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Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a

by retox 9y ago

Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? I have a deep dread and fear of numbers and mathematics in general because I don't understand them and I have never learned. Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation (the same equation) anywhere in the world.

meken 9y ago
I think the answer is that such a language would be overly verbose. You could ask the same question about human language and the same answer applies.
- heavenlyblue 9y ago
  It probably is also the case that the language used in sci articles is constantly changing, too. We may take advantage of libraries in programming (but even then they may not work on all of the physical platforms), but it's not that well defined, what is "the fundamental language" of mathematics; at least not as of today. Although I find it cool that tools like Coq are getting increasingly popular, so we will eventually get there.
Smaug123 9y ago
Why is there not a universal way of writing anything that is not ambiguous and can be read by anyone that understands the 'language'? Because humans have a tendency to defy attempts at classification and constraint. Constructed real-world languages (Lojban, Esperanto) haven't really taken off, while non-constructed languages mix and match whenever they feel like it ("le parking", "Schadenfreude"). The same is true in maths: people use what is easiest.
- erasemus 9y ago
  Because whenever we read something we have to guess the meaning. To the human mind, all communications are inherently contextual and metaphorical. To put it another way each individual has his own internal private language into which everything he reads must be translated.
  - emacsgifs 9y ago
    It's as if we truly believe that 0 is positive charged and 1 is negative charged. Or to take us a step further, that a certain arrangements of electrons is "negative" and the inverse arrangement is "positive" I often tell CS people that everything in computing is metaphor and it honestly disturbs me that 80% of the time I get blank looks.
    - sevensor 9y ago
      > I often tell CS people that everything in computing is metaphor and it honestly disturbs me that 80% of the time I get blank looks This is why people think Machine "Learning" means that the Singularity is near.
- Aaargh20318 9y ago
  > Why is there not a universal way of writing anything that is not ambiguous and can be read by anyone that understands the 'language'? You mean like Lojban (https://en.wikipedia.org/wiki/Lojban https://en.wikipedia.org/wiki/Lojban) ?
remus 9y ago
> Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation (the same equation) anywhere in the world. It's not as bad as that! You're very unlikely to hit this sort of issue until you're reading research papers (unless you decide you're very interested in mathematical history). Pretty much all the material youll come across up to ~undergrad level will use very, very similar notation.
- pbhjpbhj 9y ago
  The first alternate notation I think in maths, I used was dot (period) notation for scalar multiplication. Next would be different versions of standard form (aka engineering notation). Later, at high-school (UK, 16-18yo) we used both d/dx type notation for differentiation and x-dot (a dot above the letter for each level of differentiation). There would be 3 versions of x= ut + (at^2)/2 that we'd know, this one, d/dx, and dot-notation; primarily the former was used in physics and the latter in maths. We'd also use suffixed numbering, u_0, u_1 on occasions and use u, v, w on other occasions. One important point - they're not different equations, just written differently. You write for your audience, in an advanced paper you can abbreviate and use shorthand because you know the reader (in general) will understand. As an undergrad I found that each department I studied in would tend to use different notation - chemistry, ecology, maths, physics, computing - for the same things. Though sometimes different professors would buck the trend ... they also had different accents and vocabulary, our QFT professor loved the word recapitulation. You also get different symbol use, like rho for density in one domain and d in another; but that's not really notation per se. TL;DR sometimes you come across different notation earlier. Different notation doesn't change the equation.
- bzbarsky 9y ago
  There are at least three different commonly used notations I'm aware of for division.... And that's not counting the long division process, for which I'm aware of at least two different notations (usage varies by country).
TuringTest 9y ago
> Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? There is one, it's called formal logic. The problem with it is that it's not adequate to explain things, only to prove theorems formally; any strict unambiguous proof written in formal logic looks more like a debug trace than a scientific article. To provide a proper explanation of a mathematical result, at many points in the proof you need to summarize the hundreds of intermediate mechanical steps and write the insight behind them in common language, using potentially ambiguous words.
rsj_hn 9y ago
What do you mean by "writing" mathematics? The glib answer is that math is a superset of programming and there are non-computable arithmetic functions, so programming is not rich enough to describe what you need to describe when you are talking about math. I think a more intuitive answer is that in math you are generally talking about relations between infinite sets of objects, whereas in programming you are living in a (in practice) finite space and are evaluating everything into an integer. Programming is about calculating integers, mathematics is about proving theorems. The language has to be very different because what is crucial about evaluating integers unambiguously is peripheral to proving theorems and vice versa. You can ask, why is it so painful to give a formal proof of correctness of your program? It's impossible to do for all but the most trivial programs. You know that a brilliant person, given years of work, might be able to come up with a proof of correctness, but you also know that the language they would use to do that would be very different from the language that you as programmer would use to write your program. So it goes in both directions. Programs evaluate numbers, they are essentially adding machines, and central to that is the erasure of state via addition. Math is about symbolic relations and erasure of state is via relations on infinite sets. For example, you want to prove that if a group has a prime number of elements, then it must be cyclic. So pick a generator, raise it to powers, and get a subgroup. Then by Cauchy's theorem, the order of the subgroup has to divide the order of the main group, and that order is prime, so therefore the subgroup has to be the whole group. There are many things to unpack in that statement. For example, Cauchy's theorem, which says that if a A is a subgroup of B, then the order of A divides the order of B. So Cauchy's theorem + Group is prime order => non-zero subgroups must be the whole group. And then every non-zero group has a non-trivial cyclic subgroup + non-zero subgroups must be the whole group => the whole group is a cyclic group. So it's bit like adding numbers, in that you forget state, but the rules are much more complicated involving quantifiers and a huge universe of sets. To build a computer that could reason that way, you'd need an infinite set of registers, rules for quantifiers, etc. and your unit of memory would be abstract set relations instead of a zero or 1. You can try to simulate special cases of that in software, say with an SMT solver. If you look at the DSL of writing code for SMT solvers, it looks different from the code that a programmer is used to writing, and the best SMT solver can't really do anything too interesting from a math point of view, with very rare exceptions. Moreover in math as practiced, no one (except some logicians) writes in formal logic. It would be too cumbersome. Simple statements like "a harmonic function on a disk achieves its maximum and minimum value on the boundary of its domain" would require thousands of pages of formal symbols and quantifiers. No one can think like that, and so we don't write math that way because we also want to forget things and make compact statements, but what we forget in math is relationships among infinite sets, things like, what are the real numbers, what is a function on the real numbers, what is a differential, what is a harmonic function, what is the maximum of a function, what is a disk, all of that is packed into our statement and replaced with a math statement that is much more compact in the same way that 1 && 0 are replaced by 0 in a computer register, but the computer register is not rich enough to model relations on sets.
jiggunjer 9y ago
I had this reaction to sign language. Seemed like deaf people missed a good opportunity, but then I realized that how we talk and think is shaped very much by our language. Similarly scholars from different backgrounds have different ways of expressing a precise mathematical concept. This one paper I read implicitly treated complex scalars as real vectors, took me some time to clear up the confusion.
pathsjs 9y ago
An exercise that I made during my degree and phd years in maths was to always take notes in a consistent notation across the various lectures, regardless of the convention used by the teacher. It forced me to pay attention and as a bonus I have a consistent set of notes for most undergraduate mathematics :-)
joppy 9y ago
Why should all mathematics be written using exactly the same language and conventions? Should all computer programs be written in the same language? People modify the language (natural, mathematical, programming) they are using to suit their problem. It is unreasonable to expect that the kind of language you use at the foundations of mathematics for example (set axioms and so on) would be efficient and useful for stating properties of much more complex objects: this exactly parallels computer programming languages. Admittedly, there are no big tables containing "definitions" of these languages. But I think it is common opinion that if you don't understand the language a paper is using, then there is a very large chance you don't know the underlying theory, and the paper will make no sense anyway.
- Firadeoclus 9y ago
  There's also a small but non-negligible chance that you could have understood the paper because you know (part of) the underlying theory under a different name, using different notation and concepts but leading to the same results.

by retox 9y ago

meken 9y ago

I think the answer is that such a language would be overly verbose. You could ask the same question about human language and the same answer applies.

heavenlyblue 9y ago
It probably is also the case that the language used in sci articles is constantly changing, too. We may take advantage of libraries in programming (but even then they may not work on all of the physical platforms), but it's not that well defined, what is "the fundamental language" of mathematics; at least not as of today. Although I find it cool that tools like Coq are getting increasingly popular, so we will eventually get there.

Smaug123 9y ago

Why is there not a universal way of writing anything that is not ambiguous and can be read by anyone that understands the 'language'? Because humans have a tendency to defy attempts at classification and constraint. Constructed real-world languages (Lojban, Esperanto) haven't really taken off, while non-constructed languages mix and match whenever they feel like it ("le parking", "Schadenfreude"). The same is true in maths: people use what is easiest.

erasemus 9y ago
Because whenever we read something we have to guess the meaning. To the human mind, all communications are inherently contextual and metaphorical. To put it another way each individual has his own internal private language into which everything he reads must be translated.
- emacsgifs 9y ago
  It's as if we truly believe that 0 is positive charged and 1 is negative charged. Or to take us a step further, that a certain arrangements of electrons is "negative" and the inverse arrangement is "positive" I often tell CS people that everything in computing is metaphor and it honestly disturbs me that 80% of the time I get blank looks.
  - sevensor 9y ago
    > I often tell CS people that everything in computing is metaphor and it honestly disturbs me that 80% of the time I get blank looks This is why people think Machine "Learning" means that the Singularity is near.
Aaargh20318 9y ago
> Why is there not a universal way of writing anything that is not ambiguous and can be read by anyone that understands the 'language'? You mean like Lojban (https://en.wikipedia.org/wiki/Lojban https://en.wikipedia.org/wiki/Lojban) ?

remus 9y ago

> Now I learn that there isn't one thing to learn but a vast array? No thanks. I had this apparently romantic view that an equation is an equation (the same equation) anywhere in the world. It's not as bad as that! You're very unlikely to hit this sort of issue until you're reading research papers (unless you decide you're very interested in mathematical history). Pretty much all the material youll come across up to ~undergrad level will use very, very similar notation.

pbhjpbhj 9y ago
The first alternate notation I think in maths, I used was dot (period) notation for scalar multiplication. Next would be different versions of standard form (aka engineering notation). Later, at high-school (UK, 16-18yo) we used both d/dx type notation for differentiation and x-dot (a dot above the letter for each level of differentiation). There would be 3 versions of x= ut + (at^2)/2 that we'd know, this one, d/dx, and dot-notation; primarily the former was used in physics and the latter in maths. We'd also use suffixed numbering, u_0, u_1 on occasions and use u, v, w on other occasions. One important point - they're not different equations, just written differently. You write for your audience, in an advanced paper you can abbreviate and use shorthand because you know the reader (in general) will understand. As an undergrad I found that each department I studied in would tend to use different notation - chemistry, ecology, maths, physics, computing - for the same things. Though sometimes different professors would buck the trend ... they also had different accents and vocabulary, our QFT professor loved the word recapitulation. You also get different symbol use, like rho for density in one domain and d in another; but that's not really notation per se. TL;DR sometimes you come across different notation earlier. Different notation doesn't change the equation.
bzbarsky 9y ago
There are at least three different commonly used notations I'm aware of for division.... And that's not counting the long division process, for which I'm aware of at least two different notations (usage varies by country).

TuringTest 9y ago

> Why is that? Why is there not a universal way of writing mathematics that is not ambiguous and can be read by anyone that understands the 'language'? There is one, it's called formal logic. The problem with it is that it's not adequate to explain things, only to prove theorems formally; any strict unambiguous proof written in formal logic looks more like a debug trace than a scientific article. To provide a proper explanation of a mathematical result, at many points in the proof you need to summarize the hundreds of intermediate mechanical steps and write the insight behind them in common language, using potentially ambiguous words.

rsj_hn 9y ago

What do you mean by "writing" mathematics? The glib answer is that math is a superset of programming and there are non-computable arithmetic functions, so programming is not rich enough to describe what you need to describe when you are talking about math. I think a more intuitive answer is that in math you are generally talking about relations between infinite sets of objects, whereas in programming you are living in a (in practice) finite space and are evaluating everything into an integer. Programming is about calculating integers, mathematics is about proving theorems. The language has to be very different because what is crucial about evaluating integers unambiguously is peripheral to proving theorems and vice versa. You can ask, why is it so painful to give a formal proof of correctness of your program? It's impossible to do for all but the most trivial programs. You know that a brilliant person, given years of work, might be able to come up with a proof of correctness, but you also know that the language they would use to do that would be very different from the language that you as programmer would use to write your program. So it goes in both directions. Programs evaluate numbers, they are essentially adding machines, and central to that is the erasure of state via addition. Math is about symbolic relations and erasure of state is via relations on infinite sets. For example, you want to prove that if a group has a prime number of elements, then it must be cyclic. So pick a generator, raise it to powers, and get a subgroup. Then by Cauchy's theorem, the order of the subgroup has to divide the order of the main group, and that order is prime, so therefore the subgroup has to be the whole group. There are many things to unpack in that statement. For example, Cauchy's theorem, which says that if a A is a subgroup of B, then the order of A divides the order of B. So Cauchy's theorem + Group is prime order => non-zero subgroups must be the whole group. And then every non-zero group has a non-trivial cyclic subgroup + non-zero subgroups must be the whole group => the whole group is a cyclic group. So it's bit like adding numbers, in that you forget state, but the rules are much more complicated involving quantifiers and a huge universe of sets. To build a computer that could reason that way, you'd need an infinite set of registers, rules for quantifiers, etc. and your unit of memory would be abstract set relations instead of a zero or 1. You can try to simulate special cases of that in software, say with an SMT solver. If you look at the DSL of writing code for SMT solvers, it looks different from the code that a programmer is used to writing, and the best SMT solver can't really do anything too interesting from a math point of view, with very rare exceptions. Moreover in math as practiced, no one (except some logicians) writes in formal logic. It would be too cumbersome. Simple statements like "a harmonic function on a disk achieves its maximum and minimum value on the boundary of its domain" would require thousands of pages of formal symbols and quantifiers. No one can think like that, and so we don't write math that way because we also want to forget things and make compact statements, but what we forget in math is relationships among infinite sets, things like, what are the real numbers, what is a function on the real numbers, what is a differential, what is a harmonic function, what is the maximum of a function, what is a disk, all of that is packed into our statement and replaced with a math statement that is much more compact in the same way that 1 && 0 are replaced by 0 in a computer register, but the computer register is not rich enough to model relations on sets.

jiggunjer 9y ago

I had this reaction to sign language. Seemed like deaf people missed a good opportunity, but then I realized that how we talk and think is shaped very much by our language. Similarly scholars from different backgrounds have different ways of expressing a precise mathematical concept. This one paper I read implicitly treated complex scalars as real vectors, took me some time to clear up the confusion.

pathsjs 9y ago

An exercise that I made during my degree and phd years in maths was to always take notes in a consistent notation across the various lectures, regardless of the convention used by the teacher. It forced me to pay attention and as a bonus I have a consistent set of notes for most undergraduate mathematics :-)

joppy 9y ago

Why should all mathematics be written using exactly the same language and conventions? Should all computer programs be written in the same language? People modify the language (natural, mathematical, programming) they are using to suit their problem. It is unreasonable to expect that the kind of language you use at the foundations of mathematics for example (set axioms and so on) would be efficient and useful for stating properties of much more complex objects: this exactly parallels computer programming languages. Admittedly, there are no big tables containing "definitions" of these languages. But I think it is common opinion that if you don't understand the language a paper is using, then there is a very large chance you don't know the underlying theory, and the paper will make no sense anyway.

Firadeoclus 9y ago
There's also a small but non-negligible chance that you could have understood the paper because you know (part of) the underlying theory under a different name, using different notation and concepts but leading to the same results.