Frege Quotes

Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

[..] I do not begin with concepts and put them together to form a thought or judgement; I come by the parts of a thought by analysing the thought.

A scientist can hardly encounter anything more desirable than, just as a work is completed, to have its foundation give way.

I conceive the proposition—like Frege and Russell—as a function of the expressions contained in it.

I went back into the house and had put on the kettle for another cup of tea when my attention was caught by a spider on the kitchen wall. As I drew nearer to look at it, the spider called out, “Hello!” It did not seem at all strange to me that a spider should say hello (any more than it seemed strange to Alice when the White Rabbit spoke). I said, “Hello, yourself,” and with this we started a conversation, mostly on rather technical matters of analytic philosophy. Perhaps this direction was suggested by the spider’s opening comment: did I think that Bertrand Russell had exploded Frege’s paradox? Or perhaps it was its voice—pointed, incisive, and just like Russell’s voice (which I had heard on the radio, but also—hilariously—as it had been parodied in Beyond the Fringe).9 D

Thus the thought, for example, which we expressed in the Pythagorean theorem is timelessly true, true independently of whether anyone takes it to be true. It needs no bearer. It is not true for the first time when it is discovered, but is like a planet which, already before anyone has seen it, has been in interaction with other planets.

If anyone tried to contradict the statement that what is true is true independently of our recognizing it as such, he would by his very assertion contradict what he had asserted; he would be in a similar position to the Cretan who said that all Cretans are liars.

Gwen went up to Cambridge and read Moral Sciences and started on a Ph.D. thesis on Frege.

In philosophy we must always resist the temptation of hitting on an answer to the question how we can define such-and-such a notion, an answer which supplies a smooth and elegant definition which entirely ignores the purpose which we originially wanted the notion for.

I don't believe I have ever invented a line of thinking, I have always taken one over from someone else. I have simply straightaway seized on it with enthusiasm for my work of clarification. That is how Boltzmann, Hertz, Schopenhauer, Frege, Russell, Kraus, Loos, Weininger, Spengler, Sraffa have influenced me.

Die Gedanken sind weder Dinge der Außenwelt noch Vorstellungen. Ein drittes Reich muß anerkannt werden. Was zu diesem gehört, stimmt mit den Vorstellungen darin überein, daß es nicht mit den Sinnen wahrgenommen werden kann, mit den Dingen aber darin, daß es keines Trägers bedarf, zu dessen Bewußtseinsinhalte es gehört.

Russell’s point is that if the set is a member of itself, then by definition it can’t be a member of itself. But if it is not a member of itself, then it is a member of itself. So it is both a member of itself and not a member of itself. And that is a contradiction. This glaring mistake, allegedly, left Frege a broken man.

The classification of some piece of knowledge as analytic or synthetic, a priori or a posteriori, concerns not “the psychological, physiological and physical conditions” that made it possible for us to grasp the relevant proposition but rather “the ultimate ground on which the justification for holding it to be true rests” (Frege, 1953, §3).

Frege's work it followed that arithmetic, and pure mathematics generally, is nothing but a prolongation of deductive logic. This disproved Kant's theory that arithmetical propositions are 'synthetic' and involve a reference to time. The development of pure mathematics from logic was set forth in detail in Principia Mathematica, by Whitehead and myself.

If number were an idea, then Arithmetic would be Psychology.

Analytic philosophy, that is to say, can very occasionally produce practically conclusive results of a negative kind. It can show in a few cases that just too much incoherence and inconsistency is involved in some position for any reasonable person to continue to hold it. But it can never establish the rational acceptability of any particular position in cases where each of the alternative rival positions available has sufficient range and scope and the adherents of each are willing to pay the price necessary to secure coherence and consistency. Hence the peculiar flavor of so much contemporary analytic writing—by writers less philosophically self-aware than Rorty or Lewis—in which passages of argument in which the most sophisticated logical and semantic techniques available are deployed in order to secure maximal rigor alternate with passages which seem to do no more than cobble together a set of loosely related arbitrary preferences; contemporary analytic philosophy exhibits a strange partnership between an idiom deeply indebted to Frege and Carnap and one deriving from the more simple-minded forms of existentialism

In order to avoid these errors, we must employ a symbolism which excludes them, by not applying the same sign in different symbols and by not applying signs in the same way which signify in different ways. A symbolism, that is to say, which obeys the rules of logical grammar—of logical syntax. (The logical symbolism of Frege and Russell is such a language, which, however, does still not exclude all errors.)

Frege ridiculed the formalist conception of mathematics by saying that the formalists confused the unimportant thing, the sign, with the important, the meaning. Surely, one wishes to say, mathematics does not treat of dashes on a bit of paper. Frege’s idea could be expressed thus: the propositions of mathematics, if they were just complexes of dashes, would be dead and utterly uninteresting, whereas they obviously have a kind of life. And the same, of course, could be said of any proposition: Without a sense, or without the thought, a proposition would be an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in order to make a live proposition is something immaterial, with properties different from all mere signs. But if we had to name anything which is the life of the sign, we should have to say that it was its use.

What is it, in fact, that we are supposed to abstract from, in order to get, for example, from the moon to the number 1? By abstraction we do indeed get certain concepts, viz. satellite of the Earth, satellite of a planet, non-self-luminous heavenly body, heavenly body, body, object. But in this series 1 is not to be met with; for it is no concept that the moon could fall under. In the case of 0, we have simply no object at all from which to start our process of abstracting. It is no good objecting that 0 and 1 are not numbers in the same sense as 2 and 3. What answers the question How many? is number, and if we ask, for example, "How many moons has this planet?", we are quite as much prepared for the answer 0 or 1 as for 2 or 3, and that without having to understand the question differently. No doubt there is something unique about 0, and about 1000; but the same is true in principle of every whole number, only the bigger the number the less obvious it is. To make out of this a difference in kind is utterly arbitrary. What will not work with 0 and 1 cannot be essential to the concept of number.

The three main mediaeval points of view regarding universals are designated by historians as realism, conceptualism, and nominalism. Essentially these same three doctrines reappear in twentieth-century surveys of the philosophy of mathematics under the new names logicism, intuitionism, and formalism. Realism, as the word is used in connection with the mediaeval controversy over universals, is the Platonic doctrine that universals or abstract entities have being independently of the mind; the mind may discover them but cannot create them. Logicism, represented by Frege, Russell, Whitehead, Church, and Carnap, condones the use of bound variables to refer to abstract entities known and unknown, specifiable and unspecifiable, indiscriminately. Conceptualism holds that there are universals but they are mind-made. Intuitionism, espoused in modern times in one form or another by Poincaré, Brouwer, Weyl, and others, countenances the use of bound variables to refer to abstract entities only when those entities are capable of being cooked up individually from ingredients specified in advance. As Fraenkel has put it, logicism holds that classes are discovered while intuitionism holds that they are invented—a fair statement indeed of the old opposition between realism and conceptualism. This opposition is no mere quibble; it makes an essential difference in the amount of classical mathematics to which one is willing to subscribe. Logicists, or realists, are able on their assumptions to get Cantor’s ascending orders of infinity; intuitionists are compelled to stop with the lowest order of infinity, and, as an indirect consequence, to abandon even some of the classical laws of real numbers. The modern controversy between logicism and intuitionism arose, in fact, from disagreements over infinity. Formalism, associated with the name of Hilbert, echoes intuitionism in deploring the logicist’s unbridled recourse to universals. But formalism also finds intuitionism unsatisfactory. This could happen for either of two opposite reasons. The formalist might, like the logicist, object to the crippling of classical mathematics; or he might, like the nominalists of old, object to admitting abstract entities at all, even in the restrained sense of mind-made entities. The upshot is the same: the formalist keeps classical mathematics as a play of insignificant notations. This play of notations can still be of utility—whatever utility it has already shown itself to have as a crutch for physicists and technologists. But utility need not imply significance, in any literal linguistic sense. Nor need the marked success of mathematicians in spinning out theorems, and in finding objective bases for agreement with one another’s results, imply significance. For an adequate basis for agreement among mathematicians can be found simply in the rules which govern the manipulation of the notations—these syntactical rules being, unlike the notations themselves, quite significant and intelligible.

for example, to see whether in the developing subject, i.e. the child, integers are directly constructed starting from class logic by biunivocal correspondence and the construction of a “class of equivalent classes” as Frege and B. Russell thought, or whether the construction is more complex and presupposes the concept of order.

The modern information age would never have been possible without the work of the great logician Frege. Female suffrage was taken seriously only after Wollstonecraft. The Enlightenment stood in need of a Voltaire, Einstein needed Newton and Newton, in turn, relied on Aristotle. The history of social, political and technological change is inextricably bound to the history of thought.

That’s the rankest psychologism, and was conclusively revealed as hogwash by Gottlob Frege in the 1890s!

Decidability is impossible. We are back in the land of paradox, with Epimenides declaring that he is a liar and Bertrand Russell upsetting Frege's applecart

Although philosophers generally consider Gottlob Frege to have dealt the death blow to a conceptualist form of realism, Frege’s objections to human psychologism—such as the intersubjectivity, necessity, and plenitude of mathematical objects—do not touch divine psychologism. That Frege could simply overlook what has historically been the mainstream theistic position with respect to putative abstract objects is perhaps testimony to how utterly detached 19th century philosophical thinking had become from the historic Christian tradition. With the late twentieth century renaissance of Christian philosophy divine conceptualism is once more finding articulate defenders.

Although philosophers generally consider Gottlob Frege to have dealt the death blow to a conceptualist form of realism, Frege’s objections to human psychologism—such as the intersubjectivity, necessity, and plenitude of mathematical objects—do not touch divine psychologism. That Frege could simply overlook what has historically been the mainstream theistic position with respect to putative abstract objects is perhaps testimony to how utterly detached 19th century philosophical thinking had become from the historic Christian tradition. With the late twentieth century renaissance of Christian philosophy divine conceptualism is once more finding articulate defenders.

Um nur das hier zunächst Liegende zu berühren, sehe ich ein großes Verdienst Kants darin, daß er die Unterscheidung von synthetischen und analytischen Urteilen gemacht hat. (Grundlagen der Arithmetik, §89)

Central to Frege's philosophy was the assertion that truth is independent of human judgment. In his Basic Laws of Arithmetic he writes: "Being true is different from being taken to be true, whether by one or many or everybody, and in no case is it to be reduced to it. There is no contradiction in something's being true which everybody takes to be false. I understand by 'laws of logic' not psychological laws of takings-to-be-true, but laws of truth...they [the laws of truth] are boundary stones set in an eternal foundation, which our thought can overflow, but never displace.

That Logic was invented by a philosopher is a significant fact. Many a profession could claim the indispensability of clear thinking for sound practice. So why was logic not invented by an admiral or a general, or by a physician or a physicist? Why indeed was logic not invented by a mathematician: why is Aristotle not the Gottlob Frege of the ancient world? Logos is nothing if not a corrective to common sense. Logos has an inherent obligation to surprise. It began with the brilliant speculations of the Pythagoreans-- the original neopythagoreans, as one wag has put it--with regard to a number theoretic ontology. Apart from the physicists, the great majority of influential practitioners of logos before Plato allowed logos to operate at two removes from common sense. The first was the remove at which speculative science itself would achieve a degree of theoretical maturity. But the second remove was from science itself. The first philosophers were unique among the practitioners of logos in that they created a crisis for logos. In the hands of the sophists, philosophy had become its own unique problem. It was unable to contain the unbridled argumentative and discursive fire-power of logos. In fact, philosophy has had this same sort of problem--the problem of trying to salvage itself from its excesses--off and on ever since. Thus, logic was invented by a philosopher because it was a philosopher who knew best the pathological problematic that philosophy had itself created. -Eds. Dov Gabbay & John Woods. (2004) John Woods & Andrew Irvine. "Aristotle's Early Logic." Handbook of the History of Logic, Volume 1: Greek and Indian Logic. PP. 27-100.

In philosophy we must always resist the temptation of hitting on an answer to the question how we can define such-and-such a notion, an answer which supplies a smooth and elegant definition which entirely ignores the purpose which we originally wanted the notion for.

[I]t is here that geometry and philosophy come closest together. In fact they belong to one another. A philosopher who has nothing to do with geometry is only half a philosopher, and a mathematician with no element of philosophy in him is only half a mathematician. These disciplines have estranged themselves from one another to the detriment of both.

The more I have thought the matter over, the more convinced I have become that arithmetic and geometry have developed on the same basis —a geometrical one in fact— so that mathematics in its entirety is really geometry.

From the geometrical source of knowledge flows the infinite in the genuine and strictest sense of this word. […] We have infinitely many points on every interval of a straight line. […] We cannot imagine the totality of these. […] One man may be able to imagine more, another less. But here we are not in the domain of psychology, of the imagination, of what is subjective, but in the domain of the objective, of what is true.

[I]t is evident that sense perception can yield nothing infinite. However many stars we may include in our inventories, there will never be infinitely many. […] For this we need a special source of knowledge, and one such is the geometrical.

The question why and with what right we acknowledge a law of logic to be true, logic can answer only by reducing it to another law of logic. Where that is not possible, logic can give no answer.

Where both [Frege and Husserl] failed was in demarcating logical notions too strictly from psychological ones… These failings have left philosophy open to a renewed incursion from psychology, under the banner of ‘cognitive science’. The strategies of defence employed by Husserl and Frege will no longer serve: the invaders can be repelled only by correcting the failings of the positive theories of those two pioneers.

In order to produce it [an infinite series] we would need an infinitely long blackboard, an infinite supply of chalk, and an infinite length of time. We may be censured as too cruel for trying to crush so high a flight of the spirit by such a homely objection; but this is no answer.

Consider propositions: the things that are true or false, that are capable of being believed, and that stand in logical relations to one another. They also have another property: aboutness or intentionality. Propositions represent reality or some part of it as being thus and so. This crucially connected with their being true or false. Many have thought it incredible that propositions should exist apart from the activity of minds. How could they just be there, if never thought of? (Frege, that alleged arch-Platonist, referred to propositions as gedanken, connected with intentionality.) Representing things as being thus and so, being about something or other--this seems to be a property or activity of minds or perhaps thoughts . It is extremely tempting to think of propositions as ontologically dependent upon mental or intellectual activity in such a way that either they just are thoughts, or else at any rate couldn't exist if not thought of. (According to the idealistic tradition beginning with Kant, propositions are essentially judgments.) But if we are thinking of human thinkers, then there are far to many propositions: at least, for example, one for every real number that is distinct from the Taj Mahal. On the other hand, if they were divine thoughts, no problem here. So perhaps we should think of propositions as divine thoughts. Then in our thinking we would literally be thinking God's thoughts after him.

Consider propositions: the things that are true or false, that are capable of being believed, and that stand in logical relations to one another. They also have another property: aboutness or intentionality. Propositions represent reality or some part of it as being thus and so. This crucially connected with their being true or false. Many have thought it incredible that propositions should exist apart from the activity of minds. How could they just be there, if never thought of? (Frege, that alleged arch-Platonist, referred to propositions as gedanken, connected with intentionality.) Representing things as being thus and so, being about something or other--this seems to be a property or activity of minds or perhaps thoughts . It is extremely tempting to think of propositions as ontologically dependent upon mental or intellectual = activity in such a way that either they just are thoughts, or else at any rate couldn't exist if not thought of. (According to the idealistic tradition beginning with Kant, propositions are essentially judgments.) But if we are thinking of human thinkers, then there are far to many propositions: at least, for example, one for every real number that is distinct from the Taj Mahal. On the other hand, if they were divine thoughts, no problem here. So perhaps we should think of propositions as divine thoughts. Then in our thinking we would literally be thinking God's thoughts after him.

So the result seems to be: thoughts are neither things of the outer world nor ideas. — A third realm must be recognized. (Frege, 1918, p.302)

The Monday meetings were in a sense a tug-of-war, in which the Schlick faction—supposedly in the name of his mentors, Gottlob Frege and Bertrand Russell—sought to drag their master over the demarcation line of the “verification criterion” (Schlick: “The meaning of an assertion lies in the method of its verification”), while a famously indefatigable Wittgenstein held his ground at the other end of the rope with Schopenhauer, Tolstoy, and Kierkegaard, waiting for the whole positivist troop to collapse.

Gottlob Frege’s Begriffsschrift.

..First, we need to get clear on the subject matter of mathematics. What is mathematics about? Is it really concerned with abstract objects..? We obviously need to listen to what mathematics itself has to say.. As Frege emphasized, however, the questions are also in part concerned with language. How should the language of mathematics be analyzed? Should apparent talk about numbers and sets be taken at face value? This concern with language means that we shall also need assistance from linguistics and perhaps also psychology. Second, we need to understand how mathematicians.. settle on their first principles (or axioms), and how do they use these to prove mathematical results (or theorems)? .. The challenge is to make our answers to these two sets of questions mesh. How is it that our ways of forming mathematical beliefs are responsive to what mathematics is about? How are the practices and mechanisms by which we arrive at our mathematical beliefs conducive to finding out about whatever reality mathematics describes? In short, why is it not just a happy accident that our mathematical beliefs tend to be true? There must be something about what we do that keeps us on the right track., Since the challenge is to integrate the metaphysics of mathematics (namely, what mathematics is about) with its epistemology (namely, how we form our mathematical beliefs), we shall call this the integration challenge.

A conception of a cognitive capacity can qualify as unrestricted in aspiration and yet be insufficiently capacious in conception. A conception of a capacity, in aspiring not to go outside the order to which the capacity belongs so as to explain the capacity, may unwittingly frame its conception of the target capacity in terms that sever it from the conditions required for its genuine possession. This is a difficult balance to strike correctly in philosophy. Frege is concerned with not admitting anything psychological into his conception of the logical. This is the mark of the unrestrictedness of his aspiration - his refusal to admit anything external to the order of logic in his account of logic. But he builds his guardrail of protection against falling into the psychological sufficiently far in from the actual danger point that he severs the unity of our capacity for knowledge. Hence the need for a de-psychologizing of Frege's conception of the psychological.

Kant already taught—and indeed it is part and parcel of his doctrine—that mathematics has at its disposal a content secured independently of all logic and hence can never be provided with a foundation by means of logic alone; that is why the efforts of Frege and Dedekind were bound to fail. Rather, as a condition for the use of logical inferences and the performance of logical operations, something must already be given to our faculty of representation, certain extralogical concrete objects that are intuitively present as immediate experience prior to all thought.

In Frege’s conception of logic, a logical law states an absolutely general truth—one whose truth every rational being must, on pain of contradiction, acknowledge. In later Wittgenstein’s practice, a grammatical remark inherits an aspect of Frege’s conception of the logical. On a proper understanding of a grammatical remark, it articulates a truism— something that admits of no contrary—hence something that every speaker of the language must acknowledge. Or conversely, if there is something in a given candidate grammatical remark that proves to admit of disagreement, then the remark in question cannot serve its methodological role. It fails to bring into view a point of (what later Wittgenstein calls) grammar. Grammatical remarks acquire their point—that is, our need for such reminders derives—from our attempting but failing to achieve a proper reflective understanding of our way around our own language. If the grammatical remark serves its purpose, what is thereby acknowledged is something that can come into view only against the background of a prior failed attempt to achieve a perspicuous overview of our own concepts. A Wittgensteinian grammatical remark comes to life as such only against the background of a philosophical confusion. Logic or grammar for later Wittgenstein, pace Frege, could qualify as a science only if philosophy is one. This also means that, for later Wittgenstein, unlike for Frege, there is no preexisting stock of propositions that constitutes all of the logico-grammatical truths there are. In potentiality there are perhaps indefinitely many, but in actuality the only remarks that actually exercise the power to disclose a philosophico-grammatical truth, for later Wittgenstein, are those that allow us to make progress with the problems that actually vex us in philosophy.

How does one do justice to what occasions philosophical wonder in us without conferring false sublimity upon it? We said that what occasions Frege’s wonder—the absoluteness of the logical order—seems to him to be such that it cannot possibly be implicated in our dependence upon language: say, in our meaning to assert p in using a proposition to say one thing rather than another, or in our using just these words rather than some others to assert it. The Tractatus (while repudiating Frege’s conception that the nature of logic may in no way be implicated in that of language) still seeks a way to hold onto the idea that in logic it is not we who express, by means of signs, what we want; rather it is the nature of the essentially necessary signs—it is logic—that asserts itself. The later Wittgenstein, as we are about to see, seeks to undo this residual subliming of the logical in the Tractatus, while in no way seeking to dissipate the sense of wonder at the illimitable depth of the logical—(what he later calls) the grammatical—that shows itself in our forms of thought and life.

For Frege, an account of what it is for a purely logical power to be in act suffices to allow us to achieve a proper philosophical appreciation of what “content,” “object,” “thought,” “judgment,” and “truth,” as such, are. These notions come to be fully in place through an elucidation of that power, considered apart from our capacity to arrive at kinds of knowledge that are not purely logical in content. Our capacity for empirical judgment, when it comes into view, will come into view as a comparatively complex joint exercise of a variety of faculties, in which the logically fundamental notions that figure in its explication (“content,” “object,” thought,” “judgment,” “truth”) are still supposed to retain the specific sense originally conferred upon them in our explication of the purely logical case, while allowing for their extension to logically impure cases of thought and proposition. A certain picture of the role of reflection on the purely logical case, inthe order of explication of kinds of knowledge, is at work here—a picture that has been enormously influential on the subsequent development of analytic philosophy. On this picture, only if we are armed with a prior account of the case of purely logical thought, supplementing it as we go along, can we come to understand what empirically contentful theoretical thought (or practical thought) is. On this picture, the spatiotemporal bearing and the self-consciousness of the thinking subject do not belong to the form of thought (and hence their treatment does not belong, as Kant held, to a suitably capacious conception of philosophical logic); rather, all such further details among various species of thought are to be subsequently specified, if at all, through the introduction of further indices figuring within the content of thought. (Thoughts are simply conceived of as occurring at a time or at a person.) These consequences of the Fregean picture are not, on the whole, something for which post-Fregean analytic philosophers argue. Rather, it involves an entire philosophical picture that is simply tacitly, and largely unwittingly, assumed—a picture that is already under attack, albeit in very different ways, in both Kant and early Wittgenstein. According to this post-Fregean picture, we can furnish an account of the wider reaches of our capacity for finite theoretical cognition only by assuming the prior intelligibility of some self- standing account of how one of the ingredient capacities in empirical cognition—the capacity for logical thought—off its own bat is able to yield a delimitable sphere of truth-evaluable, object-related thoughts with judgable content, without its yet having entered into any form of co- operation with our other cognitive capacities.