All too often we find that we lack a means by which to avoid misunderstandings in abstract areas of the sciences, a deficiency which may also lead to errors in our own thinking. This is due to the imperfection of language, and yet, we rely on external signs for thinking.
It is in our nature to direct our attention outwards. Sense impressions surpass memory-images in their vividness—so much so that, initially, they almost exclusively determine the sequence of our ideas (as they do in animals). Seldom would we break free from the grip of sense perception if we did not possess some degree of control over the external world. Most animals have an influence on their sense impressions: being able to move, they can flee some impressions and seek out others. They can also affect change in their environment. Of course, human beings have this ability to a far greater extent, but this is not yet sufficient for the sequence of our ideas to gain complete freedom from sense impressions. Without the great invention of signs—which make present what is absent, invisible, perhaps non-sensory—we would be limited to what our hands can form and what our voices can sound.
I admit that, while perceiving something, we can assemble a sphere of memory-images without the use of signs. However, we can’t retain these images for long. A new perception arises: previous images sink back into night and new ones surface. However, when we are reminded of an idea through perception, we can produce the sign for it, and in doing so create a new fixed center around which ideas are grouped. Among these, we again choose one in order to produce its sign. Step by step we penetrate the inner world of our imaginations and move around in it as we like, using the sensible itself to free ourselves from its coercion. Signs have the same significance for thinking as the invention of using the wind to sail against the wind has for navigation. Therefore, we shouldn’t look down upon signs! So much depends on how they are chosen. After long training we no longer need to produce signs externally or speak them out loud in order to think. However, this does not diminish the value of signs. After all, we think in words (and if not in words, then in mathematical notation or in other signs).
Without signs, we would scarcely rise to conceptual thinking. When we assign the same sign to different but similar things, we do not signify the particular things, but what the particular things have in common—the concept. And this we gain only by signifying it; since it is not itself given in intuition, it needs a perceptible representative in order to be able to appear to us. In this way, the senses open up the world of the non-sensual.
The merits of signs are not hereby exhausted, but this may suffice to demonstrate that they are indispensable.
However, language turns out to be defective when it comes to preventing errors in thought. It doesn’t even satisfy the first requirement one has to demand from it in this respect, which is to be unambiguous. The most dangerous are those cases in which the meanings of a word are only a little different, with slight, but not insignificant variations. Many such ambiguities recur. For instance: the same word is used to signify a concept and a single instance of it. Generally, there is no differentiation between a concept and a single instance of it. “The horse” can signify the individual, but it can also refer to the species, as in the sentence: “The horse is a plant-eating animal.” Finally, “horse” can signify the concept, as in the sentence: “This is a horse.” Language is not controlled by logical laws in such a way that correct grammar usage necessarily guarantees the formal correctness of thought. The forms in which deduction are expressed are so diverse, so loose, and so flexible that premises can easily sneak in without being noticed. These are then left out of the list of the necessary conditions for the validity of the conclusion. In this way, the conclusion will acquire a higher generality than it deserves by right. Even an author as scrupulous and rigorous as Euclid often implicitly uses premises which he neither lists among his axioms nor among the preconditions of the specific theorem.
For example, in the proof of the 19th theorem in Book I of the Elements (in every triangle, the larger side is opposite the larger angle) he implicitly uses the following statements:
1) If a line isn’t longer than another one, it is shorter or equal to the first.
2) If an angle is equal to another one, it is not larger than the latter.
3) If an angle is smaller than another one, it is not larger than the latter.
However, the reader will notice the omission of these statements only when paying particular attention, especially since they seem to be so near to the laws of thinking themselves in their naturalness that they are used like those themselves. A rigorously defined set of forms of inference simply does not exist in language. Therefore, we cannot distinguish a sequence of inferences without gaps from one in which intermediate steps are omitted. One could say that such an uninterrupted sequence hardly ever occurs in language; such strictness runs contrary to our intuitive use of language because it entails an unacceptable verbosity.
The written word has only one advantage over the spoken word, that of duration. We can review a line of thought several times without fear of it changing, making it possible to check its consistency more thoroughly. Since linguistic writing systems do not in themselves provide a guarantee of logical clarity, the rules of logic are applied extrinsically, as a guideline. But even so, mistakes can easily escape the eye of the verifier, especially those that arise from slight differences in the meaning of a word. Despite this, we manage to get by both in everyday life and in academic disciplines thanks to the many means of verification at our disposal most of the time. Experience and spatial intuition save us from many mistakes. The rules of logic, however, provide little protection, as is evidenced from such areas in which the means of verification are beginning to fail. These rules have not saved even great philosophers from committing errors, nor have they kept higher mathematics free of errors in all cases because the rules always remain extrinsic to the content.
The highlighted shortcomings are due to a certain softness and mutability of language. On the other hand, it is precisely this softness and mutability that gives language its versatility and its ability to develop. In this respect, language can be compared to our hands’ ability to adapt themselves to diverse tasks. However, our hands are insufficient for us, so we created artificial hands—tools for special purposes—and these allow us to do the precise and detailed work which our hands would not be able to do.
This exactness is facilitated precisely through rigidness, the immutability of the tool’s components. It is the absence of this rigidness that allows our hands to be so versatilely skilled, but imprecise.
Likewise, verbal language lacks precision. We need a system of signs from which any ambiguity is dispelled, from whose strict logical form the content cannot escape.
The question now is whether signs for the ear or for the eye are to be preferred. The generation of the former depends more on external circumstances, and in that it has an advantage. Moreover, the closer affinity of sound to inner processes can be brought to bear. In both cases, the form of appearance is that of a temporal sequence, and both are equally transient. Sound has a deeper connection to emotion than shape and color does, and the human voice in its infinite pliability is able to convey the finest blends and variations of feeling. But no matter how valuable these advantages might be for other purposes, they are irrelevant for the stringency of deductions. Maybe this close affiliation of audible signs to the corporeal and psychic conditions of reason just entails the disadvantage of keeping the former more dependent on the latter.
The properties of the visible, especially of shapes, are by nature different. Generally, they are sharply defined and clearly distinguishable. The definiteness of the written sign will cause the signified to take a more defined shape as well. This kind of effect on ideas is what we want in precise deduction. But it can only be achieved if the sign directly signifies its object.
A further advantage of the written is its longer duration and immutability. In this it more closely resembles concepts as they are supposed to be, and the less it resembles the restless flow of our real movement of thought. Writing offers the possibility of keeping many things present at the same time. And even if at any moment we can only bear in mind a small part of those things, we still retain a general impression of the rest as well, and this will immediately be at our disposal once we need it. We can use the positional relationships of the symbols on the two-dimensional writing surface in a far more diverse way to express internal relationships than the mere following and preceding in one-dimensional time, and this makes it easier to find whatever we want to devote our attention to at a given moment. And indeed, the simple sequence doesn’t correspond to the multitude of logical relations by which thoughts are connected to each other.
So it turns out that these very properties by which writing distances itself further from the progression of ideas are most suitable to compensate for certain shortcomings of our dispositions. If the aim is not to represent natural thought as it has evolved in the interplay with verbal language, but to compensate for the one-sidedness that results from its close affiliation to the sense of hearing, then writing will have to be preferred to sound. In order to make use of the specific advantages of visible signs, such a notation has to be totally different from all verbal languages. It hardly needs mentioning that these advantages barely surface in verbal writing systems. The relative position of words on a writing surface mainly depends on the length of the lines and is, in this respect, insignificant. But there are already other types of notation which make better use of the advantages of visible signs. The arithmetic formula notation is a conceptual notation since it expresses the content directly, without the mediation of sounds. As such, it achieves the brevity that allows for the content of a simple proposition to be expressed in a single line. Such contents – in this case equations or inequalities – are written one below the other according to the sequence in which they are derived from each other. If one follows from two others, the third is separated from the other two by a horizontal line that can be translated as “therefore.” In this way, the two-dimensionality of the writing surface is utilized to gain clarity. Deduction here is very uniform and almost always is based on the fact that equal changes applied to equal numbers lead to equal results. This is certainly not the only means of deduction in arithmetic.But if the logical argument progresses in a different way, in most cases it will be necessary to express it with words. So the arithmetic formula notation lacks expressions for logical operations and therefore, it does not deserve the name of a conceptual notation in the full sense. In the notation for logical operations going back to Leibniz and renewed in recent times by Boole, R. Graßmann, St. Jevons, F. Schröder and others it’s the other way around. Although here we have the logical forms, albeit not quite complete, the content is missing. Any attempt to replace the simple letters with expressions of content here, e.g. analytical equations, would demonstrate by lack of clarity the low suitability of this notation for the formation of a true conceptual notation, for the resulting formulas would be clumsy, even ambiguous. Of a conceptual notation, I would require the following: It must have simple ways of expressing logical operators which, being restricted to the minimum number necessary, can be easily mastered. These forms must be suitable to blend with the content in the closest way. Hereby we should strive for brevity such that the two-dimensionality of the writing surface can be used to achieve clarity of representation. The signs signifying content are less essential. Once the general forms are present, such signs can easily be created as necessary. If we do not succeed in dissecting a concept into its last components or if it does not appear to be necessary, we can make do with preliminary symbols.
One might worry unnecessarily about the feasibility of the matter. It is impossible, one might say, to invent a conceptual notation in order to promote science because inventing the former requires the latter to exist already. However, the same difficulty also arises in the case of language. Language was supposed to have facilitated the development of reason, but how could human beings have created language without reason? In order to investigate the laws of nature, physical devices are used. These can only be created by an advanced technology which in turn is based on knowing the laws of nature. In each case, the circle is solved in the same way. A progress in physics causes a progress in technology and this in turn enables us to build devices which promote physics. The application to our case is obvious.
I have attempted to supplement mathematical symbolic notation with signs for logical relationships. Primarily, this is supposed to yield a conceptual notation for the area of mathematics, which I have demonstrated to be desirable. The use of my signs for other areas is not thereby excluded. Logical relationships recur everywhere and signs for special types of content can be chosen in such a way that they fit into the framework of the conceptual notation. Whether this happens now or not, in any case an intuitive representation of forms of thought has significance beyond mathematics. May philosophers pay some attention to the matter as well!
 Non inelegans specimen demonstrandie in abstractis. Erdm. S. 94.
 Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens. Halle a. S., 1879.
Copyright for the English translation © Andreas Keller and Tina Forsee, 2016, all rights reserved.
- The text is a translation (version 2.0) of:
Gottlob Frege: Ueber die wissenschaftliche Berechtigung einer Begriffsschrift, Zeitschrift für Philosophie und philosophische Kritik, 81, 1882, p. 48 – 56.
Reedited in: Gottlob Frege, „Begriffsschrift und andere Aufsätze“, Zweite Auflage, Mit. E. Husserls und H. Scholz‘ Anmerkungen herausgegeben von Ignacio Angelelli, 1964, Wissenschaftliche Buchgesellschaft Darmstadt. p. 106 – 114.
The German text has also appeared in:
Patzig, G. (ed.), 1962, Funktion, Begriff, Bedeutung: Fünf logische Studien, Göttingen: Vandenhoeck & Ruprecht.
- You can find the German text (based on Angelelli’s edition), here: https://editionnannus.wordpress.com/2016/01/03/gottlob-frege-ueber-die-wissenschaftliche-berechtigung-einer-begriffsschrift/
Since Frege died in 1925, the German text is in the open domain.
- There are two footnotes in the original text. In the original text, these are in both cases indicated by “*)” in the text and an asterisk on the same page, marking the footnote. Since the text given here in the internet does not preserve the original page divisions, we have replaced the asterisks with numbers and put the footnotes at the end of the text, so there is now a footnote 1 and a footnote 2. Footnote 1 refers to a fragment by Leibniz edited by Johann Eduard Erdmann. See also Peckhaus, Volker, “Leibniz’s Influence on 19th Century Logic”, in:The Stanford Encyclopedia of Philosophy(Spring 2014 Edition), Edward N. Zalta (ed.), URL = <http://plato.stanford.edu/archives/spr2014/entries/leibniz-logic-influence/>.
- There are two different English translations already, one by Bartlett (1964) and one by Bynum (1972), see http://plato.stanford.edu/entries/frege/catalog.html. We have not used any of these translations for preparing our own. A comparison of our translation with these previous ones has not yet been done, but we are planning to do so.
- The current translation replaces a previous one, prepared by Andreas Keller, published here before. The current translation is the result of a joint effort by Andreas Keller and Tina Forsee, who are native speakers of German and English, respectively. Some mistakes in the previous translation have been removed and the style and language of the text has been improved.
- Some conceptual differences between the two languages provided some special challenges for the translation. One is the term “Wissenschaft” which has no direct counterpart in the English language. It is often translated as “science”, but has a wider meaning than the English term, comprising not only the sciences but also mathematics, philosophy, and the humanities. We translated it as “science” (and the reader should bear the wider meaning of the German term in mind here) and also as “academic discipline”.
Another term used by Frege that has no direct counterpart in English is the word “anschaulich”. We translated it as “intuitive”. The German term “anschaulich”, derived from the verb “anschauen” (to look at) and related to Kant’s term “Anschauungsformen” (“forms of intuition”) means that something is intuitively understandable. For example, three-dimensional shapes are “anschaulich”, i.e. we can imagine them, but four-dimensional shapes are “unanschaulich” (non-anschaulich): we can calculate them but we cannot intuitively imagine them since our ability to imagine shapes is limited to three dimensions. Likewise, many abstract concepts are “unanschaulich”. The word “intuitive” in our translation is meant as a translation of this concept, so it relates to the common translation for Kant’s “Anschauungsformen”. Other uses of the word “intuitive”, especially emotional connotations, are not meant here.
- Frege’s “Begriffsschrift”, a notation for predicate calculus, is a two-dimensional, tree-like notation. The following picture is an example (pages 58/59 of “Begriffsschrift”, as reprinted in Angelelli’s edition (see above)). It is reproduced here so that the reader can better understand Frege’s discussion of the use of the two-dimensional writing surface. This notation did not really catch on. There are several notations in use today, all of them less graphical and more linear. The difficulties of typesetting such a notation and the use of typewriters in the preparation of scientific and mathematical papers during most of the 20th century might have contributed to this fact.