Frege in Context

In anybody’s short list of significant philosophers and mathematicians of the nineteenth and twentieth centuries, Gottlob Frege will appear near the top. His direct and indirect influence extends to many philosophers and groups of philosophers, including those who were influenced by him in the sense of reacting against him.

Although Frege’s work may be characterized as late modern or early contemporary, he nonetheless has a historical context. His seminal work symbolic logic and the foundations of mathematics might mislead us into thinking that he does not have a connection to great thinkers of earlier centuries. We will see, however, that to understand Frege in his historical context, we must view him in terms of John Locke and John Stuart Mill.

In his Essay Concerning Human Understanding, Locke address, among many other topics, language and reference. Locke asserts that the only possible referent of a word is an idea in the mind of the speaker (or writer). But, he writes, men mistakenly assume that their words can also refer to two other things: the ideas in the minds of others and the (physical) objects themselves. Leaving aside the latter comment about objects, the former comment about ideas in the minds of other leads us to considerations about reference.

The radical skeptic can issue all manner of objections to talk about ideas in the minds of others, let alone words referring to such ideas. But even the less-than-radical skeptic, one who is sympathetic to such a scheme of reference, will have concerns: Locke is such a one, and he raises such concerns. He imagines that it’s acceptable to posit that the speaker’s words may refer to ideas in the minds of others. But even if this is allowed, further problems remain.

If the speaker’s words can simultaneously refer to ideas in the speaker’s mind and idea in the listener’s mind, how can we know that any one word will refer to the same idea in the speaker’s mind and the listener’s mind? If the speaker says the word ‘x’ and it refers to the idea x in the speaker’s mind, how do we know that it also refers to the idea x in the listener's mind, and not, e.g., to the idea y?

This question gives rise to Locke’s famous inverted spectrum example, and also to later examples like those given by Quine in his book Word and Object. These, and other types of qualia inversion problems, isolate the question of whether a word can refer reliably to an idea in more than one mind. Locke writes:

Words are often secretly referred first to the ideas supposed to be in other men’s minds. But though words, as they are used by men, can properly and immediately signify nothing but the ideas that are in the mind of the speaker; yet they in their thoughts give them a secret reference to two other things.

Although acknowledging the problem of inverted qualia - indeed, originating the problem - Locke also seems to be content to rely on experience, to the extent that for the practical purpose of daily life, language suffices. Whether or not the qualia be actually inverted - Locke points out that we could never know this - language would work as long as the inversion were consistent. He continues:

First, they suppose their words to be marks of the ideas in the minds also of other men, with whom they communicate: for else they should talk in vain, and could not be understood, if the sounds they applied to one idea were such as by the hearer were applied to another, which is to speak two languages. But in this men stand not usually to examine, whether the idea they, and those they discourse with have in their minds be the same: but think it enough that they use the word, as they imagine, in the common acceptation of that language; in which they suppose that the idea they make it a sign of is precisely the same to which the understanding men of that country apply that name.

Finally, Locke returns to his initial assertion, that words refer properly only to the ideas in the mind of the speaker. The fact that we do, in practical daily life, act as if words also referred to ideas in the minds of listeners, or to physical objects themselves, is something other than reference properly construed. Locke therefore distinguishes between reference and what he calls the ‘common acceptation’ of an utterance (or of a writing). Ian Hacking writes:

Does Locke support this doctrine of ‘secrete reference’? I think the very phrase is loaded with Locke’s characteristic irony. ‘Give me leave here to say’, he continues in case anyone has failed to take the point, ‘it is a perverting the use of words, and brings unavoidable obscurity and confusion into their signification, whenever we make them stand for anything but those ideas we have in our own minds.’ Signifying, remember, is a relation of precedence-or-consequence. So what Locke says seems correct. Note that another term is used in this passage, ‘the common acceptation’ of a word in a language. This is something quite different from signifying. Let us take, for example, the role of a name in some speech. Had the real Mark Antony said, ‘I have come to bury Caesar, not to praise him’, there would have been the idea of Caesar present in Mark Antony’s mind. That is what the name ‘Caesar’ signifies, for Antony. The third plebeian, despite the inane responses Shakespeare puts in his mouth, doubtless also has some idea (possibly not Antony’s) of Caesar. That is what the name signifies for him. In contrast, there is the actual person referred to, recently deceased. Finally there is, perhaps, something else in the public domain: everyone realizes that Rome’s tyrant is under discussion. This shared cognition we might, borrowing Locke’s phrase, call the common acceptation of the name ‘Caesar’. Common acceptation enables Antony to address the multitude. It is whatever is ‘public’ about the established use of a word.

With this Lockean framework, the work of Gottlob Frege is seen in context. One of Frege’s many accomplishments was to express more specifically and more technically that at which Locke had only hinted, or expressed with less exactitude. Ian Hacking continues:

The distinctions required were not made efficiently until the late nineteenth century, when Gottlob Frege had to legislate a distinction into the German language in order to avoid confusion. He took the word Sinn - which has been translated ‘meaning’ but in English is nowadays called ‘sense’ - as common acceptation. By way of contrast he used the word Bedeutung - equally translated ‘meaning’ but now fixed with the English word ‘reference’.

In of the famous passages which earned Frege his place in the history of philosophy, he introduces his distinction between sense and reference - the distinction between Sinn and Bedeutung.

The regular connexion between a sign, its sense, and its reference is of such a kind that to the sign there corresponds a definite sense and to that in turn a definite reference, while to a given reference (an object) there does not belong only a single sign. The same sense has different expressions in different languages or even in the same language. To be sure, exceptions to this regular behaviour occur. To every expression belonging to a complete totality of signs, there should certainly correspond a definite sense; but natural languages often do not satisfy this condition, and one must be content if the same word has the same sense in the same context. It may perhaps be granted that every grammatically well-formed expression representing a proper name always has a sense. But this is not to say that to the sense there also corresponds a reference. The words ‘the celestial body most distant from the Earth’ have a sense, but it is very doubtful if they also have a reference. The expression ‘the least rapidly convergent series’ has a sense; but it is known to have no reference, since for every given convergent series, another convergent, but less rapidly convergent, series can be found. In grasping a sense, one is not certainly assured of a reference.

Here arises an interpretive question in the history of philosophy. Shall we spin the credit in Locke’s favor or in Frege’s favor? If we tilt it toward Locke, we might say that Locke conceptualized and expressed an important distinction about language, and that Frege merely tidied up the vocabulary used to express it. If we tilt it toward Frege, we might say that Locke was only vaguely aware this distinction, and that Frege uncovered and discovered and clearly articulated it. Ian Hacking writes:

Frege also considers something analogous to what Locke would have called the idea signified by a sign. Frege speaks of the ‘idea’ associated with a word, in contrast to the word’s sense and reference. There has been much philosophy between Locke and Frege, and the word ‘idea’ has not stayed in its place, especially when translated into German as Vorstellung and back again. But enough has been preserved to take the continuation of this translated passage as read:

Hacking then goes on to cite this passage from Frege. Over the years, competing translations of Frege’s works have been published, and the translation of some of his central vocabulary items has become a question. For example, Sinn and Bedeutung have been rendered variously as sense, reference, referent, meaning, and significance. Thus the question of which translation one is reading becomes important. In Max Black’s translation, the text reads:

The reference and sense of a sign are to be distinguished from the associated idea. If the reference of a sign is an object perceivable by the senses, my idea of it is an internal image, arising from memories of sense impressions which I have had and acts, both internal and external, which I have performed. Such an idea is often saturated with feeling; the clarity of its separate parts varies and oscillates. The same sense is not always connected, even in the same man, with the same idea. The idea is subjective: one man’s idea is not that of another. There result, as a matter of course, a variety of differences in the ideas associated with the same sense. A painter, a horseman, and a zoologist will probably connect different ideas with the name ‘Bucephalus’. This constitutes an essential distinction between the idea and the sign's sense, which may be the common property of many and therefore is not a part of a mode of the individual mind For one can hardly deny that mankind has a common store of thoughts which is transmitted from one generation to another.

With Locke, Frege is content to leave the word ‘idea’ somewhat ambiguous. According to Frege, it can include sense-impressions, acts, memories of sense-impressions, memories of acts, and even objects of perception:

We can include with ideas the direct experiences in which sense-impressions and acts themselves take the place of the traces which they have left in the mind. The distinction is unimportant for our purpose, especially since memories of sense-impressions and acts always help to complete the perceptual image. One can also understand direct experience as including any object, in so far as it is sensibly perceptible or spatial.

Yet Frege is clear that ‘sense’ must be distinct from ‘idea’ - even if ‘idea’ is left a bit vague. What is here rendered as ‘idea’ is Vorstellung which had served as a piece of Kantian jargon, and its appearance raises questions about to which extent Frege might be importing Kant’s psychology. For Frege, Kant’s metaphysics might be largely irrelevant, because Frege seems more concerned with logic and language. Frege's word Gedanke is here rendered as 'thought' and would become the topic of another famous essay by Frege. In any case, Frege's main occupations were Sinn and Bedeutung, and ‘idea’ and ‘thought’ received less of his attention. He is clear that Sinn and ‘idea’ were quite distinct:

Hence it is inadvisable to use the word ‘idea’ to designate something so basically different.

The foundational role which Frege played in the beginning of contemporary philosophy consisted in his emphasis upon language and logic, and in his task of clarifying mechanisms by which both work. To be sure, Frege was a mathematician, and much of his work was with an eye toward mathematical uses of language and logic. Yet the philosophical implications of his own work did not escape him. The Fregean flavor of analytic philosophy permeates its various subdivisions. Milton Munitz writes:

The development of analytic philosophy is a central and important aspect of twentieth-century thought. The term ‘analytic philosophy’, as we have remarked, does not designate a single, tightly organized, unified movement whose adherents subscribe to a well-defined body of commonly shared principles. There is much diversity - sometimes sharp antagonism of thought - among the various subgroups or individual thinkers collected under this heading.

In the diversity of contemporary analytic philosophy, Milton Munitz argues, the unifying thread is Frege. Frege is the point of departure: either one agrees with Frege and extends his work, or one disagrees with Frege and argues for some alternative position, or one explores some topic which has come to attention because of Frege.

Yet despite this diversity, the utility of using the label ‘analytic philosophy’ derives from two principle considerations. First, there is enough (even if not total) unity to warrant the use of a common classificatory label for those philosophers whose broad orientation is ‘analytic’ in contradistinction to those falling outside this group. Second, and perhaps more to the point, the movement we are calling ‘analytic philosophy’ has special links to the germinal ideas of Frege. The multiple interconnections and influences among analytic philosophers are illuminated when seen as so many diverse offshoots of Frege’s work. This is not to say that all analytic philosophers are 'Fregeans' in one form or another. This is emphatically not so. Many, to be sure, were directly influenced by Frege. They undertook to assimilate, interpret, and adapt what they found of value in Frege’s work in their own thought. But even among these some found points on which they diverged from or criticized Frege. (Frege himself, for that matter, did not have a monolithic, unchanged set of doctrines to which he adhered.) Nevertheless, within the analytic movement there is by and large a direct line from Frege to other thinkers in which we can note a carrying out of an interpretation or amplification of Frege's themes, doctrinal emphases, and philosophic viewpoint. On the other hand, as we trace the unfolding of analytic philosophy, there are other component movements or individual thinkers whose characteristic theses and claims are best seen as reactions to those who had a more direct or sympathetic relation to Frege’s views. Here the picture gets more complicated as we seek to follow the multiple crisscrossing interconnections. Suffice it to say for the moment that even the latter post-Fregean analytic movements are better understood when seen in terms of their place in relation to the Fregean matrix and starting point. I shall not now attempt to work out the relevant details of this observation, but will do so later at appropriate stages of our exposition. Yet it will be helpful to illustrate, in a preliminary way, the sort of thing I have mind.

Although Frege may be the foundation for contemporary analytic philosophy, he did not arise ex nihilo, as we have seen: he arises, in part, in a Lockean context. Saul Kripke notes that Frege also arises in the setting of John Stuart Mill’s thought about proper names:

Now, what is the relation between names and descriptions? There is a well known doctrine of John Stuart Mill, in his book A System of Logic, that names have denotation but no connotation. To use one of his examples, when we use the name ‘Dartmouth’ to describe a certain locality in England, it may be so called because it lies at the mouth of the Dart. But even, he says, had the Dart (that’s a river) changed its course so that Dartmouth no longer lay at the mouth of the Dart, we could still with propriety call this place ‘Dartmouth’, even though the name may suggest that it lies at the mouth of the Dart. Changing Mill’s terminology, perhaps we should say that a name such as ‘Dartmouth’ does have a ‘connotation’ to some people, namely, it does connote (not to me - I never thought of this) that any place called ‘Dartmouth’ lies at the mouth of the Dart. But then in some way it doesn’t have a ‘sense’. At least, it is not part of the meaning of the name ‘Dartmouth’ that the town so named lies at the mouth of the Dart. Someone who said that Dartmouth did not lie at the Dart’s mouth would not contradict himself.

Even as contemporary philosophers who are staunchly anti-Fregean are nonetheless working in a Fregean context, so also Frege himself was working to some extent in Mill's context. Frege argues directly against Mill. Saul Kripke writes:

But the classical tradition of modern logic has gone very strongly against Mill’s view. Frege and Russell both thought, and seemed to arrive at these conclusions independently of each other, that Mill was wrong in a very strong sense: really a proper name, properly used, simply was a definite description abbreviated or disguised. Frege specifically said that such a description gave the sense of the name.

Mill’s view of proper nouns was that they were directly linked to their referents: that they referred directly, not by means of a Fregean sense. Saul Kripke will side with Mill against Frege on this question:

The modern logical tradition, as represented by Frege and Russell, seems to hold that Mill was wrong about singular names, but right about general names. More recent philosophy has followed suit, except that, in the case of both proper names and natural kind terms, it often replaces the notion of defining properties by that of a cluster of properties, only some of which need to be satisfied in each particular case.

While Saul Kripke embraces Mill’s view of proper nouns, he rejects Mill’s view of common nouns - or he rejects what he takes to be Mill’s view of common nouns. Some scholars question whether Kripke understood Mill properly on this point. Stephen Schwartz writes:

Saul Kripke in his revolutionary and influential series of lectures from the early 1970s (later published as the book Naming and Necessity) famously resurrected John Stuart Mill’s theory of proper names. Kripke at the same time rejected Mill’s theory of general terms. According to Kripke, many natural kind terms do not fit Mill’s account of general terms and are closer to proper names. Unfortunately, Kripke and his followers ignored key passages in Mill’s A System of Logic in which Mill enunciates a sophisticated and detailed theory of natural kind terms that anticipates and is in some ways superior to Kripke’s.

Frege’s notion of Sinn was originally formulated as the way in which reference was made to an object. His famous example was the planet Venus, which is known both as ‘the morning star’ and as ‘the evening star’ - the same referent with two different senses. Thus, for Frege, Sinn was determined by a criterion: how one referred to an object. Later Fregeans would formulate Sinn as a set of criteria, from among which several could be chosen. Kripke writes:

The modern logical tradition, as represented by Frege and Russell, disputed Mill on the issue of singular names, but endorsed him on that of general names. Thus all terms, both singular and general, have a ‘connotation’ or Fregean sense. More recent theorists have followed Frege and Russell, modifying their views only by replacing the notion of a sense as given by a particular conjunction of properties with that of a sense as given by a ‘cluster’ of properties, only enough of which need apply. The present view, directly reversing Frege and Russell, (more or less) endorses Mill’s view of singular terms, but disputes his view of general terms.

According to Frege, the Sinn of a proper noun was a compressed description. A proper noun like ‘Aristotle’ contained, argues Frege, in a hidden way, a description like “the teacher of Alexander” or “a student of Plato” - Frege states:

The sense of a proper name is grasped by everybody who is sufficiently familiar with the language or totality of designations to which it belongs; but this serves to illuminate only a single aspect of the reference, supposing it to have one. Comprehensive knowledge of the reference would require us to say immediately whether any given sense belongs to it. To such knowledge we never attain.

Working out the example of ‘Aristotle,’ Frege continues:

In the case of an actual proper name such as ‘Aristotle’ opinions as to the sense may differ. It might, for instance, be taken to be the following: the pupil of Plato and teacher of Alexander the Great. Anybody who does this will attach another sense to the sentence ‘Aristotle was born in Stagira’ than will a man who takes as the sense of the name: the teacher of Alexander the Great who was born in Stagira. So long as the reference remains the same, such variations of sense may be tolerated, although they are to be avoided in the theoretical structure of a demonstrative science and ought not to occur in a perfect language.

While it is proper to view Frege as foundational to contemporary analytical philosophy, it is also proper to view him in the context of Locke and Mill. In this way, contemporary philosophy is seen in a way which emphasizes its continuity with the past.

Is Boltzmann a Phenomenologist?

Anyone familiar with the secondary literature on Ludwig Boltzmann knows that it is unusual, even odd, to refer to him as a phenomenologist. Yet a case can be made for doing precisely that. Arguments for viewing Boltzmann as a phenomenologist might even be plausible or persuasive. Such arguments will touch upon those points which empiricism and phenomenology have in common, or those points at which they come into contact with each other.

Ludwig Boltzmann, whose influence extended to Einstein and Planck, centered his notion of physics, and more broadly of natural sciences and of observational sciences, around phenomenology. Given his phenomenological leanings, he rejected any extensive role for the a priori in sciences, and rejected much of Kant’s version of physics.

Yet in some ways he seems to explicitly reject phenomenology. There is a tension within Boltzmann's thought: while arguing against phenomenologists, and eschewing the label of phenomenologist, he appears to embrace and even rigorously apply some phenomenological principles.

Both in science and in mathematics, Boltzmann sought a synthetic method which he considered to be simpler. Simplicity as a decision procedure for choosing between competing theories, which otherwise map nicely to the same data sets, is a familiar concept in the philosophy of science. Ernst Mach was roughly contemporary to Boltzmann. Alluding to Mach, Boltzmann wrote in 1892 about an economy of effort, which might be an analogue to simplicity as the concept is used in the philosophy of science concerning competing theories:

In mathematics and geometry the return from purely analytic to constructive methods and illustration by means of models was at first occasioned by a need for economy of effort. Although this seems to be purely practical and obvious, it is just here that we are in an area where a whole new kind of methodological speculations has grown up which were given most precise and ingenious expression by Mach, who states straight out that the aim of all science is only economy of effort.

The ambiguity in Boltzmann's relationship with phenomenology may arise from his use of some form of the principle of verification. Boltzmann's career path touched Ernst Mach's at a number of points, and Mach was a founding member of the Vienna Circle. The Circle was, of course, strongly linked to the verificationist understanding of meaning. Boltzmann was perhaps less interested in meaning and language, and used the verificationist principle more in the context of the philosophy of science. Inasmuch as verificationism is part of a radical empiricist program, and because both empiricism and phenomenology take raw sense-data as their foundation or point of departure, there is an overlap between verificationism and phenomenology, and this overlap gives rise to Boltzmann's ambiguity regarding phenomenology.

The more radical versions of empiricism and phenomenology take sense-data not only as a point of departure, but as the only possible point of departure - not as a foundation, but as the only possible foundation. The overlap between the two is highlighted in their more radical versions. Because much of Boltzmann's work in physics centered around particles invisible to the naked eye, the usual caveats about "indirect" or "in principle" observation apply to empirical verification of scientific theories. Paul Pojman writes that Ernst Mach

became embroiled in a long-standing dispute with Boltzmann, propounder of the kinetic theory of gasses. Boltzmann and Mach ended up agreeing in essence: if atomic theory was fruitful it should be used, but adopted what today might be considered an anti-metaphysical stance toward a theory that was still largely unsubstantiated. It is generally agreed that it was not until 1905 with Einstein's study of Brownian motion that the kinetic theory of molecules found full verification.

Consistently applying the phenomenological method, while seemingly denying that he was a phenomenologist, Boltzmann offered a critique of Friedrich Wilhelm Ostwald. Ostwald had wanted to relegate matter to the status of a construct; for Ostwald, matter was not among the atomic or foundational concepts of the world. As Joachim Schummer wrote,

The more Ostwald became convinced that thermodynamics is the fundamental theory of science — for which he saw evidence in the pioneering works of the American physicist Josiah Willard Gibbs and others — the more he engaged in natural philosophy. Two aspects may roughly characterize his philosophy. First, he asserted the primacy of energy over matter (matter being only a manifestation of energy) in opposition to widespread scientific materialism. Ostwald reformulated older concepts of dynamism dating back to the 17th-century German polymath Gottfried Leibniz with the principles of thermodynamics to form a new metaphysical interpretation of the world that he named “energetics.” Second, he asserted a form of positivism in the sense of rejecting theoretical concepts that are not strictly founded on empirical grounds. Although energetics found few adherents, the latter position found many contemporary proponents, such as the physicist-philosophers Ernst Mach in Austria and Pierre Duhem in France. As a consequence of his beliefs, for some 15 years Ostwald rejected atomism and was heavily involved in philosophical debates with his atomist colleagues, such as the Austrian physicist Ludwig Boltzmann, before he acknowledged the growing experimental evidence for the atomic hypothesis in 1909.

Detecting an inconsistency in Ostwald’s views, Boltzmann noted that it was equally unfounded to take energy as foundational or atomic to the universe. The arguments leveled against matter, or against the reality of matter, can be effectively and equally reformulated and leveled against energy. Boltzmann, again with Mach, sees matter and energy as equally derivative. What is foundational, as far as anything can be foundational in phenomenology, are sense-data. In a consistent phenomenological system, sensations are taken as the given. In 1904, Boltzmann wrote:

Mach pointed out that we are given only the law-like course of our impressions and ideas, whereas all physical magnitudes, atoms, molecules, forces, energies and so on are mere concepts for the economical representation and illustration of these law-like relations of our impressions and ideas. These last are thus the only thing that exists in the first instance, physical concepts being merely mental additions of our own. Ostwald understood only one half of this proposition, namely that atoms did not exist; at once he asked: what then does exist? To this his answer was that it was energy that existed. In my view this answer is quite opposed to Mach’s outlook, for which energy as much as matter must be regarded as a symbolic expression of certain relations between perceptions and of certain equations amongst the given phenomena.

Although Boltzmann was not in the mainstream of phenomenology, he is nonetheless a phenomenologist of some sort. Admittedly, some scholars refuse to classify him as such. Neither is he an idealist, in the sense of George Berkeley, but he veers close to some type of idealism. Manfred Grunwald corroborates Boltzmann’s influence, but disputes his phenomenology and idealism, when he writes:

Der berühmte Physiker und materialistiche Denker hat auf die weltanschaulichen Auffassungen Plancks und Einsteins großen Einfluß ausgeübt.

The question of whether radical empiricism and radical phenomenology are finally identical lies at the core of the question of whether or not Boltzmann is to be counted as some type of phenomenologist. His rejection of anything which he saw as metaphysical nudges him in the direction of empiricism. If it nudges him far enough into empiricism, and if empiricism at some point becomes one with phenomenological approaches, then we might be correct to label Boltzmann a phenomenologist, despite his own objections to that label, and despite a large body of respectable secondary literature which also declines to call him a phenomenologist.

Framed more broadly, the question emerges as this: is every empiricist a phenomenologist? is every empiricism a phenomenology? If we answer 'no' then we must articulate the distinction between phenomenology and empiricism. Alternatively, we can see phenomenology as a methodology within empiricism.

A surprising connection between Boltzmann's anti-Kantian view of physics and his social and ethical views arises from his interest in Darwin. Boltzmann's affection for Darwinism, however, gives rise to some methodological questions. Does Darwinism require Boltzmann to abandon the rigor with which he usually follows his phenomenological method? Can the arguments which Boltzmann directs against Kant and against Kantianism also be directed against Darwin and Darwinism? Manfred Grunwald continues:

Von Interesse ist Boltzmanns Stellungnahme zum Apriorismus. Er kritisierte den Apriorismus vor allem von dem einem Standpunkt, der sich aus der Entwicklungstheorie Darwins ergibt. Die sogenannte Denkgesetze hätten sich nach dem biologischen Gesetzen der Evolution gebildet und allmälich die Festigkeit aprioristischer Sätze angenommen. Die Einseitigkeit des nur durch die biologische Evolutionstheorie geprägten Herangehens und die Gefahren sozialdarwinistischer Deutungen werden in den Stellungnahmen Boltzmanns zu ethischen Fragen deutlich. Handlungen sind für ihn richtig und gut, soweit sie der Fortentwicklung der lebenden Materie dienen. Das Grundproblem der Ethik sah Boltzmann in der Beziehung zwischen der Behauptung oder Unterordnung des Willens eines Individuums und der zu fördernden Existenz eines Ganzen (Familie, Stamm, Menschheit). Einerseits behauptete er, daß sich aus den primitiven Formen der Materie, in Pflazen und Tieren, anzutreffenden Fähigkeiten (Vererbung, Zuchtwahl, Wahrnehmung, Willen, Lust, Schmerz) und ihrer quantitativen Steigerung das Denken und Wollen, das künstlerische Schaffen, die wissenschaftliche Forschung und das moralische Verhaltung der Menschen hinreichend erklären ließen. Andererseits erklärte er, daß er nur die »;naturwissenschaftliche begreifliche Seite« betrachte. Boltzmann lehnte die Verbindung naturwissenschaftlicher und religiöser Begriffe ab, weil beide aus gänzlich verschiedenen Bereichen stamman. Es zeigt sich die bei bürgerlichen Naturwissenschaftlern verbreitete Tendenz, der Religion das Gebiet der Moral zuzuweisen.

To the nature of Boltzmann's phenomenology - for phenomenology it was, if an unorthodox phenomenology - the following passage is in a longer text written by Byong-Chul Park:

As one of the founders of statistical thermodynamics, Boltzmann could not fully endorse the phenomenological method in physics. It is nevertheless an undeniable fact that Boltzmann uses the term 'phenomenology' in such a way that phenomenology in his sense is only concerned with our experience without any hypothesis in dealing with phenomena.

However one might characterize Boltzmann's phenomenology, and however it may compare and contrast to the phenomenological systems of other philosophers, Darwinism would entail a number of hypotheses which Boltzmann's understanding of phenomenology would not admit. In 1886, Boltzmann wrote:

Nowhere less than in natural science does the proposition that the straight path is the shortest turn out to be true. If a general intends to conquer a hostile city, he will not consult his map for the shortest road leading there; rather he will be forced to make the most various detours, every hamlet, even if quite off the path, will become a valuable point of leverage for him, if only he can take it; impregnable places he will isolate. Likewise, the scientist asks not what are the currently most important questions, but “which are at present solvable?” or sometimes merely “in which can we make some small but genuine advance?”

It would seem, then, from Boltzmann's philosophy of science, that the formulation of a grand system explaining the origin of life from lifeless matter, and this billions of years prior to any possible observation, would have no place in sober science. Boltzmann's phenomenological sense of science rejects the grand speculative systematic drive, present already in Kant and manifested grossly in Hegel. But does consistency demand that this sense of science likewise reject Darwinism with its hypotheses about processes which may have happened billions of year prior to any possible observation, with its hypotheses about life arising spontaneously from lifeless matter, and with speculations about matter and time emerging ex nihilo? Has Boltzmann violated his own methodological principles in embracing Darwin?

Boltzmann may have had some sense of the tension between his phenomenological and his non-phenomenological leanings. Note the contrast between the words 'internal' and 'external' in this passage, which he wrote in 1890:

I am of the opinion that the task of theory consists in constructing a picture of the external world that exists purely internally and must be our guiding star in all thought and experiment; that is in completing, as it were, the thinking process and carrying out globally what on a small scale occurs within us whenever we form an idea.

In Boltzmann's philosophy of science, then, theory is a speculative generalization - a projection - a writing large of our thoughts onto the universe. By thus defining 'theory' Boltzmann perhaps wished to avoid the methodological tensions between his attempt to apply a rigorous if idiosyncratic phenomenology and his affection for Darwinism. While not a phenomenologist in any usual sense of the word, he was certainly influenced by phenomenological methodologies. Albert Moyer writes:

Boltzmann did not agree that generalized, phenomenological theories were better equipped to reestablish order than were specialized, atomomechanical hypotheses. Though appreciative of phenomenological theories, he denied that such theories were free from hypotheses or idealizations and hence irrefutable. "Without some departure, however slight, from direct observation," he maintained, "a theory or even an intelligibly connected practical description for predicting the facts of nature cannot exist." Moreover, Boltzmann felt that the usefulness of phenomenological theories was limited merely to summarizing or developing "knowledge previously acquired." On the other hand, specialized and admittedly tentative hypotheses "give the imagination room for play and by boldly going beyond the material at hand afford continual inspiration for new experiments, and are thus pathfinders for the most unexpected discoveries." Consequently, he rejected the accusation that the "development of mathematical methods for the computation of the hypothetical molecular motions has been useless and even harmful." He also felt that the recent web of experiments involving cathode rays and radioactivity added credence to the atomistic viewpoint. Boltzmann was particularly optimistic about statistical mechanics.

Despite his dissimilarity to other phenomenologists, and despite what seems to be his explicit rejection of phenomenology, Boltzmann's sense of methodology retains features of phenomenology. Perhaps aware of this internal tension, some of his statements reflect perhaps an attempt to resolve such possible inconsistencies.

Ludwig Boltzmann Praises Michael Faraday

Ludwig Boltzmann, a significant physicist in his own right, was much influenced in his thinking by a physicist of a previous generation, Michael Faraday. Boltzmann routinely praised Faraday as major thinker. Boltzmann’s admiration of Faraday is interesting because the two men differed both in temperament and in method. Boltzmann was a morose man with little hope or cheer. Faraday was an optimistic and cheerful man. More to the point, Boltzmann, like most physicists of his era, emphasized quantitative methods and quantitative descriptions as ensuring that the propositions they considered were observable and measurable. By contrast, Faraday worked in a largely non-quantitative manner; to be sure, his propositions were observational and empirical, and were perhaps quantitative in the most minimal sense of the word, but he demonstrated a singular lack of interest in equations or in any algebraic expressions of his discoveries.

While Boltzmann praised Faraday’s work, he also recognized the profound differences between himself and Faraday, and the differences between Faraday and a later generation of physicists. He was quick to acknowledge these differences, and believed that these later physicists wrongly ignored Faraday because of his alternative methods. Boltzmann was patient enough to look at Faraday’s writings for their valuable insights, despite Faraday’s departure from Boltzmann’s preferred methods. In 1892, Boltzmann wrote:

Several scientists, amongst whom Faraday is foremost, had fashioned for themselves a quite different representation of nature. Whereas the old system regarded force centers as the only reality while treating forces as mathematical concepts, Faraday saw these latter as clearly operative from point to point of intervening space; the potential function, previously a mere formula facilitating calculation, he regarded as the really existing link in space and cause of the action of forces. Faraday’s ideas were much less clear than the earlier hypotheses that had mathematical precision, and many a mathematician of the old school placed little value on Faraday’s theories, without however reaching equally great discoveries by means of his own clearer notions.

During Boltzmann’s lifetime, various philosophers were developing what would come to be known as the “picture theory” of representation. Boltzmann considered that a picture could be, if not non-quantitative, at least not primarily quantitative, but rather qualitative. While Boltzmann preferred quantitative methods, he saw Faraday as constructing new and significant “pictures” in physics in a way which was not primarily quantitative. In 1890, Boltzmann wrote:

It is a peculiar drive of the human spirit to make itself such a picture and increasingly to adapt it to the external world. If therefore we may often have to use intricate formulae to represent a part of the picture that has become complicated, they nevertheless always remain inessential if most serviceable forms of expression, and in our sense Columbus, Robert Mayer, and Faraday are genuine theoreticians. For their guiding star was not practical gain but the picture of nature within their intellect.

Boltzmann perhaps regarded Faraday as a diamond in the rough, but a diamond nonetheless. The praise which an otherwise very restrained Boltzmann heaps on Faraday is noteworthy, considering Boltzmann’s acquaintance with other major physicists like Ernst Mach. In 1899, Boltzmann wrote:

Our factual knowledge of electricity and magnetism was enormously increased by Galvani, Volta, Oerstedt, Ampere and many others, and was brought to a certain finality by Faraday. The latter, using rather limited means, had found such a wealth of new facts that it long seemed as though the future would have to confine itself merely to explaining and practically applying all these discoveries.

Faraday’s major contributions, in the standard accounts of his activity, include the discovery of benzene and the process by which gases can be liquified. But his major work was in the field of electromagnetism. He discovered the relation between magnetism and electricity, designing and constructing the world’s first electrical generator and the world’s first electrical motor. He further discovered the field of electro-chemistry, and documented the effects of electricity on chemical reactions. Faraday also discovered the effects of magnetism on light, working with polarized lenses.

Michael Faraday belonged to a movement known variously as the Sandemanian or the Glasite movement. To which extent his involvement in this movement affected his work in physics and electromagnetics is not obvious, but he was, in any case, an active leader in that organization.

Faraday greatly influenced James Clerk Maxwell. James Maxwell’s significant career in physics can be said perhaps to have consisted of a systemization of Faraday’s discoveries. Maxwell expressed mathematically what Faraday discovered observationally. James Blackmore writes:

Perhaps the easiest way to make these differences clear between how different groups of people think would be to comment on Michael Faraday’s two most famous followers, Maxwell, who put much of Faraday’s work in mathematical form and who became a hero among theoretical physicists especially Boltzmann, and Thomas Alva Edison who devoured Michael Faraday’s “merely qualitative” book Experimental Researches in Electricity (1839-1855) to lay the groundwork for inventing or improving to the point of practicality the electric light bulb, the phonograph, the dictaphone, moving pictures, and numerous other electrical instruments. Maxwell and Boltzmann might doubt the existence of power and force or consider them subordinate or superfluous factors, especially in idealized or statistical aspects of theoretical physics, but Edison and other practical inventors had no such luxury. Power or force were vital means toward virtually all practical ends in applied science, and valid methodology of science had to acknowledge that primacy.

Boltzmann was not the only thinker to praise Faraday. Jearl Walker, in a widely-used textbook, describes Faraday:

The new science of electromagnetism was developed further by workers in many countries. One of the best was Michael Faraday, a truly gifted experimenter with a talent for physical intuition and visualization. That talent is attested to by the fact that his collected laboratory notebooks do not contain a single equation. In the mid-nineteenth century, James Clerk Maxwell put Faraday’s ideas into mathematical form, introduced many new ideas of his own, and put electromagnetism on a sound theoretical basis.

Faraday’s largely intuitive approach in electromagnetics may be compared to the intuitive discoveries in mathematics made by Srinivasa Ramanujan. In any case, Faraday’s discoveries are significant not merely because he was the first to observe a specific behavior in a laboratory, but also because he was able to correctly conceptualize what he observed. He not only was the first to see these things: he was the first to understand them.

Ernst Mach: Phenomenology and Physics

Ernst Mach worked in a wide variety of subfields within physics and philosophy, with occasional but significant forays into other disciplines. His name is perhaps most widely known from the unit of measurement named after him.

Perhaps the unifying thread among his diverse activities was a type of phenomenology tinged with his strong dislike of metaphysics. Therein lies both a strength and a weakness within Machian thought: a methodological strength from the perspective of phenomenology, and a weakness arising from an internal tension between Mach’s anti-metaphysical sentiment and the rigorous methodological demands of phenomenology. Such a tension would arise from the notion that phenomenology is tabling of questions about metaphysical realities - not a denial of such realities - and an exclusive attention to the metrics and patterns of appearance.

To be sure, Mach or a Machian would respond by pointing out that, as David Woodruff Smith has indicated, there is a variety of types of phenomenology, some of which might allow assertions or speculations about the reality or irreality of the Ding an sich. Mach wants to use the methodology of phenomenology while at the same time making a dogmatic statement about the nonexistence of certain metaphysical constructs. The mainstream of phenomenology, typified by Edmund Husserl, strictly demands that one refrain from assertions about metaphysical entities. While there is room for Mach to consider himself a phenomenologist, he certainly would be outside the mainstream of phenomenology.

We might compare three approaches: Husserl won’t - chooses not to - occupy himself with questions about the nounema behind the phenomena: this is a conscious methodological choice to refrain from discussion of the noumena. The Vienna Circle and the logical positivists assert that language about the noumena is nonsense, i.e., that it is not possible to discuss it. Mach seems to start from something like a logical positivist approach, but occasionally slips into making assertions that the noumena does not exist. Such assertions would both violate Husserl’s policy of voluntarily restraining one’s self, as a methodological principle, from making statements about the noumena, and violate the Vienna Circle’s notion that such assertions are meaningless.

Some of Mach’s remarks, taken in isolation, fall within traditional phenomenology. (Mach’s spelling sometimes appear idiosyncratic to eyes accustomed to twenty-first century German, but a variety of spelling reforms were being discussed and circulated in his day; those proposals have left their fingerprints on his texts.) To say that metaphysics is an idle pursuit, that it should be eliminated from the natural sciences, or that such questions cannot be meaningfully pursued in discussion and debate, are statements with which, with perhaps a little fine-tuning, Husserl’s disciples could agree.

Meinen erkenntnisskritisch-physikalischen und den vorliegenden sinnesphysiologischen Versuchen liegen dieselbe Ansicht zu Grunde, dass alles Metaphysische als müssig und die Ökonomie der Wissenschaft störend zu eliminieren sei. Wenn ich nun hier auf die abweichende Ansichten nicht ausführlich kritisch und polemisch eingehe, so geschieht dies wahrlich nicht aus Missactung derselben, sondern in der Überzeugung, dass derartige Fragen nicht durch Discussionen und dialectische Gefechte ausgetragen werden.

As a physicist, Mach applied the phenomenological method to the concept of bodies enduring over time. This version of the problem of identity is not new, but Mach develops the argument in a novel direction. He links our tendency - in his view, our mistaken tendency - to see a stronger identity over time than evidence warrants as the expression of a fear. It is our fear of death, and more generally our fear of a loss of identity’s duration over time, which predisposes us to embrace a metaphysical notion of identity, i.e., a notion which, in Mach’s view, goes beyond empirical sense data. Mach links our inclination to posit the identity of an object over time to our inclination to posit the duration of our own “self” or “ego” over time; he seems to indicate that the one inclination arise from the other, and that our desire to perceive our own existence’s duration arises from a fear of death.

Das Ich ist so wenig absolut beständig als die Körper. Was wir am Tode so fürchten, die Vernichtung der Beständigkeit, das tritt im Leben schon in reichlichem Masse ein.

Mach’s strong drive to deny any metaphysic leads him in psychology to deny any “ego” or “self,” fearing that any such psychological construct would ultimately open the door to dualism or metaphysics. In physics, that same drive leads him to construct a descriptive model of natural science, or observational science, along the lines of phenomenology. Science, for Mach, is the collecting and measuring of experience. Science describes sense data in the most simple and objective manner - which usually means quantitatively - and describes patterns which exist among those data, using the tools of geometry and algebra. While starting with a program which seeks to exercise admirable restraint - along the lines of Husserl’s phenomenology or a sober Kantianism - Mach seems to get carried away, and rather than simply refraining from passing judgment on metaphysical assertions, wants rather to deny them. His heirs, the logical positivists and the Vienna Circle, will attempt this restraint as they declare metaphysical propositions to be, not false, but meaningless. Mach seems at times to fall short of that development, and rather is eager to deny the existence of Ding an sich. Allan Janik and Stephen Toulmin write:

Mach’s reduction of all knowledge to sensation forms the base on which all of his thinking is founded. The task of all scientific endeavor is to describe sense data in the simplest or most economical manner. Mach actually prefers to designate sense data by the more neutral and noncommittal term “elements”; it is the feature of simplicity or economy which is distinctively scientific. So Mach’s point of view is that of a thoroughgoing phenomenalist; the world is the sum total of what appears to the senses. So, dreams constitute “elements” in the world just as much as any other class of elements, for “inner” experience is experience quite as much as “outer” experience is. Abstract conceptions, ideas, representations, are similarly reduced to sense data, by being identified as species concepts which enable us to deal with groups of “elements” efficiently.

Reflecting philosophy’s insights into, and concerns with, language, Mach notes that our use of names, and the thoughts to which they refer or for which they are symbols, reveals our tendency - both as a desire and as simply a habit of thought - to posit the continued existence of objects over time. He notes that our inclination towards such concepts is strong enough that “Ship of Theseus”-type objections from Aristotle or Heraclitus make little impact on it.

Die zweckmässige Gewohnheit, das Beständige mit einem Namen to bezeichnen und ohne jedesmalige Analyse der Bestandteile in einen Gedanken zusammenzufassen, kann mit dem Bestreben die Bestandteile zu sondern in einen eigentümlichen Widerstreit geraten. Das dunkle Bild des Beständigen, welches sich nicht merklich ändert, wenn ein oder der andere Bestandteil ausfällt, scheint etwas für sich zu sein.

In a classic phenomenological formulation, Mach tells us that objects simply are a collection of sense data. The classic formulation, however, leaves him with a classic problem. If we think of each sense datum as a point - maybe structured with three places: time, location, and which of the five senses obtained the input - then we think of an object as a set of such points. With Mach’s denial of any enduring identity of an object beyond the positing of such a set, it seems that he might be forced to acknowledge as an ‘object’ some very counterintuitive juxtapositions of sense-data, e.g., non-contiguous sets.

By his own restrictions, Mach may not say that an object is the collection of my “sense-data of it,” because the phrase “of it” imports the very metaphysical notion which he opposes. He must rather say that an object is a collection of sense-data. He may be forced to acknowledge some unlikely objects consisting of arbitrary collections of sense-data. I might take my sense-data of listening to a piano in Michigan in 1983 and place it into a set with my sense-data of flying in an airplane over Nebraska in 1975, yielding a set with more than one sense-datum in it, and thereby an “object” according to one understanding of Machian phenomenology.

Das Ding, der Körper, die Materie ist nichts außer dem Zusammenhang der Elemente, der Farben, Töne u.s.w. außer den sogenannten Merkmalen. Das vielgestaltige vermeintliche philosophische Problem von dem einen Ding mit seinen vielen Merkmalen entsteht durch das Verkennen des Umstandes, dass übersichtliches Zusammenfassen und sorgfältiges Trennen, obwohl beide temporär berechtigt und zu verschieden Zwecken erspriesslich, nicht auf einmal geübt werden können. Der Körper ist einer und unveränderlich, so lange wir nicht nötig haben, auf Einzelheiten zu achten.

Thus Mach responds by saying that an object’s - a body’s - unity and continuity are illusions and dissolve under disciplined analysis of our observations of it. Allan Janik and Stephen Toulmin write:

As a positivist, Mach was absolutely opposed to any sort of metaphysical speculation. He equated metaphysics with mysticism and consequently with obfuscation in science. In psychology he was a relentless opponent of all those who posited the “ego” as an entity; he rejected any position which smacked of the slightest hint of dualism, for he said that all dualism culminates in metaphysics. Indeed, as an ardent positivist, he did not recognize philosophy to have any legitimacy apart from science, and he continually insisted that he was not a philosopher. David Hume, the destroyer of all metaphysical claims to truth, and Georg Christoph Lichtenberg, the enemy of a pseudo science, were his philosophical heroes. Mach was, in fact, the first man to draw attention to the philosophical significance of Lichtenberg, whose writings soon became popular and influential in the artistic and intellectual circles of Vienna.

Using a classic example of the optical illusion of a straight stick or pencil partially submerged in water, and of its apparent bentness due to refraction despite our “knowledge” that it is straight, Mach wishes to reformulate our conclusion about the pencil. He rejects the usually conclusion that the pencil is straight but appears bent. He rejects that formulation because it smacks of metaphysics inasmuch as it contrasts being to appearance, and he fears that this is the toehold of metaphysics. He rather wants to formulate the conclusion thusly: the pencil is optically bent but haptically straight. (If we close our eyes and put our hands into the water, there is no evidence or sense of bentness.) By rephrasing this classic example according to his phenomenological program, he hopes to deny entry to metaphysics.

Man pflegt in der populären Denk- und Redeweise der Wirklichkeit den Schein gegenüber zu stellen. Einen Bleistift, den wir in der Luft vor uns halten, sehen wir gerade; tauchen wir denselben schief ins Wasser, so sehen wir ihn geknickt. Man sagt nun in letzterem Falle: Der Bleistift scheint geknickt, ist aber in Wirklichkeit gerade. Was berechtigt uns aber eine Tatsache der andern gegenüber für Wirklichkeit zu erklären und die andere zum Schein herabzudrücken? In beiden Fällen liegen doch Tatsachen vor, welche eben verschieden bedingte, verschiedenartige Zusammenhänge der Elemente darstellen. Der eingetauchte Bleistift ist eben wegen seiner Umgebung optisch geknickt, haptisch und metrisch aber gerade.

In physics, Mach’s phenomenology led him to oppose Newton’s view of space as substantival. Newton had argued that there is a thing, an object, called ‘space’ - that when we use the word ‘space,’ we are in fact referring to a physical reality. Historically, Leibniz opposed Newton and argued that space does not have the status of a Ding an sich, but exists merely as the description of the relative locations of physical objects. Newton would have argued that if one removed all matter and energy from the universe, one would still have empty space. Leibniz would have argued that the removal of all matter and energy would leave nothing. For Newton, empty space and nothing are two different situations; for Leibniz, they are the same. Mach winds up being closer to Leibniz than to Newton. Mach accuses Newton of dabbling in metaphysics when the latter posits space as real. Lawrence Sklar writes:

Ernst Mach was simultaneously a working theoretical physicist and a positivist philosopher. He made, in his Science of Mechanics, a heroic attempt to replace Newton’s theory by an alternative theory. The alternative is supposed to be adequate to account for the inertial forces Newton took as the primary data supporting his doctrine of substantival space, but to lack any such “metaphysical” elements that infected, according to Mach, the Newtonian scheme.

Mach seems to indicate that while physicists are well aware of the tendency of prejudices to sneak into their work, and are therefore alert to look out for them, psychologists are by contrast not as aware of the subtle influences which preconceived notions can have in their work. Here, Mach is perhaps thinking of the notion of an object or a body from physics working its way into psychology as the notion of the ‘ego’ or ‘self’ or ‘mind.’ Part of Mach’s program, then, is to purify psychology and structure it so that it will keep such prejudices at bay. A psychology structured according to Machian phenomenology will smell the metaphysics which clings to the concept of the ‘self’ and reject that concept.

Der Physiker hat oft Gelegenheit zu sehen, wie sehr die Erkenntnis eines Gebietes dadurch gehemmt werden kann, dass anstatt der vorurteilslosen Untersuchung desselben an sich, die auf einem andern Gebiet gefassten Ansichten auf dasselbe übertragen werden. Weit bedeutender ist die Störung, welche durch solche Übertragung vorgefasster Meinungen aus dem Gebiet der Physik in jenes der Psychologie entsteht.

Accordingly, Mach rejects the notion that psychological terms like ‘will’ refer to anything beyond a set of sense-data. One might detect traces of Mach’s encounter with Schopenhauer’s books here. In any case, Mach is quite clear that the word ‘will’ does not refer to a psychological object - indeed, Mach would reject the very notion of a psychological object beyond a collection of experiences - and sees rather the ‘will’ as something physical, organic, and biological:

Ich verstehe unter dem Willen kein besonderes psychisches oder metaphysisches Agens, und nehme keine eigene psychische Kausalität an. Ich bin vielmehr mit der überwiegenden Zahl der Physiologen und modernen Psychologen überzeugt, dass die Willenserscheinungen aus den organisch-physischen Kräften allein, wie wir kurz aber allgemein verständlich sagen wollen, begreiflich sein müssen.

Mach is, then, a phenomenologist of some type. The question is: of which type? To which extent would Mach harmonize with Husserl? Having established himself as a phenomenologist, Mach wishes to apply his phenomenology to empirical and observational science - he would hardly allow the existence of any other type of science. Mach wishes to restructure physics and psychology according to his phenomenology, and in so doing, is in conflict with Newton. To which extent was Mach successful? On the one hand, he had a significant influence on Einstein; on the other hand, Newton remains a powerful influence.

Ludwig Boltzmann - Philosophy and Physics

Brilliant and creative thinkers are found at the intersection of philosophy and physics. Topics like time, space, and the structure of scientific theories make fertile ground in which the thoughts of philosophers and physicists flourish.

Ludwig Boltzmann is one such thinker. He is largely responsible for the fact that the second law of thermodynamics has become associated with the concept of entropy. Boltzmann worked with thermodynamics generally, and with statistical approaches to physics. His work is in many ways foundational to certain subtopics within physics.

In approaching statistical definitions in physics, Boltzmann touched upon a central topic in the philosophy of science in general, the underdetermination of theories. Both hypothetically and in practice, we are confronted with competing theories, and in some cases, more than one theory fits the data. Our choice of theory, then, in such circumstances cannot be made by the data or with regard to the data. Boltzmann wrote, in 1899, that

we remain faithful to our principle that for the time being we are not aiming at a single best account of science, but that we regard it as expedient to try as many accounts as possible, each of which has its peculiar advantages but also its drawbacks. Again we must focus our main attention on avoiding all inconsistencies and logical mistakes and on not smuggling in tacit concepts or assumptions, and ensure on the contrary that we become most clearly aware of all hypotheses we rely on.

In arguing against Schopenhauer’s assertion that the law about the conservation of matter, a law often associated with the name Lavoisier, is true and knowable a priori, Boltzmann wrote in 1905 that

doubts about this law have arisen in connection with the behavior of radium. I am convinced that these experiments too will confirm the law, but that proves the law to be other than a priori: were it not to hold, we could retort nothing from a logical point of view.

Correctly foreseeing that the radium example, what we now understand to be the decay of an unstable isotope, would not violate the law, but rather refine our understanding of it as the conservation of matter and energy, Boltzmann indicated its empirical nature. (Lavoisier was probably not the first to stumble upon the conservation of mass, but he publicized it and so his name became associated with it.) Despite his emphasis upon mathematical methods, Boltzmann maintained that physics had an essentially empirical aspect:

Thus we must change all laws of thought in such a way that they lead everywhere to the same goal, that they correspond to experience and that overshooting the mark is kept within proper bounds. Even if this ideal will presumably never be completely realized, we can nevertheless come nearer to it, and this would ensure cessation of the disquiet and the embarrassing feeling that it is a riddle that we are here, that the world is at all and is as it is, that it is incomprehensible what is the cause of this regular connection between cause and effect, and so on. Men would be freed from the spiritual migraine that is called metaphysics.

We see here a connection between Boltzmann’s more technical results - dealing with the dynamics of molecules and their speeds, especially in gasses - and contact with questions of philosophical interest - it is a riddle that we are here, it is a riddle that the world is at all, and it is a riddle that the world is as it is and is not otherwise. He hoped, not to answer such questions or solves such riddles, but rather to sidestep them by directing physics as a whole toward empiricism. To some extent, Boltzmann may have here anticipated logical positivism and the Vienna Circle - to the extent that we can interpret him to be saying that such riddles and questions do not need to be answered, but rather need to be recognized as nonsense. Allan Janik and Stephen Toulmin offer a synopsis of Boltzmann’s work. They write that the influence of Kant and of Heinrich Rudolf Hertz

is evident also in the ideas of Ludwig Boltzmann, the man who founded the “statical mechanics” which lies at the basis, not only of the twentieth-century approach to thermodynamics, but of the modern attitude toward theoretical physics generally. Boltzmann took Hertz’s account of mechanics as defining a system of “possible sequences of observed events,” and made it the starting point for a general method of theoretical analysis in physics itself. He did so, by treating each independent property of a physical system as defining a separate coordinate in a multidimensional system of geometrical coordinates. All the possible locations of each separate body in the physical system, for instance, were ordered along three spatial “axes of reference”; all values of, say, temperature along another axis; all values of, say pressure, along a fifth; and so on. The totality of theoretical “points” in the resulting multidimensional coordinate system gave one a representation of the “ensemble of possible states” of the physical system in question; and any actual state could be defined, by specifying the particular point in this “multidimensional space” whose coordinates correspond to the actual values of all the variables. The general problem for statistical mechanics was then to discover mathematical relations governing the frequencies with which - on various assumptions and conditions - the actual states of a physical system would be distributed among its possible states; and, so, to compute the relative probabilities of finding the system, in actual fact, in one overall physical state rather than another.

There may be some hint of an internal tension in Boltzmann’s thought. On the one hand, he wants to be a hard-nosed empiricist and deal only with the data given to us by experience. On the other hand, he wants to create a matrix to cover all possible situations, including those which have not been experienced; in fact, his matrices may easily include configurations - states of affairs - which have never and will never be actualized. While there is a standard empiricist defense for talk about such unrealized possibilities - this defense will argue that such discourse is still meaningful because criteria for verification exist - , these uninstantiated possible states of affairs also present a opportunity to sneak some metaphysical thought into physics. And Boltzmann does not like metaphysics.

John Blackmore sees a slightly different type of internal tension in Boltzmann’s thought. Blackmore writes:

As a physicist, one might naturally suppose that he considered what he called methodology of science more important than natural philosophy, but in fact, as the reader has no doubt already observed, judged in terms of what he actually taught, the opposite was the case. He increasingly taught the ideas of Schopenhauer, Kant, and Brentano on logical, epistemological and ontological topics. He was never quite sure what natural philosophy was, but his lectures became much closer to traditional philosophy than to methodology of science. In short, Boltzmann may have compartmentalized what we would call his intellectual outlook into an expanding series of more or less isolated cells starting with science and what he thought was methodology of science and gradually extending to and through physiology, biology, and Darwinism on the one side and various forms of epistemology, ontology, and “metaphysics” on the the other. It may be no wonder that he had such a hard time obtaining peace of mind or a coherent, inclusive philosophy or world view. One compartment of thought seemingly didn’t know what the other compartments were doing.

The tension Blackmore sees between Boltzmann’s science and Boltzmann’s scientific methodology - between the way Boltzmann did physics and the way he talked about physics - is reflected in the way Boltzmann simultaneously endorses aspects of Schopenhauer’s philosophy while rejecting Schopenhauer’s hypothesis about the rational foundation for the second law of thermodynamics.

What are Numbers?

Consider a common question: “how much is 7 + 5?” We can quibble about the phrasing: “how many are 7 + 5?” or “what is 7 + 5?” In any case, the answer is 12. But what is 12? We are asking about what a number is. What is a number? We use numbers all the time without considering what they are. We might well know what three apples are, or what three pencils are, but what is three? What is three, all by itself, when it’s not three apples or three pencils?

This question arises, in part, because of the representative nature of language. Words, whether spoken or written, are commonly understood to be symbols which refer to things. Nouns refer to persons, places, objects, or ideas. Verbs refer to actions or states of being. Adjectives and adverbs refer to properties and qualities; prepositions refer to relationships. What about numerals? When I see the numeral ‘3’ written on paper, it is a symbol, and I might well ask if it refers to something, and if so, what that something is.

If numerals refer to something, that something would be numbers. Numerals are physical symbols, consisting of ink on paper. What type of things would numbers be? Numbers would be non-physical things, i.e. metaphysical things, at least according to philosophers like Plato. Plato's view, adopted by a number of other significant philosophers over the centuries, is called dualism and realism. The word ‘dualism’ indicates that Plato is positing two levels of reality, a physical level and a metaphysical level. The word ‘realism’ indicates that Plato posits that mathematical statements are statements about an independently-existing state of affairs, that mathematical words refer to independently-existing things, and that mathematical propositions are true or false to the extent that the correspond, or do not correspond, to such an independently-existing state of affairs. Author Rui Vieira writes:

What exactly are the objects of mathematics, and how do they relate to our knowledge of them? Since Plato (427 BC–347 BC) such questions have been central to the philosophy of mathematics. Plato realized that mathematics seems to involve perfect circles, triangles, and so on. But as Plato also noticed, there are no perfect circles or triangles to be found in the world, only imperfect approximations. Imagine a polygon [shape] with an ever-increasing number of equal sides. As the number of sides approaches infinity, the polygon will become a circle. Thus a perfect circle may be conceived of as a regular polygon with an infinite number of infinitesimally small sides. So no matter how accurate our or our computer’s rendering of a circle may be, it will only be an imperfect approximation. Finite humans and their computers cannot create objects with infinite mathematical features, such as the infinite sides of our ideal circle. Plato concluded that since there are no perfect mathematical objects to be found in the world, the objects of mathematics – perfect circles, triangles, and indeed numbers themselves – must somehow exist as eternal abstract entities beyond space and time in some otherworldly Platonic heaven called the world of Forms (or Ideas). Plato’s particular type of mathematical realism (ie, of attributing objective reality to mathematical objects), has been one of the most prevalent views of mathematics among both philosophers and mathematicians ever since.

Opposed to Plato’s view, however, are several groups of philosophers, called variously ‘formalists’ and ‘conventionalists’ and including brilliant mathematicians like David Hilbert, Heinrich Eduard Heine, and Carl Johannes Thomae. These thinkers regarded numerals as non-referential. Phrased another way, they thought that numerals are numbers and numbers are numerals. A numeral, according to formalism, is a mark - ink on paper - which is related to other marks by means of a set of rules. Famously, the analogy to games arose. Consider a card game. The seven of spades, or the eight of clubs, does not refer to, or represent, anything. Its role in a card game is simply that if it is played, then certain other things must happen in the game.

According to the formalists, 7 + 5 = 12 because the rules of the game are such that when four symbols ‘7 + 5 =’ are written, then the fifth symbol must be ‘12’. By contrast, the mathematical Platonist considers that 7 + 5 = 12 because when the object called ‘7’ is added to the object called ‘5’, another object, called ‘12’, is formed.

The formalists - conventionalists are similar - thought that they had made a contribution to the progress of mathematics, because they had freed mathematicians from mysterious discussions of what type of ethereal thing a number might be. They felt that mathematicians should not have to be hindered by discussions of metaphysical objects.

The Platonists, on the other hand, saw that formalism contained certain weaknesses. If the formalist views were adopted as the foundations of mathematics, and those weaknesses emerged, the entire superstructure might collapse. A leading critic of formalism was Gottlob Frege, who wrote:

The question now forces itself upon us: Is calling these signs numbers enough to ensure that they have the properties of the numbers proper which we have previously been accustomed to regard as quantitative ratios?

To be fair to David Hilbert, he and some of the formalists did not make a blanket ontological assertion that there are no metaphysical objects to which numerals refer. Hilbert was not so much saying that numbers as Platonic objects do not exist as much as he was saying whether or not numbers exist as Platonic objects, we will go ahead and do mathematics based only on numerals as otherwise meaningless symbols manipulated according to the rules of mathematics. The only relevant meaning had by numerals, for Hilbert, is the meaning given to them by their place in the rules. Whether or not they referred to Platonic numbers, and whether or not those Platonic numbers really existed, was for Hilbert not relevant.

There is a detectable difference between formalists and conventionalists. Formalists treat numerals as meaningless symbols to be arranged and manipulated according to a set of rules. Conventionalists treat mathematical propositions, and perhaps also meta-mathematical propositions, as true by agreement: they are the stipulations of the mathematical community. But for the purposes of discussion the ontological status of numbers, formalists and conventionalists are very similar, and we may conveniently lump them into the same category.

In any case, Frege sees formalism and the formalist project as a failure. He alleges that it cannot hold to its own program. While the formalist lays the foundations of the mathematics promising to treat numerals and all other operators as meaningless patterns of ink on paper, and to simply do no more than rearrange those symbols according to a set of rules, the formalist will in fact later attribute meaning to those symbols. Frege asserts that the formalist, having kept meaning and reference out of mathematics at the foundations, will later sneak little bits of meaning and reference into mathematics in its more advanced stages of development. Frege writes:

This attempt at formal arithmetic must be considered a failure, since it cannot be pursued consistently. In the end numerical figures are used as signs after all.

Specifically, Frege noted the use of the operators ‘greater than’ and ‘less than’ and asked, in which way a numeral, as opposed to a number, could be greater or less than another numeral. True, the formalists can reply that numeral are assigned an arbitrary order, and that the operators simply reflect that order; but this line of thought is complicated by the fact that there is an infinity of numerals, in fact an uncountable infinity, and how would an order be assigned to them all? Likewise, the formalists wants to maintain that an expression like ‘3 ÷ 0’ is meaningless, but if the formalists have already declared all symbols meaningless at the foundation, in which way is ‘3 ÷ 0’ especially meaningless? Frege continues:

We saw that terms and expressions were borrowed unconsciously and without explanation from arithmetic that has a content (e.g. ‘larger than’ and ‘smaller than’) and that their role in the calculating game remained obscure, although it seemed to be highly important. Formal arithmetic proved unable to define the irrational, for it had only a finite number of numerical figures at its disposal.

Frege stands with Plato and rejects the formalist interpretation of mathematics. The matter is not resolved. The formalist camp will charge Frege with violating Ockham’s Razor. While Frege has succeeded in demonstrating that there are difficulties for formalism, he has not conclusively demonstrated the existence of Platonic numbers.

Language and Analyticity

Contemporary philosophy - or contemporary analytic philosophy, or late modern philosophy - has inherited from Kant, for better or for worse, a schema in which we divide propositions into those which are by their structure either analytic or synthetic, which are by their epistemology either a priori or a posteriori. Placing these dichotomies on a two-by-two grid, a system with four compartments results - a system which has seemed to some philosophers just a bit too conveniently tidy.

The question can be raised whether all propositions fit properly into one of these pigeon-holes, whether the dichotomies are exhaustive (is there a third option?), and whether the distinctions involved are absolutely binary opposites with no overlap or ambiguity between them.

One type of question which can be raised is about the nature of analytic propositions. The tautological nature of analytic propositions seems clear, especially when examples are taken from mathematics or abstract physics - the types of examples Kant gives in his Kritik der Reinen Vernunft. Yet more everyday examples from natural language can seem less clear. The question can be posed, whether Kant and others have selectively chosen their linguistic examples.

Philosophers, mathematicians, and logicians tend toward idealized versions of language when they look to investigate the structure of propositions. Gareth Evans writes:

Frege was the first to formulate a systematic theory of meaning for a fragment of natural language; systematic in that it sought to provide an explanation of how the significance of complex expressions, particularly sentences, depends upon the significance of their parts. Unsurprisingly, given Frege’s larger purposes in investigating the foundations of mathematics, the fragment which concerned him was free of many of the characteristic features of natural language; in particular, indexical expressions like ‘I’, ‘now’, ‘here’, etc. However, Frege did offer suggestions as to how his apparatus could be brought to bear upon such devices.

Language interested Frege, initially and primarily, to the extent that it bore upon his investigations into logic and mathematics. Frege’s ruminations upon language tended to restrict themselves to a small and overly tidy subset of linguistic phenomena.

Let us consider a more typical proposition, and the sentence which represents it: Lee Harvey Oswald is the man who shot JFK. While this proposition does not contain any of the words which are normally considered indexicals, its analysis will nonetheless be conducted with a view to the time in which such analysis in executed.

In the year 1900, neither of the men who are referents of terms in the proposition had been born - although there may have been, at that time, a man whose initials were JFK, and there might conceivably even have been a man named Lee Harvey Oswald. Yet in conducting an analysis, we know that whoever generated the proposition meant this JFK and not that JFK, and meant this Oswald, and not that Oswald. A whole host of problems arise with such natural language sentences, sentences which are quite messier than uncluttered sentences about “bodies having extension” or “7 + 5 = 12” and so forth. Questions arise about ostensiveness, about meaning, about proper nouns, about referents which do not exist, and about synonymy.

Although some philosophers have been hastily dismissive toward Frege’s symbol of assertion in his system, writing that assertion is a merely psychological phenomenon, one might yet acknowledge that Frege, however misguided his assertion operator might have been, was responded perceptively to the complexities which natural language offers.

Our proposition about JFK and Oswald will seem different if we analyze it in 1962, or in 1964. By a much later time - let’s analyze it in the year 2013 - it will be very much like what Kant calls an analytic sentence. Considered from the standpoint of the knowing subject, many knowing subjects know about Oswald only that he shot JFK, and nothing else. For such individuals, our proposition will seem almost like the quasi-mystical a = a of the post-Kantian German idealists.

Simply put: for many natural language sentences, the question of whether they are analytic or synthetic seems to be a temporal question and a psychological question. Its analysis may be different if such analysis is carried out before or after specific events. Its analysis may depend on the epistemological state of the analyzer - whether he knows certain facts.

The murkiness resulting from the analysis, or attempted analysis, of natural language sentences has caused some to doubt the framework of analytic / synthetic and a priori / a posteriori. Willard Van Orman Quine writes:

I am not satisfied that a clear general distinction has yet been drawn between analytic and synthetic. I am even more in the dark on the Kantian distinction between analytic and apriori. This much, nevertheless, I can say: If the statements of the usual higher mathematical logic are analytic, then so are such platonistic statements as ‘There are classes’, ‘There are numbers’.

In Quine’s book Word and Object, his linguistic example begins with a word which has a fully established place in language; the task of the linguist is to discover that word’s use and meaning. An alternative example might begin with the coining of a word: astronauts return from a distant planet, bringing a sample of an unknown substance. They hand it to a scientist in a laboratory, whose task is to analyze the substance. As the scientist takes notes on the sample, he has need to name it, for the mere purpose of writing his observations.

The scientist begins by naming the sample: he calls it the ‘ABCXYZ sample’ and uses this term consistently in his laboratory journals. We might ask the question: which sentences using the name ‘ABCXYZ’ are analytic, and which are synthetic?

At first, a small handful of simple sentences, mainly observations recording sense-data, will be analytic: “the ABCXYZ sample is solid at room temperature” or “the ABCXYZ sample has a density of so-and-so many grams per cubic centimeter.” As the scientist conducts his research, synthetic sentences will emerge as new and not-directly-perceptible information emerges: “the ABCXYZ substance is composed mainly of iron and silicon” or “the melting point of the ABCXYZ sample is so-and-so many degrees Fahrenheit.”

As data about ABCXYZ becomes more clear, it is printed in reference books, and eventually in textbooks. It is taught to undergraduate students and later even to high school students. Eventually, a generation of students arises, a generation which has never seen an ABCXYZ sample, or even a photograph of a sample, and doesn’t know the heroic tale of astronauts who brought it back from a distant planet. For this later generation, ABCXYZ has always been found in all reference books and textbooks. For this later generation, the proposition that ABCXYZ is composed mainly of iron and silicon may be analytic; indeed, it may be the only thing that this generation knows about ABCXYZ.

Quine discusses a similar example:

Suppose a scientist introduces a new term, for a certain substance or force. He introduces it by an act either of legislative definition or of legislative postulation. Progressing, he evolves hypotheses regarding further traits of the named substance or force. Suppose now that some such eventual hypothesis, well attested, identifies this substance or force with one named by a complex term built up of other portions of his scientific vocabulary. We all know that this new identity will figure in the ensuing developments quite on a par with the identity which first came of the act of legislative definition, if any, or on par with the law which first came of the act of legislative postulation. Revisions, in the course of further progress, can touch any of these affirmations equally.

If we accept Quine’s objections, we have at least two options: we might take a moderately Quinian approach, and say that any decision about a proposition’s analyticity is relative to time and circumstance. Along this route we might say that a proposition is at one point in time, for one analyzer, analytic, while at some other point in time, for some other analyzer, it is synthetic.

A radically Quinian - Quine himself apparently preferred the spelling ‘Quinian’ to ‘Quinean’ - position might abolish the distinction between analytic and synthetic altogether, declaring it illegitimate.