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A comment on an answer I posted asserted that "Mathematics is NOT always about counting".

My thoughts were that if there's a unit (inches / milligrams / light years etc), then someting is being counted.

An exception would be if you're describing something within maths/arthimetec itself like the concept of 2+2=4. That defines itself without needing units, but only makes a statement about itself.

So my question is: When is mathematics NOT about counting something ?

EDIT: Just to be clear, We have discrete data like the number of people in a room, or continuous data like number of miles to the nearest curry house (a measurement, ideally very small in this case).

My intention with the question is that both are "counting" - discrete data is hopefully obvious and continuous data is, I would say, still "counting" in that your'e couting a number of miles (etc).

So i'm not talking about the difference between discrete data and continuous. I'm asking more whether/when mathematics does not refer to something in the (or a) 'real' world. Having thought further about it, I think I mean "when are there no units?"

For example:

E=mC^2 has units, or a type of unit:

E=Energy = watts/calories/whatever

m=Mass = kg / lbs etc

C = speed of light (mph etc)

So to arrive at that through some presumably tricky maths, was there ever a point where a formula didn't have a unit of some sort ?

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    You are identifying Arithmetic with Mathematics... Math includes Geometry, Set theory, Mathematical Logic, and a lot of other disciplines which are not strictly involved with natural numbers, which are the base of counting. – Mauro ALLEGRANZA Oct 15 '14 at 14:50
  • Thanks - but I don't think I am, not quite. There are always units arent there ? Even if it's just the notion of a distance, or a speed (regardless of imperial / metriec units etc) - I guess a measurement "type".. eg a distance of 4, as opposed to 4 with no connection to any context.. Am I wrong there ? – user2808054 Oct 15 '14 at 15:36
  • does it always consist of quantities? – user6917 Oct 15 '14 at 16:21
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    @user2808054 Metric geometry is only a part of geometry, there is no measurement, and therefore no units, in projective geometry or topology. Category theory considers structures up to isomorphism only, so there can be no counting in principle. – Conifold Oct 17 '14 at 19:06
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    When I tell you that 7 is a prime number, what units are involved? – WillO Nov 6 '14 at 19:59
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The best example of mathematics that involves no numbers comes from philosophy. Propositional logic is mathematics. How is it somehow "really" about numbers?

But modern mathematics consists mainly of things that are not numeric, but are made up of sets of rules.

As an extreme case consider topology. The easiest form of topology to describe is graph theory. This discipline is largely about how complex connections between things can be, and still have relatively simple descriptions. The ordinary representation of a graph is a set of points that can be moved around arbitrarily, and lines connecting some of them to one another.

An early basic result determines the conditions one must put on a graph in order to draw it in the plane. Geometry is involved in an abstract way, but no measurement. So this is a rather pure example. The only number or measurement relevant to the statement of the problem is "two", and then only as the dimensionality of a plane.

Sure, the graphs have nodes, and you could count them. The descriptions often contain numbers and the most basic ones amount to things like "draw three points on the left and two on the right and connect each point on one side to all of those on the other." But even here, arithmetic is just used as part of language, not as the main actor. In general graph-theoretic computations are seldom numeric -- they are about handling symbols that represent nodes and edges. (In this way it is kind of like propositional logic. They are both parts of the general field of "Symbolic Logic and Combinatorics".)

Important results are, for instance, about whether we can find instances of a graph with one compact description in another network with an unrelated compact description. Applications are to things like computer networking, or telephone line maintenance. The products are not numbers, but sequences of operations like a computer program.

Numbers usually enter in only after a problem is solved, to compare the efficiency of different solutions.

  • "Sure, the graphs have nodes, and you could count them. The descriptions often contain numbers and the most basic ones amount to things like "draw three points on the left and two on the right and connect each point on one side to all of those on the other." -- So you agree. Graph theory is about counting. Give me a harder one! I say OP is right, math is always about counting. General topology. Arbitrary unions and finite intersections. So you have to count the cardinality of the intersections. Topology is counting. Next! – user4894 Oct 15 '14 at 16:56
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    Then so is absolutely everything else in the universe. I can count apples, so eating is about counting... I pay for my house with money, so shelter is about counting... – jobermark Oct 15 '14 at 17:07
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    If "X is true or false" is taken to mean "I counted X and there is one or zero" then you win by default. There is no way to address the question. You have cast meaning as language and language as arithmetic. But that is so artificial a framing that you leave no room for reality. People don't mean that. So if you assume they do (a false premise) you can prove what you will. Attacking a bad definition with reductio ad absurdum is not arguing the converse. – jobermark Oct 15 '14 at 18:12
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    " Propositional logic is mathematics. How is it somehow "really" about numbers?" Yes, but do you know any scholars who engaged in scholarship for the purposes of studying propositional calculi? I've found or read a few. Some of their names are Jan Lukasiewicz, Arthur Prior, John Halleck, Carew Arthur Meredith, and Dolph Ulrich. Their highest achievements lie in finding a, or the complete set of shortest axioms for propositional calculi. So, even in propositional logic counting still gets used. – Doug Spoonwood Oct 17 '14 at 5:43
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    So we are back to counting apples making food being about numbers. Circling the drain on this one. Yes, numbers are everywhere, WE PUT THEM THERE, but having numbers apply to something do not make that thing about the numbers. The best way to lay out a silicon chip involves numbers, but it is about the chip, not the numbers. – jobermark Oct 17 '14 at 16:16
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As a math major, it caused me great pain when my family thought I was just learning how to do addition really well...

Generally, pure math (i.e., excluding applied math) can be thought of as having two three main branches (although, this is likely an over-simplification):

  1. Algebra - how to use operations on sets of elements to combine two elements into another (potentially different, potentially not) element
  2. Geometry - deals with the distance between points and the things that fall out from that
  3. Fundamentals - logic and set theory, which serve as the basis for the rest of mathematics.

Certainly, counting is used for examples all fields, but in proper mathematics one usually operates in a more abstract setting. That is to say, one often does not work with numbers, arithmetic or counting directly, but rather considers things that follow the same rules, and reasons about things in that abstract setting.

For example, take the set of functions with certain technical limitations (e.g., measurable, or integrable, or differentaible... any "well-behaved" set of functions). You can define operations on them to combine them in different ways (algebra). You can define a metric that puts a distance on the elements of this set (geometry). But the idea of "counting" in this set is very unnatural.

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    Most folks split math in three, not two. You left out the best example in your division of math: Logic and Combinatorics. You can think of propositional logic as an algebra, if you want, but the point is not solving for anything, or describing multiple models with similar solution procedures. It is on keeping deductions about a single large structure well-founded, as in geometry. This puts it in a confusing position exactly halfway between your choices, and it usually earns a place as a separate branch. – jobermark Oct 15 '14 at 16:42
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    If Boolean logic qualifies as a sophisticated meaning of 'counting'... But it doesn't. Truth value is not about counting just because we can write it as binary digits when we wish to. That is like saying writing is about paper. – jobermark Oct 15 '14 at 17:03
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    Our typing is a bitstream. But our conversation isn't. Focus hard enough on the form of anything, and it will seem numerical. But that is because we think numerically, comparing quantities is what nerves are made for. But focusing that close misses the point of everything. It was cute when Pythagoras did it, but he made his point. Since then science has made it even better, and we can move on. – jobermark Oct 15 '14 at 17:13
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    From a different perspective, even our arithmetic is no longer about counting. Considering the continuum hypothesis is not about counting. It is about symbol processing, and idealization, and the notion of succession. But omega and aleph-one are not numbers, even if we call them that for comfort's sake. A diagonalization argument is not about counting just because it sorts things, otherwise sorting your clothes is about counting. We fall back on numbers for solidity, but modern math really is beyond them. – jobermark Oct 15 '14 at 17:19
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    Of course they are NOT, nor are the finite simple groups, nor are graphs or modules, and neither are truth values. Again, numbers model everything, that does not mean those things are about those numbers. – jobermark Oct 15 '14 at 17:28
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Geometry is the simplest math without numbers. Only uses a pen, ruler (ungraded, used to make straight lines) and compass.

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    This won't hold up against @user4894's scrutiny at least, because those edges have lengths, shapes have a number of points, and corners have a number of degrees between them. – Magus Oct 17 '14 at 19:07
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    Maybe I was a little cryptic. Just because you can "grade" and angle in degrees doesn't mean it is made up of those numbers. Constructing a solution using a compass for example, always leaves for you to decide the length which you take into it, it never grades that length as "1" or "2" it just uses concepts of "same"ness to measure length. – unom Oct 17 '14 at 19:23
  • @Magus (chuckle) Also, Euclidean Geometry includes compass in addition to ruler. The chuckle refers to user4894 – Nick R Oct 17 '14 at 19:23
  • @unmircea You are correct. Geometry does include constructions which are not arithmetizable. You forgot to include compass in your spec. – Nick R Oct 17 '14 at 19:25
  • @unmircea +1 for adding compass. Geometry has enjoyed an enormous revival in the last 100 years. Well, non-Euclidean geometry mostly. – Nick R Oct 17 '14 at 19:29
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Anything that is computable algorithmically can be modelled as counting. That is a lot of mathematics.

The general idea of counting was formalized mathematically using sets called ordinal numbers. Most (if not all) mathematical objects can be modelled as sets, and we know that any well-ordered set is isomorphic (equivalent) to an ordinal number. (Here, well-ordered means "has a least element" - i.e., there's a place to start counting.) Therefore, if you wish to lose sight of counting you need to be dealing with sets which are not well-ordered.

So far we have excluded mathematics which is computable and that which can be modelled as a well-ordered set.

There are no doubt further restrictions, but that's all that come to mind now.

If you belong to a school that insists that all mathematics be computable, then I guess you have excluded everything.

EDIT There are a number of comments on other answers that show some confusion about the nature of counting. One particular confusion concerns a well known mathematical conjecture called the Continuum Hypothesis.

As I mentioned in my original answer (above), Cantor formalized the concept of counting by defining the ordinal numbers. The Continuum Hypothesis asks, what is the cardinality of the continuum. All cardinalities are defined as certain types of ordinal numbers. The cardinality of the continuum is given by the cardinality of the well-ordered set [0,1] ( = the set of real numbers between 0 and 1). So the continuum hypothesis is absolutely about counting. Is asks how many ordinals do I need to count in order to count the cardinality of the continuum.

Another confusion appears to be the assertion that a model says nothing about the nature of that which it is modelling. Plainly, anything that can be modelled as counting is mathematically isomorphic (equivalent) to counting. There is not getting around it.

  • That's the most clear and well-explained example so far, thank you! – user2808054 Oct 17 '14 at 11:03
  • @user2808054 Cheers dear! I see from your profile that you are into computers (and guitar - mine are 2 nylon string guitars, 1 of which is electric). Computation is incredibly powerful. There is a thesis, called the Turing Thesis, that basically says that anything computable by a human using pencil and paper (and ignoring limitations of time and resources) is computable - i.e. can be computed by a Turing Machine. So that's an f-of-a-lot of mathematics. We have only scraped the surface of what computers can do. – Nick R Oct 17 '14 at 18:46
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Pretty much never...

I think the best way is to compare mathematics to a natural language, and the equivalent question is "When is language NOT about spelling?".

Counting is to spelling, as algebra is to making sentences, as proofs are to essays.

You can figure out the answer from there.

There are very complicated ways of "counting", when it comes to infinities, combinatorics and so on, but most questions that math solves revolve around a finite set of basic rules, most which don't involve "measure", but rather an abstract, yet (hopefully) intuitive concept. These are called axioms. Would you say that the statemet "two lines may never meet" is a form of "measure" ? It is however a mathematical statement, and we define such lines as parallel (or orthagonal in other contexts).

Would you say that the proof thereof, or the conditions for this to happen from geometry/cartesian algebra is some sort of counting? The answer is probably no.

All the examples you gave are merely math applied to physics and the real world, math does cater only to the real world, and for the most part, it does not care about units.

For example, the idea that there are infinetly many primes uses (unitless) numbers, logic and properties of these to add a new fact to a knowledge-base which was built on these axioms.

So to answer your question, very little of math is actually about "counting".

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If you start learning mathematics seriously, you will quite quickly run into the concept of a continuous function. No counting. Slightly later, you will learn about open sets, compact sets and the like. No counting. Later, holomorphic functions. They just blow your brains away. The idea that something like holomorphic functions could exist is just mindblowing. No counting.

  • Hmm a continuous function (according to wikipedia en.wikipedia.org/wiki/Continuous_function ) is still about a measurement- a number of things. it's just not necessarily an integer. – user2808054 Oct 16 '14 at 10:16
  • @user2808054 Also, a continuum can be counted transfinitely according to Cantor's 3 principles of ordinal generation. It takes a long time to count from 0 to ω1, but it can be done using the three principles. – Nick R Oct 16 '14 at 23:06
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    @user2808054: Feel free to call anything "counting". Then all mathematics and everything else is counting, but it's a rather vacuous statement. You can't count or measure real numbers. While wikipedia articles occassionaly contain nonsense, that article says nothing to justify saying that continuous functions have anything to do with measurement. – gnasher729 Oct 17 '14 at 0:15
  • @gnasher729 Please, acquaint yourself with measure theory then come back and correct your statement. – user4894 Oct 20 '14 at 19:43
  • @user4894: Studied it for about a year, passed my exam, so I know a bit about it. You seem to have the problem that you think mathematical terms mean the same as their normal English equivalents. Measure theory has nothing to do with measuring. As an indication, in German it's "Masstheorie" and not "Messtheorie". Same as you can't count countable sets, for example. – gnasher729 Oct 21 '14 at 11:56
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Lattice theory is not about counting.

  • An abstraction of the divisibility relation on the natural numbers is not about counting? I'm constantly struck by poor quality of the examples put forth here. – user4894 Oct 20 '14 at 23:52
  • @user4894 One definition of a lattice consists of an algebraic system which has certain theorems. Another definition tells us that lattice consists of a partially ordered set where each element has a supremum and an infimum. Counting tells you how many of something you have. On the other hand, telling you something about the ordering structure and how ordering structure works works out as telling you something quite different. Consider say {0, 5.55, (55+(3-e))} or any set of numbers under the maximum and minimum operations and determine if they're lattices. Did you count anything? – Doug Spoonwood Oct 21 '14 at 4:23
  • I'm using a more general meaning of counting. Surely it's not true that math is literally about grade school arithmetic. So we have to look at the more general sense. Partially ordered sets are motivated (as Wiki says) by divisibility in the natural numbers. So 5 divides 4, 10, 15, 20, 25, etc. and 3 divides 3, 6, 9, 12, 15, ... and you get a nice lattice. That's the sense in which I mean counting. Abstracted, generalized counting. The conversation is trivial if we take counting in its most literal meaning. – user4894 Oct 21 '14 at 6:53
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    @user4894: I think there was another thread here a while ago about how to argue with people who refuse to understand. You can feel free to call everything and anything "counting", and then maths is all about counting, but it would be all about "user4894 counting", not real counting. – gnasher729 Oct 21 '14 at 11:59
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    @user4894: Divisibility is used as one example to produce a partially ordered set. It is actually quite an unusual example. Much more common are two, three, infinitely dimensional sets. And divisibility, once you leave the natural numbers, has again nothing to do with counting. Maybe to you, but not to any mathematician. – gnasher729 Oct 21 '14 at 12:01

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