Gottlob 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.

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.

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

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.

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

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.

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.

[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.

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.

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.