Mathematical Truths: Discovered or Invented?

[zoobyshoe]

The fact that you can speak math in Russian and German doesn't disqualify it from being a language. Pig Latin is another example of a language within a language. We're talking about different kinds of language here.

http://en.m.wikipedia.org/wiki/Language_of_mathematics

I'm not disputing that math has (its own) language (sophisticated jargon). I'm disputing that it is a language. Trivial, homely kind of proof: If math is a language, translate the following sentence into math: "I trained my German Shepherd to growl at the biker who lives next door."

Neuroscience now has a sophisticated jargon. Can we say "Neuroscience is a language,"? To say it about math opens up the door to saying it about any field with a sufficient body of experts speaking that field's jargon, "Physics is a language, Biology is a language, Economics is a language, Politics is a language."

But, we can't separate math from the "natural language" within which it's being used. The natural language is required to explain the math symbols and relationships. Math is communicated by language without, itself, being a language.

 

[Pythagorean]

I'm not disputing that math has (its own) language (sophisticated jargon). I'm disputing that it is a language. Trivial, homely kind of proof: If math is a language, translate the following sentence into math: "I trained my German Shepherd to growl at the biker who lives next door."

Neuroscience now has a sophisticated jargon. Can we say "Neuroscience is a language,"? To say it about math opens up the door to saying it about any field with a sufficient body of experts speaking that field's jargon, "Physics is a language, Biology is a language, Economics is a language, Politics is a language."

But, we can't separate math from the "natural language" within which it's being used. The natural language is required to explain the math symbols and relationships. Math is communicated by language without, itself, being a language.

It's not a requirement of language that it translate to every other language. The other fields you mention all pertain to the study of phenomena that exist in the natural world independent of the discipline itself (economy existed before economics, politics existed before political science - math did not exist before mathematics). In many cases, mathematics is the language we use to quantify the phenomena in those fields.

Mathematics primary function is a finer grain description than natural languages. Instead of saying there are a lot of soldiers on the battlefield, a scout reports the exact amount to the general. Like natural language, we can delve into what definitions really mean and what their implications are and discover new things about the language (as, for instance, Noam Chomsky's "Syntactic Structures").

 

[zoobyshoe]

It's not a requirement of language that it translate to every other language.

I'm not presenting it as a requirement. I'm presenting it as a property we observe in all undisputed languages.

The other fields you mention all pertain to the study of phenomena that exist in the natural world independent of the discipline itself (economy existed before economics, politics existed before political science - math did not exist before mathematics). In many cases, mathematics is the language we use to quantify the phenomena in those fields.

You've described a difference, but one that has no bearing on whether or not we could call any of those other fields a language.

Mathematics primary function is a finer grain description than natural languages. Instead of saying there are a lot of soldiers on the battlefield, a scout reports the exact amount to the general.

A finer grain description of quantity. Which is why you can't translate non-quantitative statements into math. Its scope is very limited.

Like natural language, we can delve into what definitions really mean and what their implications are and discover new things about the language (as, for instance, Noam Chomsky's "Syntactic Structures").

This touches on the more abstruse reasoning you had in the other thread. I thought you were onto something to the small extent you went into it, in the sense that I thought you could probably eventually make a good case for analyzing math as if it were a language.

Math uses a special subset of language, which is fine to call, "The Langauge of Mathematics," but, while math has its own language, it is not, itself, a language. (Except in a Marshall McLuhan kind of way. But no one seems to be presenting that viewpoint here.)

 

[jerromyjon]

We are splitting hairs for no constructive purpose here, so I'll just say one last thing... a language in my opinion doesn't have to be capable of describing a certain amount or scope of information, merely communicating information of any type should qualify. Just my opinion! Thanks to all for the discussion.

 

[Pythagorean]

I'm not presenting it as a requirement. I'm presenting it as a property we observe in all undisputed languages.

You've described a difference, but one that has no bearing on whether or not we could call any of those other fields a language.

A finer grain description of quantity. Which is why you can't translate non-quantitative statements into math. Its scope is very limited.

This touches on the more abstruse reasoning you had in the other thread. I thought you were onto something to the small extent you went into it, in the sense that I thought you could probably eventually make a good case for analyzing math as if it were a language.

Math uses a special subset of language, which is fine to call, "The Langauge of Mathematics," but, while math has its own language, it is not, itself, a language. (Except in a Marshall McLuhan kind of way. But no one seems to be presenting that viewpoint here.)

I contest your first point. Undisputed languages do have mixtures of other languages where translation fails. The Japanese have a whole subdivision of language dedicated to it called Katakana, we call sushi sushi in English.

I'm not clear how your second point disqualifies it from being a language. But I also dispute the statement of scope - in some ways, mathematics scope is limited, but it takes the function of natural language where natural language is limited, not just in quantification, but in the nature of operations (of which verbs are sub. There are languages that have limited scope though, like pidgins, as well as being an example of a mixture of languages, and pidgins are never a first language learned either, they only augment their two parent languages.

 

[russ_watters]

This is an attribution of intent to a deterministic process. The description of motion by classical mechanics is an approximation only, especially as it uses the system of real numbers (no physical device can measure a real number quantity, and there are other problems that are mentioned below).

That's not correct. Measurement error doesn't have anything to do with whether the mathematical relation you use to operate on data is valid. Pi is an exact ratio even if representing it in numerical form is difficult. And 2-1=1 is an exact operation that you can execute in real life by eating a grape.

 

[slider142]

That's not correct. Measurement error doesn't have anything to do with whether the mathematical relation you use to operate on data is valid. Pi is an exact ratio even if representing it in numerical form is difficult. And 2-1=1 is an exact operation that you can execute in real life by eating a grape.

The attribution of limited measurement capability to error is interesting, but not supportable by any physical experiment. There is no physical device that will measure a length to be Pi, by which we can check the so-called "error" of our physical measurement. The attribution of this inability to error, by believing that pi is actually present but the device is just not measuring it properly, is an unnecessary assumption.
One may retort that this is heresay, by appealing to perhaps statistical arguments: that repeated measurements of approximations of Euclidean circles by physical materials or processes shows the internal error remains small. However, this is not support for the exact number Pi. This statistical data also supports the hypothesis that the physical ratio is actually Pi - 1/(Skewes'[/PLAIN] number). There are many models of geometry that will produce ratios of many different amounts, and an infinite amount of them are close enough to pi to be below the ability of any physical measure to detect. Therefore, the determination that it must be exactly pi that is being measured is demoted to a personal preference. It is convenient to use pi because it is associated with a popular form of geometry that we have been raised on for most of our lives, and most of our colleagues share knowledge of various theorems in that particular geometry. However, a matter of convenience is very different from establishing a universal truth.
In fact, when measuring triangles on curved surfaces, such as the surface of the Earth, we already know our approximation is just only that, from our study of differential geometry (specifically, spherical geometry).
The idea of proving subtraction exists by eating a grape is also an interesting conflation of physics with mathematics. What do you think of the "proof" or "illustration" by first adding 2 drops of water to each other, then removing 1 drop of water, thereby "proving" that 1 + 1 - 1 = 0 ? This is also an exact operation you can execute in real life.

 

[russ_watters]

The attribution of limited measurement capability to error is interesting, but not supportable by any physical experiment.

Huh? That's the definition of error: deviation of the measurement from a known value. So:

There is no physical device that will measure a length to be Pi, by which we can check the so-called "error" of our physical measurement.

That deviation from Pi is the error! You are suggesting we don't know the real value of Pi because no device can measure it. That isn't correct. We can calculate the value of Pi as exactly as we want (or leave it in equation form and make it actually exact) and the deviation of a measurement from that vaue is the measurement error.

The rest of your post describes the fact that the commonly used value of Pi is onlly true for certain types of geometry. That isn't a problem with it being exact, that's just part of the definition of the exact value we're discussing.

 

[zoobyshoe]

I contest your first point. Undisputed languages do have mixtures of other languages where translation fails. The Japanese have a whole subdivision of language dedicated to it called Katakana, we call sushi sushi in English.

It's a fact of cultures that there exist things in one culture which don't exist in another, like sushi, and a translation of a Japanese sentence containing the word "sushi" would require some sort of aside definition of the word for any audience unfamiliar with it. At some point in my past someone had to define it for me, but their definition was in English. There's no failure of translation, it just takes more effort to translate concepts that don't exist in the culture of the audience language. Translation is never as simple as a one-to-one correspondence of words from one language to the other.

I'm not clear how your second point disqualifies it from being a language.

I can't say, "I trained my German Shepherd to growl at my biker neighbor," in "math".

But I also dispute the statement of scope - in some ways, mathematics scope is limited, but it takes the function of natural language where natural language is limited, not just in quantification, but in the nature of operations (of which verbs are sub.

I don't think it takes the function of natural language. It extends natural language to give that "finer grain description" of the limited subject of quantity. Every math book I've ever read is about 95% English sentences and 5% formulas (which are nothing but shorthand for English sentences). Open a math book and what do you see most of? Words, words, words. All about quantities.

There are languages that have limited scope though, like pidgins, as well as being an example of a mixture of languages, and pidgins are never a first language learned either, they only augment their two parent languages.

I don't know any pidgin, but, from what I've read about it, you could express that you trained your dog to growl at your neighbor in it. I think Hamlet's "To be or not to be," soliloquy could also be expressed in pidgin, the bald meaning, at least (with some substantial loss of poetic nuance, obviously).

 

[russ_watters]

I can't say, "I trained my German Shepherd to growl at my biker neighbor," in "math".

No one claimed you could, zooby. Math is a language that is designed to describe specific and limited things.

Zooby, do you at least recognize that your claim here is non-mainstream? That math is generally regarded to be a language?

From a previous post:

F=ma is a physics concept, arrived at by experiment and observation. It's not a math concept.

Math is the language of physics. That's why physics laws are nothing more than mathematical relations.

We didn't learn simple multiplication from accelerating masses.

I didn't claim we did. But if you are saying that developing math had to happen as a way to describe something in nature in order to be a language, then:

Multiplication was invented to make repeated addition easy and fast.

You're arguing against your point. Yes, multiplication was invented to make repeated addition of quantities of objects found in reality easier.

For Newton's laws of motion (in particular, gravity and orbits), Calculus was invented to help describe them.

 

[slider142]

Huh? That's the definition of error: deviation of the measurement from a known value.

The problem is that we do not know the value. We assume the value is pi out of convenience.

So:

That deviation from Pi is the error! You are suggesting we don't know the real value of Pi because no device can measure it.

This is incorrect. I demonstrated that we do not know that the object you are measuring has a length or ratio of pi. I did not imply anything about the value of pi or its calculation. Its ability to be objectively observed as a known value of any physical property, promoted distinctly above every other real number that lies close to it, was called into question. It exists as the "known value" for anything roughly related to things that sort of look like they should be modeled after ideal Euclidean circles (that have also never been observed) entirely as a popular idealism.

 

[russ_watters]

The problem is that we do not know the value. We assume the value is pi out of convenience.

That's just plain wrong. Do you not know that Pi can be calculated without utilizing measured values of real objects in nature? Those digits aren't just pulled out of the air.

This is incorrect. I demonstrated that we do not know that the object you are measuring has a length or ratio of pi. I did not imply anything about the value of pi.

No, I'm sorry, but all you've demonstrated is that you don't actually know what Pi is. The value of Pi is not found by taking measurements of objects.

 

[slider142]

That's just plain wrong. Do you not know that Pi can be calculated without utilizing measured values of real objects in nature? Those digits aren't just pulled out of the air.

As I've noted above, the ability to calculate pi has never been called into question. The idea that it is an absolute truth related to any physical observable has.

No, I'm sorry, but all you've demonstrated is that you don't actually know what Pi is. The value of Pi is not found by taking measurements of objects.

The ad hominem is unnecessary. Your second sentence supports my argument. It has no relation to physical reality other than convenience.

The last few replies make it clear that my argument may have been misinterpreted. What do you believe the proposition is that I proposed, the one that you are primarily arguing against ? It seems like we actually agree on some things.

 

[Danger]

The simplest thing that comes to mind right now is that "nothing" existed in the days of the Romans, who had no "0" in their numbering system.

 

[russ_watters]

As I've noted above, the ability to calculate pi has never been called into question. The idea that it is an absolute truth related to any physical observable has.

Ok, backing up:

There is no physical device that will measure a length to be Pi, by which we can check the so-called "error" of our physical measurement. The attribution of this inability to error, by believing that pi is actually present but the device is just not measuring it properly, is an unnecessary assumption.

It appears to me that what you are really trying to say is that we have no real/absolute proof that the real world obeys Euclidian geometry (agreed). That if a measurement deviates from Pi we have no way of knowing whether that is because of an error in measurement or because the universe doesn't obey Euclidian geometry. While it isn't exactly correct (we can always build a new, better measurement device that tells us that most of the previous device's deviation was measurement error), it isn't relevant to the OP's question. The OP created the thread and used the word "Truths" -- it may not have been my choice because I recognize that in science we can never prove anything completely. But I don't think your quibble is useful for answering the OP's question. If we say, "sorry, we can't be sure we've found any Truths", I'm fine with that, but we're still left to answer whether the relations we've found were invented or discovered.

Still, backing-up more:

[f=ma isn't math?] No, it is not. A mathematical proposition is a purely logical one: it can be proven true or false solely on the basis of assumptions and certain laws of thought .

Aside from the self-evident that f=ma is an equation and an equation is a mathematical statement ... and without knowing its history of how it came to be all you would know is that f=ma is a purely logical mathematical proposition (so, yes, f=ma is math), what I think you're really quibbling with here is whether f=ma is true in nature and/or can be called a Truth.

This additional side issue of whether f=ma was derived via rigorous mathematical procedure (it may not have been: it may basically just have started as a curve fit) is both irrelevant and in general wrong. f=ma was originally discovered by Galileo, who was not a mathematician of the quality of Newton, but in general physics is mathematically rigorous.

That is, a mathematical textbook or academic council will never request a student to necessarily perform an experiment in order to prove a theorem. Newton's assumption that F=ma could be made into a mathematical theorem if we make certain other assumptions (ie., the Newton-Laplace Determinacy Principle and certain assumptions about the manifold that best models physical processes). However, that is not the spirit of the equation: it is meant to be supported by its application to physical processes, not by mere internal self-consistency. Any internally consistent model can be made into a mathematical theory, including many that have no analogues in any physical process.
That is, if a single physical process disagreed with F=ma in any way that could not be removed by reasonable further assumptions, F=ma would be replaced by another model. This can never happen for a mathematical statement: a mathematical statement's proof depends only on logical argument and is thus always true when those assumptions are true. No interaction with physical verification is ever necessary (ie., see various abstruse theorems such as Banach-Tarski ).

You're splitting a hair that doesn't exist. Whether f=ma applies properly to nature doesn't have anything to do with whether it is a valid mathematical statement. But better examples would be Newton's Law of Gravity (called a law and not a theory because it is a mathematical relation based on assumptions) and Relativity. Relativity was mathematically derived by taking observations that were believed to be universally true and making them postulates for the purpose of mathematically correct logic. Newton's Law of gravity is known not to be true in nature, yet it is still mathematically correct and is still used in cases where its postulates are close enough to true in reality to be acceptable to assume.

Perhaps a mathematic council would not require a student to perform an experiment on a purely mathematical derivation that has no believed/assumed connection to the physical universe, but a physics council would require them to use mathematical rigor in constructing their equations. I think much of the argument here is based on the false assumption of a reciprocity: that because math doesn't require experiments, physics doesn't require mathematical rigor. It does.

 

[zoobyshoe]

No one claimed you could, zooby. Math is a language that is designed to describe specific and limited things.

Or, maybe it's not a language.

What I perceive happening here is the same thing that happened with the "Physics doesn't do 'why' questions" meme that infected PF for a while. Someone makes a forceful or otherwise attention-getting statement that doesn't get properly disputed in a timely way, and it takes off and starts having a life of its own. Everyone repeats it as if it's gospel, doctrine, and starts rationalizing their own support for it.

Zooby, do you at least recognize that your claim here is non-mainstream? That math is generally regarded to be a language?

Citations? I'm going to need a slew of them to support the claim of "generally regarded".

In the meantime here are some definitions of math, none of which characterizes it as a language:

Definitions of mathematics
Main article: Definitions of mathematics
Aristotle defined mathematics as "the science of quantity", and this definition prevailed until the 18th century.[29] Starting in the 19th century, when the study of mathematics increased in rigor and began to address abstract topics such as group theory and projective geometry, which have no clear-cut relation to quantity and measurement, mathematicians and philosophers began to propose a variety of new definitions.[30] Some of these definitions emphasize the deductive character of much of mathematics, some emphasize its abstractness, some emphasize certain topics within mathematics. Today, no consensus on the definition of mathematics prevails, even among professionals.[7] There is not even consensus on whether mathematics is an art or a science.[8] A great many professional mathematicians take no interest in a definition of mathematics, or consider it undefinable.[7] Some just say, "Mathematics is what mathematicians do."[7]

Three leading types of definition of mathematics are called logicist, intuitionist, and formalist, each reflecting a different philosophical school of thought.[31] All have severe problems, none has widespread acceptance, and no reconciliation seems possible.[31]

An early definition of mathematics in terms of logic was Benjamin Peirce's "the science that draws necessary conclusions" (1870).[32] In the Principia Mathematica,Bertrand Russell and Alfred North Whitehead advanced the philosophical program known as logicism, and attempted to prove that all mathematical concepts, statements, and principles can be defined and proven entirely in terms of symbolic logic. A logicist definition of mathematics is Russell's "All Mathematics is Symbolic Logic" (1903).[33]

Intuitionist definitions, developing from the philosophy of mathematician L.E.J. Brouwer, identify mathematics with certain mental phenomena. An example of an intuitionist definition is "Mathematics is the mental activity which consists in carrying out constructs one after the other."[31] A peculiarity of intuitionism is that it rejects some mathematical ideas considered valid according to other definitions. In particular, while other philosophies of mathematics allow objects that can be proven to exist even though they cannot be constructed, intuitionism allows only mathematical objects that one can actually construct.

Formalist definitions identify mathematics with its symbols and the rules for operating on them. Haskell Curry defined mathematics simply as "the science of formal systems".[34] A formal system is a set of symbols, or tokens, and some rules telling how the tokens may be combined into formulas. In formal systems, the word axiomhas a special meaning, different from the ordinary meaning of "a self-evident truth". In formal systems, an axiom is a combination of tokens that is included in a given formal system without needing to be derived using the rules of the system.

-wiki

noun
1.
(used with a singular verb) the systematic treatment of magnitude,relationships between figures and forms, and relations betweenquantities expressed symbolically.

http://dictionary.reference.com/browse/mathematics

: the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

http://www.merriam-webster.com/dictionary/mathematics

1. (Mathematics) (functioning as singular) a group of related sciences, including algebra, geometry, andcalculus, concerned with the study of number, quantity, shape, and space and their interrelationships by usinga specialized notation

http://www.thefreedictionary.com/mathematics

noun The definition of mathematics is the study of the sciences of numbers, quantities, geometry and forms.

http://www.yourdictionary.com/mathematics

The word "science" crops up most. I think math as language is the concept that's not mainstream. Math is not a communication system.

From a previous post:

Math is the language of physics. That's why physics laws are nothing more than mathematical relations.

No, the language of physics is the large system of very specifically defined terms, a few of which I mentioned earlier. The jargon of physics. Physics employs math as its main tool, to keep track of quantities. That's not communication, it's accounting, as Feynman (sort of) put it. And, physics laws are not "nothing more than" mathematical relations. The physics in F=ma lies in the hard won grasp of the concepts of force, mass, and acceleration. The mathematical relation, as such, a = bc is no revelation. The formula great meaning derives from the concepts that were found to have that simple relation. As I have said a few times on the forum, Newton's laws were the result of about 2000 years of struggle to get traction on the phenomenon of motion. It was not at all apparent how to parse the situation. Starting from scratch and defining mass, for example, in the presence of the confounding phenomenon of weight was not easy. Figuring that, and the other concepts, out is the physics. Once that was done, the mathematically trivial relationship eventually became apparent. For every success like Newton's 3 laws, there are uncounted failures that aren't in the books. We cherry pick the things we've found a hand hold on and forget about the failed attempts that lead nowhere. Physics is about figuring out what to account for.

Saying mathematical truths are discovered is like saying chess truths are discovered. Both statements ignore the fact you're making discoveries about a human mental invention and falsely imply you're making discoveries about nature.

Does that mean the the universe didn't know how to make objects move properly until Galileo(?)discovered f=ma?

F=ma is a physics concept, arrived at by experiment and observation. It's not a math concept. We didn't learn simple multiplication from accelerating masses. Multiplication was invented to make repeated addition easy and fast.

I didn't claim we did.

Yes you did, indirectly. You were denying math was invented and claiming it was discovered in nature. Therefore, we must have learned multiplication from nature, perhaps by observing F=ma, or some such. Your point was some vague claim math has always existed in nature before we 'discovered' it, was it not?

Yes, multiplication was invented to make repeated addition of quantities of objects found in reality easier.

Yes, and not to "describe what's around you". To avoid getting ripped off at the market. All the early evidence is that early arithmetic was commerce driven; accounting. But, at least you're now admitting it was invented.

I'm really surprised that you don't seem to appreciate that math doesn't get cooking until you abstract it from real world representation and deal with quantities as quantities. That's where all the advances are made. Read Euclid. There's no hint of a mention in the whole book about practical applications. The ancient mathematicians worked forward fascinated by the logic in and of itself, with no particular concern whether or not there might be a mundane use for any of it.

For Newton's laws of motion (in particular, gravity and orbits), Calculus was invented to help describe them.

Again, glad your referring to it as "invented."

 

[zoobyshoe]

Aside from the self-evident that f=ma is an equation and an equation is a mathematical statement ... and without knowing its history of how it came to be all you would know is that f=ma is a purely logical mathematical proposition (so, yes, f=ma is math), what I think you're really quibbling with here is whether f=ma is true in nature and/or can be called a Truth.

F=ma is not a mathematical statement. It is a physics statement. The variables stand for variable quantities of specific physical phenomena, not abstract quantities in general. It's a statement about the relationship of force, mass, and acceleration. There's no point in invoking someone who doesn't know that, we all do. We know the intent of it is to describe nature, not abstract number relations.

This is a math statement:

(a + b)² = a² + 2ab + b²

It holds true for any two numbers a and b.This is a physics statement:

V = IR

Only holds true when V = voltage, I = current, R = resistance. It does not hold true when V = 20, I = 100, R = 365. It has no significance, conveys no meaning when divorced from the proper physics concepts.

 

[Pythagorean]

@zoobyshoe Mathematics is not defined as a language, it's language status is a matter of classification. Looking up definitions and not finding "is a language" is essentially an appeal to ignorance, but I'll gladly provide you with citations.

It's very common in science and mathematics to talk about math as a language, in the classroom, in research meetings, etc - I say this anecdotally as someone who participates in academia as a researcher, student, and teaching assistant.

Anyway, here are some citations:

http://www.cut-the-knot.org/language/MathIsLanguage.shtml

http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx

https://www.dpmms.cam.ac.uk/~wtg10/grammar.pdf

http://www.tandfonline.com/doi/abs/10.1080/00131720008984764?journalCode=utef20#.VFuO__nF8qI

http://www.jstor.org/discover/10.2307/20205297?uid=3739448&uid=2&uid=3737720&uid=4&sid=21104975748287

 

[zoobyshoe]

@zoobyshoe Mathematics is not defined as a language, it's language status is a matter of classification.

All the definitions I posted encompass classifications. Math was most often classified in them as a science, then as a field of study. None classified it as a language.

It's very common in science and mathematics to talk about math as a language, in the classroom, in research meetings, etc - I say this anecdotally as someone who participates in academia as a researcher, student, and teaching assistant.

Anyway, here are some citations:

http://www.cut-the-knot.org/language/MathIsLanguage.shtml

http://www.ascd.org/publications/books/105137/chapters/Mathematics-as-Language.aspx

https://www.dpmms.cam.ac.uk/~wtg10/grammar.pdf

http://www.tandfonline.com/doi/abs/10.1080/00131720008984764?journalCode=utef20#.VFuO__nF8qI

http://www.jstor.org/discover/10.2307/20205297?uid=3739448&uid=2&uid=3737720&uid=4&sid=21104975748287

Your anecdotal report and the citations are some support for Russ' assertion that it is "generally regarded" as a language by people involved in math. Which is what I requested.

However, on reading the citations, I find a lot of assertion without support, assertion with incomplete support, assertion with incompetent support, and generally, a confused shift from speaking about the 'language of mathematics' (its jargon) to 'mathematics is a language.'

The first link is the worst, having the largest amount of inarticulate puffery. The links with more serious attempts to get people to regard math as a language are misguided attempts at trying to formalize an informal "manner of speaking": the way mathematical concepts are expressed is extremely specific and rigorous, and that can be off-putting. You have to enter into the spirit of it to make sense of it, and you might informally encourage someone to do that by suggesting they consider it a separate language unto itself, until such time as they get the hang of it. However, it looks like that heuristic is being misunderstood as a fact, and the links show linguistics being scoured for possible similarities between language and math, in the attempt to make a case by confirmation bias, as in the last link. There's no point to this. All you need to do it advise students they have to pay particular and sustained attention to terms and concepts.

 

[collinsmark]

I'm in the camp that mathematics is a language. But it's more than that.

The Embalse Nuclear Power Station in Argentina, a pressurized heavy water (PHWR), is a building. Yes, Embalse is a building, but it's more than just a building. Similarly, mathematics is a language, but it's more than that.

Mathematics is a language, but it's a very special, very precise language. It is also a tool of logic (tool being the key word here). And in addition to those, it is an embodiment -- an abstract repository so to speak -- of a particular body of knowledge: the very logic itself of previously proven ideas.

Language:
I think it was Stephen Hawking that related an anecdote told to him by his publishers when writing one of his lay-person targeted books. They told him something to the effect that every equation he put in his book would drop the book's sales by 10% (or some-such). I guess most lay-people are scared away by equations and will put the book down if they see one.

I find that sad, because communicating certain ideas are so much clearer in my opinion if they are communicated with equations and math. One could say, "the gravitational force between two bodies is proportional to the area of a rectangle who's length takes on the mass of one body and who's width takes on the mass of the other body; and the force is also inversely proportional to the area of a square who's sides are the distance between the two bodies." Gah! Are you kidding me? Just say,
$$F = G \frac{m_1 m_2}{r^2}.$$
That's so much more clear to me, and communicates the same idea! Then again I've taken the time to learn what that notation means, with the multiplication and the division and the squaring operation. So I understand that maybe some others less well versed in mathematical notation would prefer the former. If only they knew what they're missing!

Tool:
Egill Skallagrímsson, born in Iceland in the tenth century, in a time and place where the common person could not read a written language (paper and parchment had not been introduced to Scandinavia yet), was a famous warrior-poet. While most people couldn't read or write, Egill on the other hand had learned the "runes." He could read and write. I can imagine Egill scratching out a poem-in-progress in the dust by his feet as he's working out the details, "Hmm, that doesn't quite fit. Let me scribble that out. Ah, that word works much better here." A written language can be a wonderful tool to organize one's thoughts before speaking them.

English is a spoken language but also has a written form, and they are different from each other. Although the written form does not have the pronunciation, and the spoken form does not have the spelling, nor the same detail of punctuation. (Before arguing that the written language simply imitates the spoken form, ask yourself why the language has words with silent letters.) Try to read a book to someone out loud, and the style is noticeably different than it would be if the story was created and spoken on the spot. It becomes quite obvious that there is more to the written word than just the fact that it is written.

This is where the written form of mathematics really shines. It's able to communicate ideas and relationships that simply couldn't be clearly spoken. It aids one in organizing one's thoughts in ways that the common, non-mathematical language fails.

With only a little effort, starting with
$$F = G \frac{m_1 m_2}{r^2},$$
I can use mathematics to reorganize that idea and say,
$$r = \pm \sqrt{\frac{G m_1 m_2}{F}}.$$
That would be difficult if the only language available was English.

[Edit: And elaborating on this tool idea, sometimes when physicists and engineers use mathematics to rearrange and combine thoughts it can help lead to new, unexpected ideas. For example, when Paul Dirac reformulated the ideas of non-relativistic quantum mechanics with the principles of special relativity, it lead to hints of antimatter. The existence of antimatter "fell out of the math" so to speak (this relates to the $\pm$ sign when you take the square root). Mathematics is not just a tool, it can be used, in part, as a predictive tool.]

Knowledge:
Pythagoras proved that for a right triangle, $c^2 = a^2 + b^2$. Brahmagupta derived the roots of a second order polynomial to be $x = \frac{-b \ \pm \sqrt{b^2 - 4ac}}{2a}$. Euler showed us that $e^{ix} = \cos x + i \sin x$ (and yes, this is a provable relationship -- not a mere definition or "trick"). Once proven, those ideas become added to the overall body of mathematical knowledge. We don't need to re-prove them from scratch when working on other things; we can leverage the results and go from there.

So mathematics isn't just the language of logic, it also embodies the very logic itself of previously proven theorems.

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I'll end this post with a love poem I wrote several years ago:

A love poem, by collinsmark:

The number of ounces per ton,
less a dozen times square fifty-one,
with the cube of neg-nine
together combine
to make three score, less one to the none.
​

$$32000 -(12)(51)^2 + (-9)^3 = 3(20) - 1^0$$