Naked Science Forum
General Discussion & Feedback => Just Chat! => Topic started by: Eternal Student on 17/06/2021 00:04:16
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Hi.
Why do you (Nakedscientists) have a thousand sections for life sciences (that was just a rough count) and not one section for Mathematics? Mathematics is everywhere but life sciences are only useful in a small number of real world situations.
Let's say Jane went shopping and picked up a thing that required her to evaluate an integral over a closed path on the complex plane. Alternatively consider John, who was out walking in the sunshine when he had a need to generate an infinite sequence of numbers that are co-prime but not primes. Where are these people going to get help with their serious problems?
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Hi.
Why do you (Nakedscientists) have a thousand sections for life sciences (that was just a rough count) and not one section for Mathematics? Mathematics is everywhere but life sciences are only useful in a small number of real world situations.
Let's say Jane went shopping and picked up a thing that required her to evaluate an integral over a closed path on the complex plane. Alternatively consider John, who was out walking in the sunshine when he had a need to generate an infinite sequence of numbers that are co-prime but not primes. Where are these people going to get help with their serious problems?
General maths is not a science, unlike economics or statistics which is a way of interpreting data through maths.
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Probably because most of us aren't that advanced when it come to mathematics ES. But yes, I agree in that it could be one of TNS offerings. Although when mathematics is needed it usually will get used here too. If you use it though it should be a priority to be very clear about what your symbols indicate, and how to read it/you. For much the same reasons.
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The principal language of science is mathematics. If this were a cricket forum, you wouldn't expect it to have a section on English.
John and Jane do indeed have serious problems, but I don't know where to look for clinical guidance. Georg Cantor never really recovered.
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Let's say Jane went shopping and picked up a thing that required her to evaluate an integral over a closed path on the complex plane. Alternatively consider John, who was out walking in the sunshine when he had a need to generate an infinite sequence of numbers that are co-prime but not primes. Where are these people going to get help with their serious problems?
Well, they could just sign up to a math forum.
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You have a good point there Origin, but it's also about what type of mathematics one use. String theory has its own f.ex.. I'm no expert on in, but I've noticed elsewhere that the mathematics used seem to differ with what discipline you work under. And the ones wondering may not want it in pure mathematics, as you might lose them. Which is why I think, and as I noticed here, most people explain their variables and define them, as well as whatever equation might be involved. I think it's important to do it that way.
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That's probably the difference between what TNS wants to be and what f.ex physicsforum look for. Haven't checked but I would expect them to have a mathematical section there. That doesn't mean that TNS couldn't have it too, but I'm not sure it fits its guidelines?
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Hi all.
Thanks for taking some time to reply.
I don't really want a Mathematics section to be honest. I'm just a little surprised that you don't have one.
I'm going to assume that no one would grumble if an applied Mathematics problem found it's way into some other section (probably the Physics, Astro and Cosmo section). It's a bit mean not to consider Mathematics a science.
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Mathematics is definitely a science ES. A very abstract one, and one that sometimes need invention to correctly describe the world. Einstein f.ex had to look for new mathematical definitions to describe GR, if I remember right that is. But it builds on logic, and without logic we would be in a magic land.
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Mathematics is definitely a science ES.
I would definitely not call math a science. However, you cannot do any meaningful science without mathematics.
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What comes first Origin, the mathematics or the experiments? Sometimes it's one sometimes it's the other.
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I differ there Origin, I think it's a science. It's logic, and that is what we use.
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I think it's a science. It's logic, and that is what we use.
Science is supposed to be the investigation of the world around us by observation and experiment.
Not sure logic fits that, pure maths doesn’t.
Maths has lots of proofs, which science doesn’t.
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Hi.
I've added a poll to this thread. I've never tried this on a forum post before.
Please note that I am not a moderator or involved with the software used on this website. It is simply not in my power to act on this information.
This is a thread in the "Just Chat" section and I'm just showing some consideration for my fellow Mathematicians. I've cast my vote already (No). It's not because I don't think mathematics is a science, I do, it's just that I wouldn't use such a section much and I remain confident that no one will object to some Mathematics questions appearing in the other existing sections sometimes.
Best wishes to all.
LATE EDITING: Ticked the box to make the poll open to guests AND also members should be able to change their vote later.
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Hi all,
So Colin/Yor on ;
I think it's a science. It's logic, and that is what we use.
Science is supposed to be the investigation of the world around us by observation and experiment.
Not sure logic fits that, pure maths doesn’t.
Maths has lots of proofs, which science doesn’t.
Its funny that there is a potential for different interpretations as to whether mathematics is a science in its own right, I don't know if there is a right or wrong resolution.
https://en.wikipedia.org/wiki/Mathematical_proof
https://en.wikipedia.org/wiki/Pythagorean_theorem
Does the geometry of the physical world cross over the two sides of the debate, or is mathematics purely a language we can utilize to describe/predict physical occurrence in the scientific method ?
https://en.wikipedia.org/wiki/Scientific_method
Either way my dogs can run and jump to catch a tennis ball mid flight without knowing anything about x^2 equations
https://en.wikipedia.org/wiki/Parabola
;)
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Mathematical proofs are absolute deductions from clearly stated axioms. So mathematical knowledge, in the form of standard formulae and solutions, grows by expansion with no culling.
Scientific proof is closer to "proof spirit" or the German Prufung - a test. Scientific knowledge is the residue of explanatory hypotheses that have not been disproved by experiment, so it grows almost by contraction towards "unified theories" and the like, and scientific data, the measurements that we use for engineering and for investigating the validity of our hypotheses, grows by expansion and refinement.
The bit in between, where we use known mathematics to model our observations and make predictions, is intellectually treacherous: as can be seen from many threads in this forum, it is tempting to believe that the model is the reality. The simplest case of the behavior of light, demonstrates that you sometimes need two very different models to predict the result of a single reality.
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Hi all,
Beautifully put Alan.
😎
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Hi.
Mathematical proofs are absolute deductions from clearly stated axioms....... etc.......
This is the rose tinted view of what Mathematics is.
The history of Mathematics did not involve first developing some axioms and then constructing some theorems and consequences from those axioms. Instead it was a rather haphazard collection of observations made from reality mixed in with some idyllic notions and abstractions about how things should work. This is pretty much how all modern science developed.
There wasn't an instant leap of faith that a structure like the Real numbers existed. The ancient Greeks were convinced that irrational numbers like √2 could not and should not exist. However, the ancient Greeks were pioneers in developing the axioms of Euclidean geometry and laying the groundwork for what would later be considered as a system of formal proof. They (the ancient Greeks) obviously were not starting from a consistent set of axioms (otherwise irrational numbers would not have troubled them) but instead they were discovering and developing Mathematics as they went along. Any notions of fundamental axioms that they may have had were inconsistent and incomplete.
Like other scientific theories, Mathematics has been developed along the lines of Classical Baconian induction rather than springing forth as a set of deductions from a declaration of axioms. There is still controversy over whether Mathematics is "discovered" or "created" - to skip such a discussion let's agree that it is both discovered and artifically developed.
Examples:
(1) Set theory is generally considered to have been established in about 1870 by Georg Cantor and Richard Dedekind and this was actually only naive set theory.
The more formal axiomatic set theory that is recognised by Mathematicians today such as the ZF(C) axiom system wasn't established until the twentieth century.
This gives us two choices:
(i) We can decide that "Mathematics" developed before the twentieth century wasn't true or formal Mathematics, instead it was just Pseudo-Mathematics, since it could not have been based on a formal set theory axiom system. So, for example, the ancient Greeks weren't doing Mathematics.
(ii) We define "Mathematics" more sensibly. It does not always have to be a formal process of making deductons from a set of axioms.
(2) It is very likely that most scientists reading this have studied some Mathematics. So you will "know" that you can do some Mathematics without really needing to worry about exactly which axiom system you have utilised. Given a quintic equation that describes the path taken by some object, you can at least use numerical techniques to find solutions for the path. In every sensible notion of the phrase "you will have done some Mathematics" and yet many of these scientists would be unable tell me if they required the axioms of Euclidean geometry to solve that equation.
More generally, the thinking processes and procedures that mathematicians use to develop new Mathematics and solve new problems do not always (I'd say quite often don't) involve any consideration of the fundamental axioms.
(3) It is just speculation that set theory is the best framework to provide Mathematicians with their fundamental axioms (which are roughly equivalent to their fundamental or irreducible particles in Physics - the things from which all other structures are made). For example, we have seen the development of Peano's axioms which are a good attempt to remove the assumption that the Natural numbers were god given and showed that they can be constructed from more elementary objects. This gave Mathematician's confidence that set theory may be our fundamental objects (e.g. Bertrand Russell wrote an entire book about this idea where he tried to re-create all mathematical proofs and statements from set theory). However, we have also been shown results such as Godel's incompleteness theorems which shatter some of our hopes. We now know that there can be Mathematical statements that are true but cannot be proved from fundamental axioms, along with the serious concern that we cannot always formally prove that an axiom system is self-consistent.
Our set theory is a good working model of what Mathematics is formally based on but that is all. Atoms are a good working model of the most fundamental particles in physics but the possibilitiy exists that atoms never were real objects (there were perhaps only fields). This would not rip up all the value of any Physics where atoms were considered as fundamental particles. In the same way, it is quite possible that the axioms of Mathematics are NOT based on set theory as we may have assumed but there is something else that is more fundamental and would be a better basis for our axiom system. This would not rip up the value of all the mathematics we have already developed or make it less useful as a model of reality.
(4) Frequently Pure Mathematician's are given the task of working backwards to find axioms from results and NOT to find results from axioms. There is a final overall statement, a result that must be considered. The pure mathematician's are given the task of assuming the result to be true and working backwards to identify the smallest set of atomic statements that would be required to hold. The most common example of this is that the axiom of choice is considered a controversial higher order axiom in set theory and it is desirable to establish that a result can hold without requiring the axiom of choice.
This sort of "reverse engineering" is every bit as much a part of Pure Mathematics as working from the axioms forwards and building up a new result.
Summary
In reality, Mathematician's are scientists trying to understand and analyse either the real world or a more abstract idealisation of something that was put together to model the real world. They use thinking processes, analytical skills and seek a level of rigour and proof that are akin to those used by Scientists. Constructing formal proofs and manipulating axioms is little more than a poetic description of what Mathematics is.
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Heh, I'm of two minds, but I think I will call it a science, and a logic, or 'logic's' anyway. What else would one define it to build on if there was no logic to it? That the logic doesn't leads to practical use doesn't state that it isn't a logic.
And if you think of it, what defines this 'universe'. Laws, properties, constants. Show me the grains and I will change my mind, possibly :)
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ES have you considered that this Baconian method also leads to indeterminacy?
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Agreed there is a lot more to maths than proofs, but the fact remains that mathematical proofs are absolute whereas scientific proofs are "moving best estimate" even if they move very slowly.
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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.
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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.
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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.
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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.
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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.
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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".
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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.
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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.
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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.
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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.
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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).
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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.
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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...)
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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.
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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.
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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?
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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
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