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Gottlob Frege (1848 – 1925) was the German founder of analytic philosophy of language and mathematics. Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician.

Gottlob Frege

Gottlob Frege, philosopher

Quotations sources: WikiQuote, GoodReads, UCSD,


Quotes from Gottlob Frege

The ideal of strictly scientific method in mathematics which I have tried to realist here, and which perhaps might be named after Euclid I should like to describe in the following way:

It cannot be required that we should prove everything, for that is impossible; but we can demand that all propositions used without proof should be expressly mentioned as such, so that we can see distinctly what the whole construction rests upon.

We should accordingly strive to reduce the number of these fundamental laws as much as possible, by proving everything which can be proved.

Furthermore I demand — and in this I go beyond Euclid — that all the methods of inference used should be specified in advance. Otherwise is it impossible to ensure satisfying the first demand.

I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.

Facts, facts, facts,’ cries the scientist if he wants to emphasize the necessity of a firm foundation for science. What is a fact? A fact is a thought that is true. But the scientist will surely not recognize something which depends on men’s varying states of mind to be the firm foundation of science.

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

A scientist can hardly meet with anything more undesirable than to have the foundations give way just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was nearly through the press.

The aim of scientific work is truth. While we internally recognize something as true, we judge, and while we utter judgements, we assert.

It really is worth the trouble to invent a new symbol if we can thus remove not a few logical difficulties and ensure the rigour of the proofs. But many mathematicians seem to have so little feeling for logical purity and accuracy that they will use a word to mean three or four different things, sooner than make the frightful decision to invent a new word.

Your discovery of the contradiction caused me the greatest surprise and, I would almost say, consternation, since it has shaken the basis on which I intended to build my arithmetic…. It is all the more serious since, with the loss of my rule V, not only the foundations of my arithmetic, but also the sole possible foundations of arithmetic seem to vanish.

There is more danger of numerical sequences continued indefinitely than of trees growing up to heaven. Each will some time reach its greatest height.

Equality [1] gives rise to challenging questions which are not altogether easy to answer. Is it a relation? A relation between objects, or between names or signs of objects? In my Begriffsschrift [2] I assumed the latter. The reasons which seem to favor this are the following: a = a and a = b are obviously statements of differing cognitive value; a = a holds a priori and, according to Kant, is to be labelled analytic, while statements of the form a = b often contain very valuable extensions of our knowledge and cannot always be established a priori. The discovery that the rising sun is not new every morning, but always the same, was one of the most fertile astronomical discoveries. Even to-day the identification of a small planet or a comet is not always a matter of course. Now if we were to regard equality as a relation between that which the names ‘a’ and ‘b’ designate, it would seem that a = b could not differ from a = a (i.e. provided a = b is true). A relation would thereby be expressed of a thing to itself, and indeed one in which each thing stands to itself but to no other thing. What is intended to be said by a = b seems to be that the signs or names ‘a’ and ‘b’ designate the same thing, so that those signs themselves would be under discussion; a relation between them would be asserted. But this relation would hold between the names or signs only in so far as they named or designated something. It would be mediated by the connection of each of the two signs with the same designated thing. But this is arbitrary. Nobody can be forbidden to use any arbitrarily producible event or object as a sign for something. In that case the sentence a = b would no longer refer to the subject matter, but only to its mode of designation; we would express no proper knowledge by its means. But in many cases this is just what we want to do. If the sign ‘a’ is distinguished from the sign ‘b’ only as object (here, by means of its shape), not as sign (i.e. not by the manner in which it designates something), the cognitive value of a = a becomes essentially equal to that of a = b, provided a = b is true. A difference can arise only if the difference between the signs corresponds to a difference in the mode of presentation of that which is designated. Let a, b, c be the lines connecting the vertices of a triangle with the midpoints of the opposite sides. The point of intersection of a and b is then the same as the point of intersection of b and c. So we have different designations for the same point, and these names (‘point of intersection of a and b,’ ‘point of intersection of b and c’) likewise indicate the mode of presentation; and hence the statement contains actual knowledge. For more go to USCD.


COMMENTS on Gottlob Frege quotations

A fair presentation of Frege would be the last paragraph quoted above, but that style of material is beyond the scope of this survey called Philosophers Squared; and yet perhaps every one of the people quoted in this survey might be better represented by such much longer quotations.

It is said by the people that ought to know that Frege had tremendous influence on Bertrand Russell, Ludwig Wittgenstein, Rudolf Carnap  and other modern giants of philosophy.

Frege’s statement, Every good mathematician is at least half a philosopher, and every good philosopher is at least half a mathematician, makes the rest of us feel bad, because we can not attain to goodness without doing the math. And yet, after studying briefly over a hundred philosophers while doing this series, it seems to me more important that a living philosophy that leads to a good life is healthier, rather than the mathematically precise one that Frege pursues. Yes, they are different goals, but as we only live once as individuals we should strive to make our life as good as possible in our own terms. Alternatively, if we choose to live for humanity—that is for our DNA rather than our individual life—then it seems rational to spend one’s efforts maximizing the health and longevity of society. Socrates gave his life to enhance his city, and its laws, which seems reasonable by that second standard, but it appears many professional philosophers give their life to abstractions, which may be eternal, and yet they evaporate without a mind to observe them, and seem to do little to enhance their personal life, or the life of humanity. This abstract sentiment is observed in Frege’s statement, The aim of scientific work is truth. While we internally recognize something as true, we judge, and while we utter judgments, we assert. I am questioning if truth is a sufficient reason for life’s quest for the general populace, as it seems only philosophers and scientists praise it, seek it, and value it.