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I'm not claiming that words are superior to symbols in all respects. I'm just curious if there is a school of thought arguing that natural language has more use than mathematics or formal language in discovering truth.

  • You will find a lot of natural language in the mathematical literature -- many standardized words and phrases, but nevertheless. Publishers don't like to see pages and pages of nothing but mathematical symbols. It's just too hard to read. – Dan Christensen May 21 '15 at 15:57
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    "Discovering Truth" and "Explaining things" are pretty vague, even by philosophy of language standards. What work do you see symbols and mathematics currently doing that you might like to substitute in spoken words for? Is this specifically in reference to formal theories of Truth, are you interested in the philosophy of Logic, or is this a more general species of explanation? Perhaps it might be of interest to read the SEP article on Scientific Explanation: plato.stanford.edu/entries/scientific-explanation – Paul Ross May 22 '15 at 10:32
  • i doubt it - how do you "explain something" without using formal languages. try it and then compare with science – user6917 May 23 '15 at 10:13
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    @MATHEMATICIAN Mathematics was successfully practiced for centuries before formal languages were invented, and even today they are largely ignored in most of mathematics. Formal languages explain nothing, that is not what they are for, which is why proofs in mathematical papers are written in a natural language. Formulas are occasionally inserted to monitor for rigor, or to get over opaque parts by brute force. The formal parts require the most effort from the reader to be understood, and above a certain threshold make the text unreadable. – Conifold May 23 '15 at 22:02
  • i don't actually know if mathematics doesn't always meet a definition of formal language; i'm sorry i seemed to have failed to communicate with you oops – user6917 May 23 '15 at 22:21
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There must be applicable quotes from (the later) Wittgenstein.

E.g. the following one seems to argue that Hegel would be interested in categories of things (and perhaps denote them by symbols like X) whereas Wittgenstein would doubt such generalization (and perhaps describe family resemblances using ordinary sentences and words instead).

Hegel seems to me to be always wanting to say that things which look different are really the same. Whereas my interest is in showing that things which look the same are really different. I was thinking of using as a motto for my book a quotation from King Lear: 'I’ll teach you differences."

Here is a similar argument from Nietzsche's On Truth and Lies in a Nonmoral Sense:

The very concept arises from the equation of unequal things. Just as it is certain that one leaf is never totally the same as another, so it is certain that the concept ”leaf” is formed by arbitrarily discarding these individual differences and by forgetting the distinguishing aspects. This awakens the idea that, in addition to the leaves, there exists in nature the ”leaf”: the original model according to which all the leaves were perhaps woven, sketched, measured, colored, curled, and painted–but by incompetent hands, so that no specimen has turned out to be a correct, trustworthy, and faithful likeness of the original model.

Ludwig von Mises' description of economic behavior in Human Action also comes to mind: if memory serves he did not use a single formula (and very few symbols such as X) in a long book whose subject area (i.e. economics) would seem to offer many opportunities for symbolic notations. I am also curious whether he was influenced by a (philosophical) school in making this choice.

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Well, there is Heidegger; he wrote a book called Language, thought and Poetry; which was probably inspired by his introduction to the poetry of Holderlin; he also wrote in an essay that 'language is the house of being; within which humans dwell'.

This is very different from Wittgenstein approach to language where he attempted to reduce it to logic.

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Heidegger and the continental tradition that followed him comes to mind, many of them denounce not just the symbology of mathematics but "exact thinking" and science in general as antithetical to "discovering the truth" in their sense. This position is related to what Snow calls "the two cultures" divide. For more see Friedman's Parting of the Ways. In 1943 postscript to What is Metaphysics Heidegger writes:

"The suspicion directed against "logic," whose conclusive degeneration may be seen in logistic [modern mathematical logic], arises from the knowledge of that thinking that finds its source in the truth of being, but not in the consideration of the objectivity of what is. Exact thinking is never the most rigorous thinking, if rigor receives its essence otherwise from the mode of strenuousness with which knowledge always maintains the relation to what is essential in what is. Exact thinking ties itself down solely in calculation with what is and serves this exclusively."

Objections to oversymbolization and formalization of mathematics were raised even by some prominent mathematicians, notably Poincare and Gödel. Poincare wrote things like "when someone does not understand a problem he writes a lot of formulas, as the understanding improves formulas are replaced with words", and mocked Peano's symbolic proof with "if it takes four pages to prove that 1 is a number how long would it take to prove a really hard theorem" (quoted from memory). As Heinzmann puts it "he doesn't understand the new logic, because — I'll argue — he wants to understand mathematics". But Poincare is no opposed to "happy innovations of language" in mathematics, he even praises it:

"Mathematics is the art of giving the same name to different things... When language has been well chosen, one is astonished to find that all demonstrations made for a known object apply immediately to many new objects: nothing requires to be changed, not even the terms, since the names have become the same."

  • Great answer, just want to note that the opposition to oversymbolization is basically Occam's razor. And obviously Occam's razor applies to non-formal arguments as well. The problem seems to be that the incentive to obfuscate is usually stronger in formal arguments, which makes it a practical issue but not a methodological one. – Yang May 26 '15 at 13:25
  • Good answer. It seems many people think mathmatics is some kind of standalone outside object which is completely unrelated with human beings' ( thoughts ) themselves. – Kentaro May 30 '15 at 8:24
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Words aren't superior or inferior to symbols -- words are symbols. "Natural language" is a systematic arrangement of symbols, just like formal logic or mathematical equations are. I think that understanding when natural language should be your "symbology" of choice over mathematics or formal language is pretty intuitive.

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    Can you provide more justification for this position? Not necessarily saying it's wrong, just needs more explaining. – James Kingsbery May 21 '15 at 20:42
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The entire non-analytic tradition ("continental") qualifies on the basis of how they choose to write themselves. Wittgenstein is sometimes classified as a "ordinary language" philosopher, but I don't think he would believe in the "primacy of words", just that some words (ordinary, intuitive) are better than others (invented, formalized). In fact, he would probably not agree to a distinction between words and symbols at all.

von Mises as suggested by another answerer is a more tricky case. He did not use math or formal symbolism (and all evidence was that he probably would have been very poor at it), but believed extremely strongly in solely the use of a priori deductive reasoning in theoretical exploration. This makes his approach equivalent to the use of symbolism, since the definitions and axioms do not have to be related in any way to actual experience or "the real world", making them essentially symbolic placeholders in a theoretical argument.

More generally, in philosophy of social science, there continues to be a debate (at times heated) about the importance of mathematical formalism for explanation. It is generally accepted now that quantitative approaches are important for empirical fact-finding, but the role of formal logic in theoretical exploration is still a contentious issue. A Model Discipline by Primo and Clarke is a very nice, middle-of-the-way treatment of this issue.

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If I must answer literally, I think yes. Kindly refer to the below link. Already back in late 19th, there was someone ( please take a look at the link. )

As I answered here,

According to my citation ( please refer to the link )

The concepts of number and figure have not been derived from any source other than the world of reality. The ten fingers on which men learnt to count, that is, to perform the first arithmetical operation, are anything but a free creation of the mind. Counting requires not only objects that can be counted, but also the ability to exclude all properties of the objects considered except their number — and this ability is the product of a long historical development based on experience. Like the idea of number, so the idea of figure is borrowed exclusively from the external world, and does not arise in the mind out of pure thought.

So, very long time ago, when I personally ( or anybody? -:) ) do not know, but possibly and may can go back to when as is in a movie 2001 A Space Odyssey, when an ape learned that a part bones of an animal can break another material so that it indicates the ape learned how to use the tool, then in only analogy, if we the human beings at some stage learned how to count using its own body, then can we guess the word, which corresponds to the notion of counting the outside object, can simutaneously might have arisen at that same time? Especially if we consider human beings as the social beings???

Sorry for my poor knowledge, but my thought is, how can a man, when he or she learned to count, because he is a social being, meaning he needs to communcate his idea ( = he learned how to count ) to another human being?

For example, it is not so much hard for me to guess, when a Chinese, learned to count the outside object with his index finger or which finger. Then simutaleously, did he not attempt to draw a line, using the same hand, in a case of Chinese,  , ( in case of 2,   ), both of which are simple lenear line or a set of it.???

  • I don't edit your posts for English. Please have a look at the changes. In the future, please make quotes with > instead of >>, and don't add irrelevant text. This is in line with the SE standards. Thanks. – user2953 May 30 '15 at 8:45
  • I apologize for it. – Kentaro May 30 '15 at 8:46

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