in Wittgenstein's later works.

6

Tractatus: 3 – 3.13.

7

In an earlier section of the Tractatus Wittgenstein uses a different analogy when he explains that "A

picture cannot, however, depict its pictorial form: it displays it." (2.172) 8

This objectivity rests on certain assumptions that Kant must make, such as the unity of thought and the

existence of god.

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rational human activity. Therefore, there could not possibly be talk of anything illogical, since this sort of talk would not be talk at all but rather meaningless mumbling. Putnam explains that in his reading of Frege he identifies a tension between a Platonist and a Kantian view. On the one hand there are many reasons why one may claim that Frege takes the Platonist approach. It may be said that for Frege the laws of logic are quantifications over all objects and concepts. On the other hand since Frege identifies the structure of thought with the structure of an ideal language, and since the laws of logic govern that ideal language, it seems that he also adopts the Kantian view whereby the laws of logic have a different status than empirical laws. It is now easy to see the similarities between the Kantian view, where there cannot be an illogical thought, and Wittgenstein's claim that statements of logic have no sense but are not nonsense. Interpreting Wittgenstein in this way allows Putnam to adopt an alternative explanation to the necessity of logical statements than the one Quine attacked in "Two Dogmas". Quine directed his attack towards the explanation of logical necessity provided by Carnap. Carnap explained that logical statements are necessary because they are no more than conventions. A change of the logical laws would be nothing more than a change to the meaning of the words. Wittgenstein's view allows Putnam to claim that the necessity of logical statements needs no explanation, nor could we ever provide such an explanation, since they do not describe things in the world but only the way we think. A Suggestion for a New Distinction When Quine rejected the analytic synthetic distinction in "Two Dogmas" he explained that the reason for this was the lack of clarity in the terms that we use to define analyticity. This unclarity in his opinion was so vast that it distracted us from seeing that analytic statements are not epistemologically deferent than empirical hypothesis. One of the main arguments used by Quine in that paper was that we take logically true sentences (sentences in which only the logical words effect the truth value of the sentence), which one may view as a narrow definition of the class of analytic statements, and extend it into a much wider definition of analyticity by substitution of synonymous words. This substitution attempt, Quine argues, is fallacious since the notion of synonymy itself depends on the notion of analyticity.

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We are now faced with two definitions of analyticity. The first is a narrow definition which only includes logical truths and the second is a wider definition which also includes sentences synonymous with logical truths. Putnam points out that Quine's famous argument is only valid when referring to the wider definition. The narrow definition escapes Quine's claws unharmed, but this is still not enough to establish that logical truths are necessary and counter Quine's main theses regarding necessity. Even with the narrow attempt at a definition of analyticity we are left with logical truths that are "contextually a priori"9, a notion that is vary alien to the line of though of Kant and Frege that Putnam is attempting to promote. The notion of revising logical truths is a hard one to swallow, even for Quine himself. After suggesting that the revision is possible, it seems like Quine attempts to defend the special status of logical truths. He gives two arguments; the first is that even though truths of logic are theoretically revisable, they are located at the far end of the continuum of revisability. This continuum has, on its one end, empirical statements of the kind we revise every day and on its other end statements like the truths of logic which we choose to hold come what may. The second argument is that the revision of logical truths only occurs during attempts at translation, and that in those cases it is only a change of meaning and not a revision in a deeper sense10. Putnam's dissatisfaction has to do with how Quine's arguments might be interpreted. Quine describes the space between the synthetic and the analytic as a continuum rather than a clear cut distinction. But it seems that for all intents and purposes he abandons the traditional notion of analytic a priori in favor of classifying all statements as synthetic a posteriori ones. The notion of "reluctance to give up" does not capture well the difference between what makes logical truths true, and what makes empirical hypothesis correct. To illustrate the matter, Putnam asks us to consider three sentences: 1. It is not the case that the Eiffel Tower vanished mysteriously last night and in its place there has appeared a log cabin. 9

The term "contextually a priori" is not the original term used by Putnam. I use it regarding stamens that

we can revise, but do not know how to give sense to their revision. This is similar the class of statements Putnam terms "quasi-necessary relative to a conceptual schema" or "necessary relative to a body of knowledge". I think that the term captures well the problematic nature of such truths. 10

This type of revision is similar to Carnap's change in convention.

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2. It is not the case that the entire interior of the moon consists of Roquefort cheese. 3. For all statements p, '~(p•~p)' is true11. Putnam explains that the difference between the first two statements and the third one does not only have to do with how much one is reluctant to give up each statement. It is a real distinction that has to do with the fact that one knows how statements one and two may turn out to be false, while one does not even understand what it means for the third statement to be false. A Relative Notion of Truth This difference between statements is significant, Putnam explains, not because it shows something new about us, but because it exposes something about the methodology involved. At first glance it seems like the notion that we cannot imagine something, is a psychological fact and that this kind of reasoning is psychologism. But what is crucial for understanding the significance is to understand what is needed for the realization that the third statement is false. And what is needed is a different method of reasoning, an alternative logic. When dealing with quasi-necessary truths, an entire alterative theory must be available before we can give a sense to the revision. In his paper "It Ain't Necessarily So" (Putnam 1975) Putnam gives several great examples of this sort of revision. Euclidian geometry was overthrown as the geometry that describes physical space when Einstein's relativity theory showed that space has attributes that are inconsistent with it. Proving that statements of Euclidian geometry are false may be viewed just like any other falsification of an empirical hypothesis in the light of contradicting observations. Putnam argues that prior to their overthrow truths of Euclidian geometry did not have a status of empirical hypothesis. In order to explain this argument Putnam introduces a relative notion of truth - “true relative to a conceptual schema” A statement true relative to a conceptual schema is necessary according to everything the scientific community knows at the time (in case we are dealing with scientific truths). 11

In his paper "There is at least one a priori truth" (Putnam 1978), Putnam argues that a minimal law of

contradiction, the statement that “Not every statement is both true and not true.” is the most certain a priory truth.

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An empirical statement by this account is a statement that it is known how to falsify, based on the knowledge inherent in that conceptual schema alone. Alternatively, statements for which the knowledge inherent in the conceptual schema is not enough to falsify are statements necessary relative to that conceptual schema. Since these statements might still be falsified by knowledge outside of the conceptual schema (for example future knowledge), they are only quasi-necessary and not absolutely necessary truths. Since Euclidian geometry was the only geometry available during the 18th century, it was a necessary truth relative to the conceptual schema of the time. Only when alternative geometries were developed by Lobachevski and Reimann, and the theory of relativity applied such geometry to physical space, was it possible to understand what it means for space to be non-Euclidian. For example, while Euclidian geometry asserted that space was infinite, Einstein's physics claimed it was not. Similar things might be said regarding the necessity of the law of contradiction, and the possibility of revising it. The motivation behind Putnam's introduction of a relativised notion of truth is not to salvage analyticity or to revive the analytic synthetic discussion. It is to point out an important distinction we can make between different types of knowledge that was eclipsed as a result of the collapse of the analytic-synthetic dichotomy. Unrevisability of Some a Posteriori Statements Wittgenstein narrows the gap between analytic and synthetic statements from a different direction. There are, he explains, statements that we commonly classify as empirical, but whose revision we fail to give sense to. Much like the analytic statements that we dealt with before, which we thought are necessary until history showed to be revisable, these new "quasi-empirical"12 statements we tend to think are revisable but do not know how that revision is possible. It seems that what makes the revision of this type of statements inconceivable is that changing them requires altering some of our fundamental assumptions about the world. Change those assumptions and language looses its orientation. One example of such a case would be for one to wake up one day and discover that his name is not really what he thought it was all his life. We can treat this matter like an 12

The term "quasi-empirical" is not in the original text.

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empirical discovery that one just made about who he really is, and say that he was mistaken about his name. But from a subjective point of view, when such a fundamental assumption about the world is lost, it makes just as much sense to think that the events after the discovery are factitious as it is to think that past experiences are. When we are faced with the discovery that a fundamental scientific assumption we have about the world is false, we are collectively faced with a similar subjective situation. An example of such a situation would be to discover that the statement "water has boiled in the past" is false. Even though this is agreed to be an empirical statement, so much of our other knowledge depends on it being true that the discovery of its falsity causes language to loose its reference point. Such a discovery would require such extensive modifications elsewhere to our web of believes that we would not know what to keep constant and what to change. Putnam explains that what Wittgenstein's examples show is that even though it is true that in a certain sense we can revise any statement (even in a trivial way, by altering the meaning of words), in another sense there are cases where it is not rational to do so. Giving Sense to Riddles When Quine examines the possibility of revising logical truths he limits his discussion to talk of statements and avoids talk of beliefs and meanings because of the problematic nature he attributes to them13. Even from Quine's angle it is possible to see why the statements described above are problematic. Any attempt at a translation of a language which contains notions such as "water has never boiled" or "7+5=13" is not something that we can presently understand. The unique situation that arises when examining such statements is that we cannot say of them that they are unrevisable, but at the same time we are unable to attach a clear sense to there revision. History shows that eventually, a sense is given to the revision of such statements after all. The non-Euclidian statement that "there is only a finite number of places to go to travel as you will" today makes perfect sense. The trick to understanding how it was possible to give such a revision a sense, may be shown by a riddle. What makes a riddle 13

In "Two Dogmas" Quine explains that this problematic nature has to do with the terms meaning,

synonymy and analyticity having interdependent definitions.

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intriguing is exactly that we do not know what sense to give to it. Only after we have arrived at a solution do we understand the sense of the question. If the riddle-question had only one clear sense, it would not be a riddle at all (or at least not an interesting one). Putnam explains that prior to knowing the answer any attempt at a translation of the riddle would be futile since the translator would be unable to preserve its intended sense. Similarly, the sense of revising a contextual a priori truth is something that is only understood after the fact. Speaking of Logic Looking back at the positions of Kant, Frege and early Wittgenstein described earlier, there are several important distinctions to be made. The idea that logical truths are true by virtue of the nature of thought is a metaphysical idea that Putnam would like to jettison. Kant describes not two, but three types of knowledge; synthetic a posteriori, synthetic a priori and analytic a priori truths. While the distinction between a priori and a posteriori truths is clear, the distinction between the status of analytic and synthetic a priori is a more subtle one. Synthetic a priori truths for Kant are true because they describe the structure of reason. They do not say something about the world, but rather what they say is that any experience is "filtered" according to the innate conceptual structures of reason common to us all. Analytic truths on the other hand are true because they are a result of the nature of thought. The difference between structure of reason and nature of thought is an important one for Kant. Synthetic a priori truths are truths that we can rationalize about, for example by explaining why they are necessary or by thinking of a world where they do not exist. We cannot rationalize, negate or explain analytic truths precisely because they are reason. For Wittgenstein the opposition between synthetic a priori and analytic truths is no longer a concern. In his view the opposition is between a posteriori (empirical) truths and a priori truths. Wittgenstein follows Frege's footsteps in identifying the nature of thought with the structure of an ideal language. The resulting view is that truths are either logical truths, now identified with the structure of an ideal language, or they are about the world. In the first case their revision is without a sense, just as they are unthinkable for Kant. In

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the second case the revision of judgments about the world is thinkable and subject to confirmation. What Putnam hopes to retain from the Kant-Frege-early Wittgenstein idea is not the metaphysical principle that guarantees the unrevisability of logic due to it being the nature of thought. What he hopes to retain is the notion that a revision of logic is something that we cannot give sense to in principle at the time of the revision. It is not the case that logical truths are absolutely unrevisable or absolute a priori. But it is the case that they hold quite a different status than empirical hypothesis. How a Change of Context Affects Statements Before we move on to consider how the discussion we had so far influences arithmetic, Putnam offers some important clarifications. In the previous paragraphs we came to the conclusion that there is a type of questions that we can only give sense to after we have the answer at hand. Putnam points out that what we mean when we talk about giving question a sense is something much broader then to simply understand what the person who asked it meant. Understanding sense in this way is something much closer to understanding the full context of the answer. Since I think that this point is crucial for the debate, I will try to clarify it further by examining the way the Chinese philosophy dealt with the similar questions. In Chinese philosophy it is said that a question that Western philosophy considers to be a yes/no question may be answered by a third alternative. When a question is answered by the term 'Mu'14 it is meant that the context in which the question was asked is too narrow to contain the answer. This is a way of explaining that the fact that the question was asked indicates a lack of fundamental knowledge. There is a story of a student who asked his master: "It is said that the nature of the Buddha is present in all things. In what way is it present in a dog?" - To which the master replied 'Mu'. Taking the case of Chinese philosophy as an analogy, I think that it illustrates well the point that there are some questions that are asked at a time when the context is too narrow to contain the answer. 14

The Chinese term 'Mu' has several philosophical uses. The one I am referring two is taken from Zen

Buddhism and is, naturally, subject to interpretations. The particular interpretation brought here is the popular one.

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After the context is broadened, usually by scientific development, we are able to understand words in both the new and old contexts. This is the case with the terms of physics after Einstein's theory succeeded Newton's. Putnam points out that when we shift from an old context to a new one, we notice that terms change their reference (a change in extension). We thought that momentum is the product of mass and velocity and today we know that this is not the case. Rather than saying that only the meaning of the word momentum has changed, we say that the old theory was wrong and that we have discovered something genuinely new. I think that this indicates a broader change than a change of reference; it indicates a change in intension. My opinion on how a change of context is related to a change of meaning differs from Putnam's. In his paper "It Ain't Necessarily So"15 he asks what we would say about the statement "all cats are animals" if one day we were to discover that all cats really are, and always were, automata. Putnam point out that the problem in this case is not with the use of the word cat (we will keep calling them cats) as much as it is with the claim that cats are animals. He explains that in this case to say that "cats turned out not to be animals" would keep the meaning of the words 'cat' and 'animal' unchanged. I think that Putnam is correct if we understand meaning as extension. But if we understand meaning as intension, than both the meaning of 'cat', and to a lesser degree 'animal' has changed. Intension, taken as a term that captures the sum of all the different uses of a word, is capable, in my opinion, of explaining how a change in context affects statements. Another clarification Putnam suggests has to do with the status of contradictions. It was noted earlier that for Frege and Wittgenstein who identified logic with the structure of an ideal language, a thought about contradiction is not really a thought at all. Putnam explains that a thought that is a contradiction, such as thinking that '(p•~p)' is what those philosophers considered to be a thought without a sense. That does not mean that we cannot make use of a contradiction in order to express something else, such as the truth value 'false' with in a sentence. We may be unable to understand a sentence even if it is "well formed" if present theories prevents us from doing so.

15

The original example is taken from Donnellan (1962).

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Is There Absolute Necessity? At least one major question still seems to stand despite all that was discussed so far; whether we can determine if a proposition of the kind that we have talked about is only quasi-necessary or if it is truly necessary. Putnam argues that that is simply not a good question. The reason for this is that the question assumes a point of view external to human knowledge from which the matter can be judged. Such an objective point of view does not exist. We can not give sense of the demand to justify or explain a contextual a priori statement because we accept those statements since we do not doubt them. The absolute-necessity or quasi-necessity of a proposition is not something to be discovered about the world but a fact about us. In order to revise a proposition of the type discussed so far, there must be an alternative, confirmable theory present. Since the invention of such an alternative theory is something that is entirely a human activity, we are the ones who determine whether a proposition will ultimately be revised or not. If we fail to conceive some alternative theory, and as a result never revise some propositions, can it truly be said that those propositions are objectively necessary? It cannot, and that is why any attempt to claim that a proposition is either absolutely necessary, or conversely that all propositions are revisable, is a dogma. There is a serious objection to consider to what was described in the last paragraph. We have concluded that the necessity of a proposition is dependent on the way in which human knowledge develops and not on some metaphysical principle. The problem arises when we consider Gödel's incompleteness theorems. What those theorems say is that all logical systems of any complexity (including arithmetic) are, by definition, incomplete; each of them contains, at any given time, more true statements than it can possibly prove according to its own defining set of rules. What makes these statements true has to be independent of human knowledge, since they cannot be proven from within our arithmetic system. These are arithmetical truths which are not conceptual truths. Does that not mean that the necessity of the truth of these statements is independent of human epistemology? To this claim Putnam replies that the fact that there are currently undecided mathematical propositions that are true does not carry any metaphysical weight. Bivalence is not a principle according to which we may draw metaphysical conclusions.

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The Need for Justification It is time to consider what we are left with at the end of the discussion. Regarding Quine’s theses in “Two Dogmas”, no objections were made against the main point of that essay; the concept of analyticity, in the sense it had for Carnap, ought to be disposed of. In other words, the attempt to base knowledge on a priori grounds has not withstood the test of time. Instead, a new distinction was offered between empirical statements and “contextually a priori” statements. The intention was not to identify this new distinction with a new metaphysical category of truths but to describe how statements face historical changes. This conclusion has ramifications for the idea that the existence of mathematical entities needs justification. There are different senses in which we can speak of the existence of mathematical objects. When we do mathematics we often come across statements such as: “There exist prime numbers greater then a million” or even “Numbers exist”. In his paper “On What There is” (Quine 1948) Quine explains that when a speaker states that “something exists”, he is ontologically committed to its existence and that in the case of mathematics objects are "intangible". Even if we adhere to the idea that we are committed to the existence of numbers as a result of doing mathematics, just as we were unable to give sense to the notion that arithmetic is wrong, so is there no sense in giving a reason why arithmetic is right. Arithmetic needs no justifications. In empirical sciences there is a strong connection between the usefulness of a theory and its truth; this is not the case in arithmetic. For example, the existence of electrons is a theory of physics conceived based on numerous observations. The fact that this theory has been able to produce accurate predictions and explain otherwise unexplained observations consistently, persuades us to except that there really are electrons. In mathematics on the other hand, it is possible for a theory to be completely useless and still be inarguably correct. The difference between empirical sciences and mathematics is quite large. The existence of mathematical objects has nothing to do with how extensively the theory is applied. It may never be applied and still this would not change a thing concerning the discussion we have about the existence of mathematical objects. The

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usefulness of a mathematical theory is an empirical issue completely separate from the unempirical question –“is the theory correct?” Quine's battle against the a priori has ended in his claim that the base of all knowledge is empirical. This attempt at "naturalizing epistemology" is problematic due to the construct of our language. Notions such as sense, truth and confirmation – in other words the majority of the terms used when the issue is debated - are normative ones. The sense of the attempt to naturalize these terms is unclear. Putnam suggests that there is a better way of describing the role of arithmetic in scientific discovery than to naturalize it, a way that does not depend on the notion of a priori. Arithmetic is the tool with which we justify; in itself it does not need justification.

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Discussion In the following section I would like to discuss some of my own thoughts on the issues expressed in this seminar. These thoughts are concerned with the general discussion about logic and necessity and not particularly with the subject of mathematical necessity. The main works that I reference are Putnam's "Rethinking Mathematical Necessity" and "It' Ain't Necessarily So", and Quine's "Two Dogmas". Revising Absolute a Priori In the aftermath of Quine's "Two Dogmas" several views surfaced in an attempt to salvage something of the notion of a priori. The reason for these attempts, the same reason described earlier in this essay, is that it seems that Quine's account does not accurately describe some of our every day experiences with knowledge, although it is convincing in most other areas. Perhaps the most objectionable of Quine's statements is the one claiming that every statement is revisable. What differentiates between statements of physics and statements of logic, he explains, is their different location within our web of believes. Putnam agrees that the old notion of absolute a priori has crumbled, and offers us a rehabilitated notion of necessity; one which he calls quasi-necessary (and I call "contextual a priori"). For this he relies heavily on the study of events from the history of science. The concept of contextual a priori is an attempt to explain the phenomenon whereby statements that are considered necessary at one time are later revised. In this section I will attempt to provide a different explanation for this phenomenon. What I wish to explain is how statements change the degree of necessity attributed to them by the scientific community over the course of time. This explanation may be used to clarify how the apparent revision of statements that are considered absolute-a priori is possible. However, I do not intend for this it to be a justification of the belief in the existence in such a priori. This explanation may also be applied towards a relativised notion of a priori of the kind Putnam promotes. I agree with Putnam that the question whether there is an absolute notion of truth or only a relativised one is unanswerable. This is so because we will never be able to determine if some truths remain constant

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because they are absolutely necessary or because we have not discovered an alternative yet16. Any attempt to give an answer to this question assumes an external point of view. As a case study I wish to examine how Einstein's physics revised notions regarded as necessary by Newtonian physics. For example, the notion of absolute simultaneity - that the statement "either two events occur at the same time or they do not" is true. This is a good notion to examine since even today laymen’s intuition is inconsistent with the way such basic concepts are presented by the theory of general relativity. Let us briefly describe a thought experiment17; in the diagram bellow the long arrow represents a train traveling at velocity v in the direction of the arrow and the dashed line represents the railway embankment. For people traveling aboard, the train is a frame of reference - they regard all events in reference to the train. Every event which takes place along the embankment also takes place at a particular point of the train. The definition of simultaneity can be given relative to the train just as it can be given in respect to the embankment.

V

M'

A

M

Train

B

Embankment

The following question arises: if two strokes of lightning emanating from points A and B are simultaneous with reference to the railway embankment, are they also simultaneous relatively to the train? Before the time of the theory of relativity it had always been assumed in physics that a statement of time had an absolute significance, i.e. that it is independent of the state of motion of the body of reference. But we can define simultaneity in this case by saying that the two lightning strokes occur at the same time if they both reach point M, located in the middle of the distance AB, at the same time (since they both travel at the speed of light).

16

A reason for not discovering such an alternative maybe that we are incapable of conceiving it.

17

This thought experiment was originally described by Einstein (1920).

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Events A and B also correspond to positions A and B on the train. When the flashes of lightning occur, point M' coincides with point M, but it moves towards the right in the diagram with the velocity v of the train. An observer sitting in the position M’ in the train is traveling towards the beam of light coming from B and away form the beam coming from A. If the train is traveling vary fast, the observer will see the beam of light emitted from B earlier than he will see that emitted from A. Observers who take the railway train as their frame of reference must therefore come to the conclusion that the lightning flash at B took place earlier than the lightning flash at A. The conclusion is that events which are simultaneous with reference to the embankment are not simultaneous with respect to the train, and vice versa. Every reference frame has its own particular time. Unless we are told the frame of reference to which the statement of time refers, there is no meaning to a statement about the time of an event. Let us now discuss how the theory of general relativity was developed by Einstein. Historically, it is said that the first time that Einstein's theory met with empirical confirmation was at an experiment conducted by Sir Arthur Eddington at 1919. Critics of that experiment claim that an empirical conformation of Einstein's theories was only achieved at a later time. What I would like to point out is that before 1919 there was no empirical confirmation for relativity. The significance of this fact is that it shows that the methods used by Einstein when developing the theory of relativity must not have been empirical methods18. It is reasonable to say that the development of the theory of relativity was done mainly by mathematical means. Sure, there were empirical observations involved and, as a consequence of those observations, inconsistencies in Newtonian physics were discovered. These inconsistencies were probably a major reason why an alternative theory was needed. But we know that those observations alone do not account for the development of the body of knowledge we call the theory of relativity. We know this because it would not have been an issue to empirically confirming the theory of relativity, even when it was first conceived, if its development was a result of empirical observations. 18

Note that I use the terms empirical and non-empirical here as if a clear distinction exists between the two.

For now my intention is only to point out our linguistic behavior. I will return to this issue shortly.

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One more point for clarification: We have already suggested that the truths of logic escape Quine's criticism. This is so because Quine's criticism is directed at the notion of synonymy when it is employed to expand the class of logically true sentences into other forms of analyticity. Logically true sentences19 are not dependant on any of the terms Quine showed to be unclear. In the description I gave so far of how the development of the theory of relativity caused the revision of statements considered to be a priori until then, I have used the terms empirical and a priori as though there is a clear distinction between them. I would now like to clarify my intentions. Suppose that there really is a distinction between a priori and a posteriori knowledge. Suppose also that a priori knowledge consists in its core of the truths of logic that have escaped Quine's judgments. We can also suppose that a priori knowledge consists of the entire class of mathematical statements in addition to statements of pure logic, but this is not essential to the point. At some point in history, someone arranged the class of statement considered a priori (according to the definition we have just supposed) by some theory he conceived. How can we explain that later on a new theory was devised and that class of a priori statement changed? A simple, almost trivial, explanation is human error. During the course of time, a person reviewing the notes left by past scholars may discover that they were wrong, perhaps by employing some deductive method in an incorrect way. This might seem like a weak explanation for the revision of a priori statements when the class of statements is small. But the larger we allow the class of a priori statements to be the more sense it makes. When that class contains the whole of mathematics, a revision of a priori statements due to human error is a common event. The class of a priori statements changes over the course of time because we continually correct mistakes in it. If what Einstein did when he developed the theory of relativity was to review the mathematical formulas governing Newtonian physics, correct and further develop them, sometimes by devising whole new mathematical methods, then in a sense what he did was to correct Newton's mistakes. The result of these mistakes was that some statements, such as the one about absolute simultaneity, were included in the class of a priori statements by mistake. When Lobachevski and Reimann developed alternative 19

For a definition of logical truth see footnote 2.

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geometries, they did not do so in order to correct a mistake in Euclidian geometry20. But the development of alternative geometries alone was not enough to demonstrate how space is non-Euclidean. That was only accomplished by the theory of relativity. Can this explanation really account for the claim that every statement is revisable? I think that the best we can say is that it does a good job at describing our struggle with knowledge. It gives a reason for why we revise knowledge of any type, empirical or other. The difference between this explanation and the other ones discussed in this essay is that it allows us to keep our belief in the existence of a priori truths (relative or not) and still explain the history of science. One problematic aspect of this explanation is that calling the theory of Newtonian physics a mistake hardly does it justice. Perhaps the word mistake (or error) is not the right one to use in this context. But perhaps the gravity of the situation has to do with the immense changes that have to occur to the scientific body of knowledge before someone can correct these "mistakes". Maybe we have talked about revisions for so long that we have forgotten that we are the ones making them. And the reason we usually have in mind for doing so when we make them is that we do so to correct a mistake, no matter what type of knowledge we are dealing with. It is possible that the truths of logic are the most basic laws there are. It is also possible that the laws of logic describe the nature of thought. But as any student of arithmetic or logic would testify, this does not prevent us from making mistakes when we apply them.

20

When Lobachevski first published his theories he called his work "imaginary geometry".

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Conclusions Quine introduced the notion of naturalizing epistemology in a paper called "Epistemology Naturalized" (Quine 1969). He did so by both showing that traditional epistemology has failed and arguing that natural science can succeed in its place. Quine thinks of traditional epistemology as being a doctrinal project concerned with identifying the foundation and deducting from it beliefs about the physical world. Quine also refers to the conceptual project of Carnap and the logical positivists, which is concerned with providing definitions for translating talk about physical bodies into talk about sense impressions. Quine thinks that not only did Carnap fail; it is in principle impossible to succeed (to prove this he describes a thought experiment about translation of the word "gavagai" to "rabbit"). Quine asserts that the quest for certainty will not succeed. The alternative he provides is for us to study the relation between sense impressions and theories about the world. He feels that we should reject both the need for justification and the quest for certainty. Rather we must study scientifically the natural phenomenon in the human brain. In essence, he has removed normativity from epistemology. In "Rethinking Mathematical Necessity" Putnam argues that Quine's account of knowledge implicitly situates mathematics as empirical and as a result misses out on some important distinctions. First, Quine's denial of a priori truths mistakenly classifies all statements as empirical. The study of arithmetical statements shows that there is an important methodological distinction between contextual a priori truths and empirical truths. This distinction is evident throughout the entire history of scientific discovery. The new distinction Putnam offers is not meant to be a new metaphysical device, a second realm of truths. Instead the distinction offers a better description of scientific behavior. The second point Putnam criticizes in Quine is the notion that the attempt at justification is useless. The criticism goes much further than to say that we do not know what sense to attach to the notion that logical truths are wrong. The criticism is that our language (and in a deeper sense the way we think) cannot account for a revision of logical truths. Normative notions dominate the way we speak about epistemology and those notions cannot be naturalized, reduced to empirical means. Borrowing from Kant, Frege and the early Wittgenstein Putnam argues that mathematics escapes the need for

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justification. Again, not meant as a metaphysical statement, but as a statement that better describes our relation with knowledge. From a different prospective, Putnam points out that a verificationist theory of meaning fails at places where a theory of meaning as uses succeeds. The first reason for this is that a verificationist theory of meaning fails to apply to many scientifically decidable truths where verification is impossible due to technical reasons. We should not be forced to give up scientific knowledge we can justify but cannot verify. Another reason is that in some cases we do not grasp the meaning of certain statements until there is a change in use. This was the case with the statements like "there are finitely many places to go to, travel as you will" before Euclidian geometry was overthrown. Only when this statement was actually used in scientific discourse we were able to give it sense. Realism cannot be settled with a verificationist point of view; that was the problem with the positivist project. But if we naturalize epistemology we miss interpret the history of science; that is the problem with Quine's view.

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Works Cited 1. Carnap, R. (1963), The philosophy of Rudolf Carnap, P. A. Schilpp (ed.), pp. 4467; 868-877; 915-922. 2. Donnellan, K. (1962), "Necessity and Criteria", the Journal of Philosophy, LIX, No. 2, pp. 647—58. 3. Einstein, A. (1920) "The Relativity of Simultaneity", Relativity: The Special and General Theory. New York: Henry Holt, pp. 168; Bartleby.com, 2000. http://www.bartleby.com/173/9.html. 4. Grice, H.P. and Strawson P.F. (1956), “In defense of a dogma”, Philosophical Review, AP 1956; 65 5. Kant, Immanuel (1787), Critique of pure reason, translated and edited by Paul Guyer, Allen W. Wood, Cambridge University Press, 1998 6. Putnam, H. (1975), “It ain’t necessarily so”, Mathematics, Matter and Method (Philosophical papers, vol. 1), Cambridge University Press. 7. Putnam, H. "There is at least one a priori truth", orig. published in 1978; reprinted in Putnam's Realism and Reason, Philosophical Papers vol. 3 (Cambridge Univ. Press, 1983) 8. Putnam, H. (1994), “Rethinking mathematical necessity”, Words and life, J. Conant (ed.), Harvard University Press. 9. Quine, W.V. (1948) “On What There Is”, The Review of Metaphysics 2, pp. 21-28. Reprint in many places including Quine, From a Logical Point of View 2nd ed. (Cambridge: Harvard University Press, 1980). 10. Quine, W.V. (1951), “Two dogmas of empiricism”, Philosophical Review JA 1951; 60 11. Quine, W.V. (1969), "Epistemology Naturalized" reprinted in Naturalising Epistemology, ed. H. Kornblith, 1985. Cambridge MIT press pp. 23-24. 12. Wittgenstein L. (1922), Tractatus Logico-Philosophicus, translated by D.F. Pears and B.F. McGuinness (Routledge and Kegan Paul, London 1961).

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