[yor_on]

Agreed Alan, sometimes I think of, and call mathematics a tool. That is when we apply it on our reality. Then it becomes our best tool for describing it. So I am of two minds, but I don't think I can ignore the possibility of mathematics being a science either. Sometimes the mathematics already exist before it's even possible to use. We sort of 'rediscover' what some mathematician already proved.

[Eternal Student (OP)]

Hi all.

Let's see if we can stir up some interest in Mathematics.  What is counting?  This should be something everyone has thought about.  You might even teach someone to count, maybe you have children of your own, what is it that you are teaching them?

Low level:
    So you think you can count things?   What are you actually doing?
Pointing at things and saying a number?   Would it be OK if you point at the same thing more than once and say a number?  Is it OK if you accidentally (or deliberately) don't point to a thing and assign it a number?
Does it matter if you change the order in which you were pointing at things (let's say going from right to left instead of left to right, or going from the left but skipping one as you move right and then coming back from the right to the left to count the ones you skipped).

Completing a primary school education we might reasonably describe the process of counting a set of things as follows:
You must point to each thing once and only once and say a number (starting with 1, then 2 and following sequentially through the set of counting numbers).   However, you can do this pointing and counting in any order you like.

Here's a quote from Wikipedia    (https://en.Wikipedia.org/wiki/Counting):
In mathematics, the essence of counting a set and finding a result n, is that it establishes a one-to-one correspondence (or bijection) of the set with the subset of positive integers {1, 2, ..., n}.

The idea of finding a bijection with the Natural numbers is a very powerfull one and if you understand it you're ready to consider the countability or uncountability of infinite sets.  However, let's not develop our discussion along these lines but instead go back for a moment.  Were you really finding bijections when you count a set of objects or were you actually doing something else?

Let's say you have counted some things - does that mean that you have assigned some ordering to them?  More specifically, could you put the things in a line starting with the object you called 1 and then have object number 2 next to it etc. etc.  ?

Medium level:   
   A total ordering of a set is exactly what you might think it is.  If you're already familiar with the idea, use that.  Otherwise I'll provide an abridged definition based on the use of  the "less than" symbol < .   Other references will allow equality in such a relation (so it's more like a ≤ sign).

 A set, S, is totally ordered if we can identify an order relation   <  and exactly one of the following is always true for any elements  a,b  in S:
     (i)  a < b
     (ii)  b < a
or, (iii) a = b    (the equals relation is understood to mean a is exactly the same as b).
   Additionally, an order relation should have a transitive property:  If a <b   and  b<c   then we can assert a< c.

Consider the following statements and decide if they are true or false:
    Every countable set can be (totally) ordered.
    There are some sets that are countable but cannot be ordered.
    There are some sets that can be (totally) ordered but are not countable.
   
(By the way, some of these are tricky to verify or falsify if we allow infinite sets.  However, it will be enough for now if you only want to consider sets of finite size).

High Level
      What have you been doing when you count a set of things?  Finding a bijection (a one-to-one correspondence) between the set and the natural numbers    OR   finding a way to order the set of things?

      Can you define a (total) order relation on the set of Complex Integers (often called "Gaussian Integers" these are the set of numbers  a+bi  where a,b are integers)?
This could be difficult and the Complex numbers don't have an order relation naturally defined on them.  For example, the modulus won't separate 1+2i  or 2+1i and they obviously aren't equal to each other.
    Would it be easier if we asked a different question - is the set of Gaussian Integers countable?

   Counting things is about seeing (or imposing) an order between the things.  It always has been and we're very good at it.  I can give you lots of different sets of things and you can count them.

[Halc]

This seemed to be the only question asked, and probably not because you don't know the answer.

Would it be easier if we asked a different question - is the set of Gaussian Integers countable?

Yes, just like the rational numbers are countable, and using the same method.
For one, there is an intuitive bijunction between the rational numbers and the the complex integers.

[Petrochemicals]

How to write "Square metres" short hand?  If you write 20^2m  or 20m2 it is 20 metres squared, which is totally different to 20 square metres.

[Bored chemist]

What comes first Origin, the mathematics or the experiments? Sometimes it's one sometimes it's the other.

Some of the most beautiful and important bits of science don't really need maths.
You can explain Darwin's theory of evolution without doing any maths.
Olber's paradox needs you to be able to count, but that's about it.
[pick a direction and keep going; if the universe goes on "forever" then eventually, there is a star in that direction, so it should be bright]

The way that DNA works is not intrinsically mathematical.
You don't need numbers to explain how you can unfasten a zip, and then add teeth + cloth to the pieces to make another zip.

[alancalverd]

What have you been doing when you count a set of things?  Finding a bijection (a one-to-one correspondence) between the set and the natural numbers    OR   finding a way to order the set of things?

You have to do the first before you can do the second.

And it's not a unique definition. If you assign a correpondence between your set and a sample of a pseudorandom number table, you can reorder the set but the largest assigned correspondence doesn't tell you anything about the size of the set. It's a useful mathematical tool for eliminating bias in an experiment, but it ain't "counting". 

[alancalverd]

Some of the most beautiful and important bits of science don't really need maths.
You can explain Darwin's theory of evolution without doing any maths.
Olber's paradox needs you to be able to count, but that's about it.
[pick a direction and keep going; if the universe goes on "forever" then eventually, there is a star in that direction, so it should be bright]

The way that DNA works is not intrinsically mathematical.
You don't need numbers to explain how you can unfasten a zip, and then add teeth + cloth to the pieces to make another zip.

Maths is about more than counting or numbers.

Nevertheless evolution depends on a random genetic change making its owner more likely to thrive than those without the mutation. "More than" is the basis of counting.

Olbers'  paradox involves the concept of a nonrecursive vector being definable in a 3-dimensional space littered with random objects. Quite a lot of mathematical ideas there, even before invoking infinity and probability.

The structure of DNA is only known because Crick and Watson happened to attend a lecture on Fourier transforms. The replication of DNA depends on the mathematical concepts of complementarity and antisymmetry, and the topological possibility of unwinding a finite helix - it wouldn't work with the requisite precision if it were a donut, and there's no chemical reason why such a compound couldn't exist. 

There are a few societies that don't have any concept of cardinal number. Can't remember the title of the book but around 20 years ago  I attended a lecture by an author who had lived  with a South American tribe that had some concept of ordinals (precedence) but lived in a virtually constant environment with no external human contact  so had no need for cardinals, and only had one vector: towards or away from the river. 

[Eternal Student (OP)]

You have to do the first (counting) before you can do the second (ordering).

Practical:    I can't count a load of coins that are thrown on the table.  I have to order them first to make sure I don't accidentally count one twice or alternatively skip one and not count it.   Even if I don't physically move the coins around with my hands I will visually decide what is the best way to start counting to avoid problems.   in most cases, I am using spatial location information about the objects to impose an order relation between the objects.  If I'm really unlucky two coins will fall one directly on top of the other and if I don't or can't order the objects by examining this third dimension then I won't sucessfully count them.   For example, if I was shown a photograph of the coins taken from the top then I cannot count them simply because there is no way to create a total ordering of the coins using the 2-dimensional spatial information available.  On a practical level I would say we must order things first and only then can we count them.

Theoretical:    It's arbitrary which is done first.  We can discover or establish the truth of one before the other but they would always have been either both true or both false simulatneously. 

And it's not a unique definition.

   Yes.  Counting can be done by pointing at the objects in any order.  It is often imposing an order and not identifying any underlying reason why that ordering is more natural than any other.
    I'm not sure what the main point or issue was here.   Counting isn't an attempt to identify the most natural relationship between things, just that they can be ordered.

If you assign a correpondence between your set and a sample of a pseudorandom number table....

  I'm not sure what this section is about.  If your random numbers happened to repeat then you've not totally ordered the set of objects or counted them.
   Once a (finite) set of objects has been ordered then there is a corresponding bijection with a set of Natural numbers that has been identified.  The "smallest" object in your ordering will be identified with the number 1, the next smallest is 2  etc. etc.    It may take the person some extra work to be certain what the terminal number, n, the total number of objects actually is but the bijection (the counting process) is established by the order relation.

    This is more complicated for infinte sets,  some infinite sets can be totally ordered (like the real numbers and the order relation < "less than"  ) but they are still uncountable.  I've left infinite sets almost completely out of previous discussions since it's not obvious how we (human beings) naturally tend to deal with them.  As it happens, I think we are happy to use the Real numbers because they are naturally ordered and that allows us to make sense of them.  We don't have a great need to actually "count" them.

[Eternal Student (OP)]

How to write "Square metres" short hand?  If you write 20^2m  or 20m2 it is 20 metres squared, which is totally different to 20 square metres.

  Hi Petrochemicals.

20²  metres     is    400 metres   and this stands apart from the other things you mentioned.  However, a merciful audience would be suspicious you had just accidentally put your squared symbol in the wrong place.
I suppose you could write  (20 m)²  and this would technically be  the same as  400  m² if you really wanted to annoy people.

Meanwhile,   20 square metres    or   20 m²      or    20 metres squared        all mean the same thing.

I think it's common enough to say  "metres squared"   but   very rare to actually write that down.  We would tend to write down either  "sq. metres"    or   m².

[Eternal Student (OP)]

Origin said:
  I would definitely not call math a science.  However, you cannot do any meaningful science without mathematics.

followed by Yor_on:
  What comes first Origin, the mathematics or the experiments? Sometimes it's one sometimes it's the other.

followed by Bored_Chemist:
   Some of the most beautiful and important bits of science don't really need maths.....

   Much of this has been discussed elsewhere already.  I was only going to remind people that there isn't an easy way to identify what came first.   Early Natural Philosophy was the Mathematics, the Science and the Philosophy of the day.   Ancient Greeks and other early civilizations didn't bother to separate their Mathematicians from their Scientists.

[Eternal Student (OP)]

Hi.

I thought I might also pick up on this:

You can explain Darwin's theory of evolution without doing any maths.

    POSSIBLY you can.
However it's interesting how much Maths is used without realising it.   For example, what sort of life could evolve if the universe had only 2 spatial dimensions (or just 1)?   The case with 2 dimensions is frequently examined.  It is suggested that an organism could not evolve an advanced digestive system - because the evolution of an alimentary canal would effectively cut the organism into two organisms.  The case with just 1 dimension is even worse - two organisms would never be able to get past one another and this causes all sorts of problems and limitations.   However we can see that 3-dimensions is enough to work with and we have a partial argument to explain why we may not be aware of more even though Physicists (especially String theorists) suggest there may be more:  Organisms such as ourselves simply don't need to be aware or utilise more spatial dimensions.   (There is more discussion and a better development of the ideas online and I think there's a few popular You Tube videos).

------
Voting Update
   The poll attached to this thread is going to close soon.  Last chance to vote.  As previously stated - there's no reward for voting and I'm not involved with TNS (except as an ordinary member).  It's just interesting to see who thinks a Mathematics section should be added.  Voting should be open to everyone including guests who haven't registered.   Thanks to all for your contributions,  bye for now and best wishes.

[evan_au]

Why do you (Nakedscientists) have a thousand sections for life sciences (that was just a rough count) and not one section for Mathematics?

I would speculate that the founder of the forum (Dr Chris Smith) is a virologist by trade, and thus attracted a lot of biology questions.

But if you listen to an episode of "Ask! The Naked Scientist", you will discover that he has an amazing breadth of knowledge across many fields, and is able to present this clearly in a "Questions without notice" talkback radio format.
For example: https://www.thenakedscientists.com/podcasts/ask-naked-scientists/why-wont-water-burn

I conclude that the categories on the forum now represent the types of questions that he has been asked by the general public, over many years.
- Sometimes people need help on their maths homework (pure maths)
- or physics homework (which usually involves maths)
- but your general mathematician would consult a specialist mathematics forum
- We do get questions about maths; the most common questions tend to be about infinity (which certainly challenged a lot of mathematicians, even after Cantor)
- Frequently associated with General Relativity, the size of the universe and Black Holes (which all involve actual or potential infinities)
- Often by people who think that they have disproved Einstein, or showed that space flight is impossible, or that black holes emit matter (rather than attract it...), or believe that they have overcome the decay implicit in thermodynamics.
- And on this forum, we often violate Euclid's 5th axiom, since there are many questions about the topology of black holes and the universe.
See: https://en.wikipedia.org/wiki/Parallel_postulate
(I see they call it a postulate, but I think that Euclid used it as an axiom?)

Some of the most beautiful and important bits of science don't really need maths.
You can explain Darwin's theory of evolution without doing any maths.

I would put it in reverse - the maths of biological systems is so complex that until recently, humans (or computers) could not start to cope with it. And so biologists used hand-waving arguments or "just so" stories.

I was interested to hear an interview with a scientist who had started the "mathification" of ecosystems by representing the interactions of the species as matrices, and making deductions about the stability of the ecosystem by looking at the properties of these matrices.

But in reality, biological system cross the critical threshold of > 3 non-linearly interacting systems, which means that they are almost all chaotic in nature. And chaotic systems are hard to model in pure mathematics. That's why applied mathematicians and physicists often have to resort to computer simulations. Some recent Nobel Prizes have been awarded for computer modeling of biological molecules.

So I think you will find more discussion here of computer analysis than of pure mathematics.
- Even when I see a question on the forum about pure maths, I will often model it on a spreadsheet, and provide a graph of the results (my training is in Engineering, not pure maths...)
- See, for example the analysis of the ancient Greek method of approximating pi with inscribed and circumscribed polygons.
https://www.thenakedscientists.com/forum/index.php?topic=81365.20

PS: If you want to do a lot of maths, you will need to master the version of LaTEX used on the forum - but that seems to be too hard for most forum participants (there is a user guide...)

[Eternal Student (OP)]

Hi.
The Poll has closed.

Conclusion:    Very little information about the preference of the users has been gained.

Brief details of the analysis:
Null hypothesis:   There is no underlying preference among the users to suggest that TNS should add, or not add, a Mathematics section.
Mathematical model:  People vote yes or no with equal probability.  Binomial distribution used.  The third option where no preference was declared will be discarded in this analysis.
Result:  Two-tailed Prob. of  5 No  and 1 Yes out of 6 tries  (or more extreme)  =>   approx. 22%.
Conclusion:  Null hypothesis cannot be rejected.
   Risks of corruption, bias and a non-representative sample are considerable.  Examples:  People could log on as a guest and vote 2 times;   There were only 7 responses   etc.
  - - - - - - -
Thanks to everyone for taking part.  Bye for now.

P.S.  It's not escaped my attention that many people would look at the poll results and just declare "most people do not want a Mathematics section".  It's interesting how formal statistics has softened the impact.

[alancalverd]

Very little information about the preference of the users has been gained.

...said the next leader of the Labour Party/Green Party/SNP/EU.

An amateur mathematician would conclude that, following an unrestricted debate, a 5:1 vote against a proposition constitutes a fairly strong rejection, but a politician would dismiss it as a meaningless blip.

So here's an application of set theory. Are professional mathematicians clandestine politicians? If so, why should we believe any mathematical proof? 

[gem]

Hi all
I think you have to be careful of the starting agenda of polls.

https://en.m.wikipedia.org/wiki/Lies,_damned_lies,_and_statistics

🤔