A language involves symbols and terms acceptable to people who use it. I find similar things in mathematics. Do you agree?
Dear Kiran,
Mathematical writing involves symbols but it also involves words.
Someone who has written about the similarity between mathematical learning and language acquisition is Seymour Papert in his book Mindstorms. This publication led to the LOGO programming and turtle graphics movement which was popular in primary schools in the 1980s and 1990s.
Mathematical is a language for science. It is famous the following quote from Galileo:
“Philosophy is written in this grand book, the universe which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the letters in which it is composed. It is written in the language of mathematics, and its characters are triangles, circles and other geometric figures without which it is humanly impossible to understand a single word of it; without these, one wanders about in a dark labyrinth.”
Mathematics is a language or at the very least a communication system. It is designed around symbols expressing concepts which is exactly what language is for. Mathematics is organised by a set of standardised rules just as spoken language. The concepts provide the foundation or explanation for ideas so yes, mathematics is a language.
This is a very good question. It is entirely appropriate to consider the similarities and differences between ordinary languages and mathematics.
In addition to the excellent answers already given, there is a bit more one might want to add.
It is true that mathematics can both be written and spoken. The writing of mathematics stems from the use of its symbols and terms as well as its axioms, theorems and proofs. In the right hands, mathematics leads to very meaningful speech that combines ordinary language such as French, Italian, English, Greek, Arabic or Turkish, Chinese or Hindi with a generous measure of spoken symbols, visualizations, examples based on specific terms, assumptions, assertions and proofs.
Here are a few differences between ordinary languages and the language of mathematics.
Symbols: Unlike ordinary language, the symbols of mathematics have sources from Greek and Arabic as well as from a long tradition of invented symbols such the integral and derivative (either the common Leibniz notation with elongated S for the integral the d-notation for derivatives or the less common Newton dot notation for derivatives), or the proximity relation symbol delta or the more recent annotated delta symbol with a London bridges superscript for strong proximity.
Visualization with Graphics and Geometry: Unlike ordinary language, mathematics has many sources of visualization of its ideas and structures, found in graphics and the rich landscape of geometry.
Technical terms: Unlike ordinary language, mathematics is awash with many beautiful technical terms that are explained with a combination of ordinary language and a mixture of symbols. Here is another source of a sharp contrast between ordinary language and mathematics, since ordinary languages is not composed of technical terms. See, for example,
https://www.researchgate.net/publication/276443539_The_Language_of_Mathematics_The_Importance_of_Teaching_and_Learning_Mathematical_Vocabulary
https://www.researchgate.net/publication/266998431_The_language_of_mathematics_Telling_mathematical_tales
Article The Language of Mathematics: The Importance of Teaching and ...
Book The language of mathematics. Telling mathematical tales
My age is twice your age.
Ordinary language: My age is twice your age.
Mathematical equation language:
y = 2x
where y is my age and x is your age
This can be further translated to a graph where x is in the axis of abscissa and y is in the axis of ordinate.
Dear Kiran, this is a very good question.
Attributes (alphabet, a set of rules, etc...) inherent in other languages, certainly included in mathematics. The main function of language is the transmission of information and its storage.
On the basis of the development of language we can say that the language of mathematics itself (as a system) accumulates information besides the fact that language is the carrier for. Like any language, math develops mainly in the interaction with the physical world. Despite the abstraction from the sense, most of its ideas (from the natural numbers) came as reflection of the physical objects. Apparatuses of mathematical theories has been developed mainly with jumps in the development of fundamental physical theories, and other branches of science. Mathematics is the strict formal system.
As for the term "strict"? The predictability of the appearance of symbols by using the language of mathematics is huge. The predictability of the appearance signs of alphabet with the message generation is directly related to the concept of entropy. It was developed with the development of information theory. But so far it is not applied directly to the analysis of mathematics, as a phenomenon.
What is the information?
Mathematics is more than a language since it has united people all over the world, even when some of them cannot understand, even do not need to study the mother tongues or languages of each other.
Moreover it is more than a language since a single Mathematical expression is used for different interpretations in different fields of arts, science, technology.
Even the NATURE uses the language of mathematics !
Mathematics is universal language, Vrajeshkumar. We are studying it.
Good question, Vasiliy: What is the information?
We can define it's quantity, but can not define it's sense.
Which sounds are understandable and which are not?
Not another, Mohamed. It is the only universal language. Can not be another matheamatics. All other languages have it's structure.
I agree with Eugene, mathematics is a universal language. Interestingly, it needs other languages to effectively communicate in it (mathematics).
Mathematics cannot express all that is important to express. We are here using English to talk about mathematics and almost nothing here that is said could be said using mathematics alone. All natural languages are much more universal than mathematics and can encompass aspects of the world that are totally impossible to express in the mathematics language. But the great strenght of mathematical language is its precision. Whatever is said in this language is true by definition and can be proven to be so. While watever I am saying here might be false and can be questioned and there is no way to proof beyond a doubt what I am saying in English. Mathematics is a dictatorial language in the sense that it is authoritative.
The question is old, and was extensively studied in the beginning of the XIX° century.
A language is a formal system which enables to share information, knowledge, and even emotions (one writes novels which are fiction). One wan expect from a language that it is rational (one cannot express statements which are not logical), precise, rich, aesthetic,...
Mathematics are different. It is comprised of several theories (mathematical logic, arithmetics, set theory, analysis,...) which are based on formal systems : a set of axioms define objects and their properties, and the theory of predicates (mathematical logic) enables then to buid rational narratives, such as demonstrations of theorems. The specificity of mathematics is not that it uses a precise and rational language (and indeed demonstrations in mathematical logic uses plain english), but it studies objects that it has invented. And these objects are not necessarily the representation of physical objects. The most obvious is the axiom of infinity : there are sets with an infinite number of elements. The issue then has been to build formal systems with as few objecy, that is as few axioms as possible. And the mathematicians were surprised to discover that, in order to have formal systems for theories such as arithmetics, there is not a unique set of axioms : one can add an infinite number of axioms to the system. They are not necessary... as of today, but perhaps one day we will find that they can be useful.
So from this point of view mathematics is not a language, this is much more, this is a science, which has the power to invent its own objects. And this was a genuine rupture with the Euclidean construct where lines, triangles,... are the idealization of real things. Today mathematical objects are nothing more than their definitions through axioms. And its inventions are not arbitrary : they have a purpose, that to provide an efficient theory. One could do without the axiom of infinity, but then the mathematics that one uses daily (and the construct which have been invented before any axiomatisation) would be less efficient.
Any language is associated with a world it help created. There are many musical languages and musical worlds and in these worlds new object such as a sonata come to be created.
Of course Louis, you are right on this point. And it is true that mathematics is a lonaguage, but as I said, this is much more. And I referred to an old debate : is mathematics a science or a language. I think that this thread is more about this issue.
Chemistry (and biology) is closer to language than mathematics, just look at interactions, combinations, context, individual expressions, etc. Parallels are more accurate.
Mathematics is more than a language. Its symbolism resembles to formal structure of a language. But that is not the whole thing. It has its own concepts to deal with and it is not just a formalism.
On the other hand, one can say that Math is the neutral language of everything.
There is a great problem connected with languages, Louis. We can express only finite amount of information by any language (discrete symbols). Is it enough for reality which is probably infinite?
We can express a lot by using languages but we must know how to interpret and we do that by using our conceptual vocabulary and inter-languages (the processes of language learning which go far beyond literal translation).
We can define quantity of information, Hristina, but how can we define it's sense?
Eugene,
All is finite. I am finite. What I have to say is finite. The time I have to say it is finite. The number of sentences that can potentially said is infinite.
''We can express only finite amount of information by any language (discrete symbols). Is it enough for reality which is probably infinite?''
Reality surely cannot be totally said in whatever language. But the point of a communication is never to say the whole of reality. That would take more time than the age of the cosmos even if it was possible. The point of communication is always very limited. We have to remember that in a natural language words are not defined like in mathematics. Take the dictionary definition of the word:
chair: noun, a separate seat for one person, typically with a back and four legs.
We learn to interpret and use this word when we are around 18 months. Our mother tell us ''seat'' on the ''chair''. She pointed it to us. Gradually we associate these chair forms and their use in paralllel that we learn to name it and designate it through language. A natural language do not objectively define itself. It is intimatly links to our experience. It would be very difficult to define a chair using a formal language.
Louis,
As usual your comments are pertinent. And they show that Mathematics is much more than a language. Mathematics is a not just a mean to share or transmit knowledge or information. Mathematics creates new knowledge.
Mathematics enables us to prove new scientific laws, in the meaning that a theorem is valid if some conditions are met, and then it will always be true, even if implemented in some special cases. This is the definition of a scientific law. Mathematics enables to create new laws. With other language such as english or music, one can create new works of art, but they will always be unique, even if some critics say that a new work has open the doors to a new style, the work of arts are not commensurable.
The theorems are new, in a profund meaning .For instance it has been proven that there is no program which could automatically prove any theorem in Arithmetics : the list of mathematical theorems (objects which are deemed true) is unending, even inside a given formal system.
Of course mathematics, with its precise definitions, enables us to communicate, to share information. But in usual languages, even in musics, the meaning of the language is found in past or shared experiences. We have the concept of a chair because we have experienced it. An opera by Wagner is telling because the sounds evoke some images. More generally art is the mean to convey emotion, but to do this efficiently we must have had previous emotions. For a mathematician, like myself, a Fréchet space is something that I can see as clearly as a chair, but it would be difficult to show any physical realisation of it. Even if this is true for many scientifc concepts (for instance it is difficult to find a definition of "life"), Mathematics stay special because they are not rooted in practical experience, or rather that, to become fully a science, they had been obliged to cut their roots. And this step is fascinating because it could not be crossed by a computer.
Any two distinct formulations of mathematics, for example one by humans and another by intelligent aliens, are nonetheless equivalent. That is, any truth expressible in "our" mathematics will have an *exact* equivalent in the "alien's" mathematics.
Not so with natural language. So I would say that mathematics is "more" than just natural language.
jean claude,
I totally agree. Mathematics is not only usefull for the sciences but it is essential to all the technological and engineering domains and to the domains demanding systematic organisation and planning.
What I am trying to put my fingers on is the essential difference between the mathematical language and the other languages.
The arts are also human languages. All these languages are intimatly link to us and to our cultural experience. Mathematic is the only language that do not refer to anything outside of itself. It does not make external references. It is a language about relational structures. Humans with their experience of the world and of societies make use of this experience in order to add new aspects to this language. What is worth adding is a subjective process , I would say an artistic process. What is adopted widely and taught is also the result of a communities and will depend on the need and aesthetic of this community. But there is no subjectivity in the use of the language. It is either a correct use or an incorrect one. And any competent mathematician can distinguish in between the two. Upstream what is worth exploring, what is worth funding as research will involve a lot of personal and social subjectivity but what is down stream, the use of the language involve none. It is interpretatively unambiguous while all the arts and what is said that is not trivial in other natural language need intrinsically ambiguous human interpretation.
I agree Louis. And as I said in my post, until the end of the XIX° century, with eucliden geometry and the numerous theories which had flourished, mathematicians still saw their concepts as some idealization of physical things. For instance it was a surprise when Lobatchevski introduced examples of non euclidean geometries : geometry should emerge from the properties of observed space, and that the same theorems could apply to a different reality was disturbing. Today most people do not realize the real step which have been accomplished in accepting that Mathematics should stand alone. This is where is the true revolution. Moreover there is no indisputable, unique, basis for the mathematics. To follow Kerry's remark one cannot be sure that aliens would have the same mathematics : it can be also rigorous and efficient, but built around another set of axioms.
About art, Aesthetics is not my forte, but I guess that Gaston Bachelard has written on the subject, in the second part of his career.
Another consequence of this aspect of Mathematics, in any theory using it as formalism appear common structures, usually see as characteristic of the physical, or social, world, but which are actually mathematical artefacts. This is the topic of one of my papers ("common structures...") and my understanding of QM. Another example : assume that one wants to check the existence of some phenomenon, which appears as often, as a singularity, or a disturbance with respect to a more common phenomenon (say a photon or a gravitational wave). Because of the methods which are used, and criteria of scientific proofs, the phenomenon itself is quantized : the singularity appears in a specific format (this is one explanation of the Planck's law).
In my opinion, mathematics is the strongest language for human being which they can understand easily and practically.
Eugene,
In mathematics an infinite set is a set such that there is a bijection between the set and one of its part (so stricly contained in the set itself), so this is an "impossible" object. The analogue for the human mind is imagination. We can conceive doing something that has never happened, or could not be experienced. We can imagine that we can abolish gravity and see a car which levitates. So we establish a correspondance between something which exists (ourself or a car) and something which has never happened. I think that a computer could not do the same. An artificial intelligence can learn the concept of car, from many data collected about cars. For it a car which levitates is not a car, it cannot be conceived.
This is in what mathematics are special : we can imagine objects wich do not exist, and use them in a rational and efficient way. And this is the triumph of the human mind.
To Eugene,
This is the same concept. Infinitesimal comes from the inverse.
I think we need to distinguish between 'Theoretical Mathematics' and 'Applied Mathematics'. Under the theoretic side, we can make statements that can be proven - and remain proven, even if the context they fit in is found to be not universal. So Euclidean geometry remains valid, however the context is found to be not universal, as there are extensions of this geometry to larger contexts ('non-Euclidean geometry'). The theoretic side does not have to correspond to reality and can be defined in an axiomatic way were we imagine objects that might exist in Plato's unshadowed world. Formal logic and symbolism primarily exist on this theoretic side and can be said to be a language - or more than a language.
On the Applied side, we require connection to reality and so must limit any theoretic aspects to those we can find that fit our direct or indirect experiences. Experiences involve finite quantities and even finite representation of quantities (while we have a precise theoretic 'value' for pi, we do not have a anything close to a precise representational value of pi). Also, on the Applied side, since we require connection to reality, we must invoke 'natural language' to connect the theoretic aspects and numeric quantities to 'real' objects. Thus Applied mathematics cannot be divorced or separated from natural language and thus does not stand on its own as a language here.
A curious aspect might be the question of whether theoretic mathematics can be entirely divorced from applied mathematics. I will suggest it cannot, since both sides require representation (symbols) of numeric values, which are directly connected with our reality. This would indicate that, while mathematics is something different and 'more than' natural language, it remains connected to our experiential reality in some aspects. This would also mean that aliens would not likely have the exact same mathematics as we do - the symbols used to connect number to representational measurements are likely unique to us and them and hence what we find will not be entirely the same. This would be especially true on the Applied side, where the objects we experience could be significantly different and hence so would be our 'sciences'.
George,
Languages advance very quickly in the domains where their use is evolving. In all domains of rapidly changing technology or scientific domains new English words are constantly introduced in order to be able to express these new realities. On the web there are so many words that are used and that did not exist prior to the use of the web. If we would quantify the number of new English words introduce each year in specialized domain , I guess that it is 100 x more than the number of new mathematical concepts introduced each year.
George,
Here are a few new words that all teenagers understand :
Selfie, unlike,tweet,inbox,hashtag,unfriend,troll,blog,bitcoin, bluetooth, bookmark, blogoshphere, e-business, microblogging, mp4, newsgroup, online banking,surfing, hacktivism
Mathematics has its exact language, but the math is more than a language. It is the result of a conscious and millennial effort of humanity.
Unfortunately, it is not given for everyone in its full glory.
Yes, I do! However, it is the language that only the experts effectively communicate with!!
We are always measure quantity (of "information") by means of a standard. This is a relative process at all levels and senses. Usually we assign or describe the standard on qualitatively different layer of information. Obviously, this process has no end, like the unfortunate race of heroes of Zeno paradoxes. Using of countable series or continuum is invariant only (relative to this process).
Louis Brassard Sir,
Your answer is supporting Language development of course, Language is advanced very quickly. Thanks a lot.
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