I'm not sure about that. Have you heard about Turbo's "black-bean hummus"? Apparently, if you eat it you need a dilithium crystal suppository to avoid exploding.jerromyjon said:I highly doubt mankind will ever invent a way to travel through space faster than 300,000 km/s...
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.russ_watters said:Does that mean the the universe didn't know how to make objects move properly until Galileo(?)discovered f=ma?
Did grapes teach us how to subtract? We invented counting and arithmetic to keep track of our grapes. Show me where counting exists in nature, where we learned it from nature. We invented the counting numbers and we impose them on our grapes in our mind. That's not about nature, it's about not getting ripped off at the marketplace in ancient Sumeria.Does that mean if I didn't have any math (subtraction) to describe my eating of grapes that eating grapes would make more grapes appear in front of me?
Math isn't a language. The sentence, "The sum of the squares of the two sides is equal to the square of the hypotenuse," is a sentence in English concerning certain quantities. It's not a language separate from English. Russians do math in Russian, and Frenchmen do it in French. Pythagorean maintains math is a language but his reasoning about that is actually quite abstruse and has nothing to do with math as a description of the world.I totally disagree with you. As people have said, math is a lanugage used to describe things that are happening around you. Those things are happening whether you have the language needed to describe them or not.
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). It's just one of the most popular systems in which calculus has a reasonably simple logical structure (non-standard analysis will present the same results using a different number system, so the popularity of the real number system is just an historical artifact).russ_watters said:Does that mean the the universe didn't know how to make objects move properly until Galileo discovered f=ma?
Subtraction is just one of many very abstract descriptions of that particular process. You can also describe the process of eating without any such great abstractions as separation of cardinal quantity from quality, as well as invoking the existence of an inverse operation between abstract cardinal quantities, as many authors and storytellers have no problem doing.russ_watters said:Does that mean if I didn't have any math (subtraction) to describe my eating of grapes that eating grapes would make more grapes appear in front of me?
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 . 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.jerromyjon said:F=ma isn't math? Move along people... nothing to multiply here...
zoobyshoe said: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.
Did grapes teach us how to subtract? We invented counting and arithmetic to keep track of our grapes. Show me where counting exists in nature, where we learned it from nature. We invented the counting numbers and we impose them on our grapes in our mind. That's not about nature, it's about not getting ripped off at the marketplace in ancient Sumeria.
Math isn't a language. The sentence, "The sum of the squares of the two sides is equal to the square of the hypotenuse," is a sentence in English concerning certain quantities. It's not a language separate from English. Russians do math in Russian, and Frenchmen do it in French. Pythagorean maintains math is a language but his reasoning about that is actually quite abstruse and has nothing to do with math as a description of the world.
Physics is our attempt to describe what's "happening around you," and it is relegated to figuring out ways to quantify things that aren't obviously quantified, and then to keep track of those quantities. Math is a tool here, not the description. The description involves concepts: mass, resistance, intensity, charge, pressure, temperature, wavelength, etc. which we believe can be quantified and treated mathematically.
Mathematics, as such, is all too often about nothing but numbers. Take the preoccupation of mathematicians with prime numbers, for example, or Fermat's Last Theorem. (Not that there's anything wrong with that.)
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."Pythagorean said: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
zoobyshoe said: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.
I'm not presenting it as a requirement. I'm presenting it as a property we observe in all undisputed languages.Pythagorean said:It's not a requirement of language that it translate to every other language.
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.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.
A finer grain description of quantity. Which is why you can't translate non-quantitative statements into math. Its scope is very limited.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.
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.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 said: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.)
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 said: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).
russ_watters said: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.
Huh? That's the definition of error: deviation of the measurement from a known value. So:slider142 said:The attribution of limited measurement capability to error is interesting, but not supportable by any physical experiment.
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.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.
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.Pythagorean said: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 can't say, "I trained my German Shepherd to growl at my biker neighbor," in "math".I'm not clear how your second point disqualifies it from being a language.
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.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 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).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.
No one claimed you could, zooby. Math is a language that is designed to describe specific and limited things.zoobyshoe said:I can't say, "I trained my German Shepherd to growl at my biker neighbor," in "math".
Math is the language of physics. That's why physics laws are nothing more than mathematical relations.F=ma is a physics concept, arrived at by experiment and observation. It's not a math concept.
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:We didn't learn simple multiplication from accelerating masses.
You're arguing against your point. Yes, multiplication was invented to make repeated addition of quantities of objects found in reality easier.Multiplication was invented to make repeated addition easy and fast.
The problem is that we do not know the value. We assume the value is pi out of convenience.russ_watters said:Huh? That's the definition of error: deviation of the measurement from a known value.
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 said: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.
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.slider142 said:The problem is that we do not know the value. We assume the value is pi out of convenience.
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.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.
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.russ_watters said: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.
The ad hominem is unnecessary. Your second sentence supports my argument. It has no relation to physical reality other than convenience.russ_watters said: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.
Ok, backing up:slider142 said: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.
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.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.
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 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 .
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.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 ).
Or, maybe it's not a language.russ_watters said:No one claimed you could, zooby. Math is a language that is designed to describe specific and limited things.
Citations? I'm going to need a slew of them to support the claim of "generally regarded".Zooby, do you at least recognize that your claim here is non-mainstream? That math is generally regarded to be a language?
-wikiDefinitions 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.
http://dictionary.reference.com/browse/mathematicsnoun
1.
(used with a singular verb) the systematic treatment of magnitude,relationships between figures and forms, and relations betweenquantities expressed symbolically.
http://www.merriam-webster.com/dictionary/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.thefreedictionary.com/mathematics1. (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.yourdictionary.com/mathematicsnoun The definition of mathematics is the study of the sciences of numbers, quantities, geometry and forms.
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.From a previous post:
Math is the language of physics. That's why physics laws are nothing more than mathematical relations.
zoobyshoe said: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.
Russ said:Does that mean the the universe didn't know how to make objects move properly until Galileo(?)discovered f=ma?
zoobyshoe said: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.
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?Russ said:I didn't claim we did.
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.Yes, multiplication was invented to make repeated addition of quantities of objects found in reality easier.
Again, glad your referring to it as "invented."For Newton's laws of motion (in particular, gravity and orbits), Calculus was invented to help describe them.
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.russ_watters said: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.
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.Pythagorean said:@zoobyshoe Mathematics is not defined as a language, it's language status is a matter of classification.
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.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