What is "Mathematics"

Edited By Frances IoM on 31/05/2022 09:56:50

Isn't that last technique - for areas - something of a forerunner of the Mid-Ordinate method of Integration (which I recall vaguely as summing rectangles of [Area = Width X Centre-line Height] ) ?

Hi Fraances

Certainly the notation and language of Maths is invented but the Maths is maybe discovered? Hard to argue that Primes were invented.

regards Martin

I see Mathematics as a tool used to promote clear reasoning in support of practical and theoretical problem solving. It's a broad discipline - whilst engineers mostly only use maths to extract information from data, scientists often use it to explore and validate new concepts.

As maths is a tool it must be man made.

Maths and the properties of nature it can be used to describe aren't the same. Maths serves the people who invented it and although number and formulae can predict and quantify nature, nature doesn't require maths to do its stuff. For example, although Maths can be used to quantify Forces and predict how they influence events, maths can't define what a Force actually is. Despite humans knowing masses are weakly attracted to each other, and understanding the relationship well enough to land a robot on Mars, we don't know what gravity is. Nature just happens to obey laws that we can describe mathematically, and therefore model nature.

As Maths is a tool it follows that people can and do live happily without it. However. I'd argue it is the most powerful tool ever invented and therefore worth learning as far as our limited time and aptitude will allow. Basic engineering may be valuable, but Maths is absolutely essential to move science and engineering forward.

Dave

Edited By Frances IoM on 31/05/2022 21:04:08

Where did you get that idea from? I used maths to model all sorts of things, including solving differential equations. Needs lots of icepacks for sure, but invaluable. I'm not pompous enough to quote the 'no maths = no engineer' so beloved of academics, but some engineers certainly do clever maths, much cleverer than I would dream of

Edited By duncan webster on 01/06/2022 00:11:33

The extract information idea can certainly be challenged: it's a generalisation, and they're always built on sand.

My experience of maths is of levels:

• We all manage money to some degree:
  • Doing ordinary shopping, I have a rough idea what I can afford, which is why I'm more likely to buy Beer than Champagne.
  • Buying a house, with a wife and young family on the way, I did a much more careful analysis, considering my reliable salary, low likelihood of generous pay settlements, and promotion prospects relative to inflation and interest rates over a decade. It led to a marital dispute, because the figures revealed my desire to prioritise the house, collided with my wife's urgent desire to fill it with carpets, cushions, and Tippets for the children.
  • At work, even more elaborate financial analysis was done before investing corporate dosh in equipment and projects. For example, Business Cases always required me to provide a Discounted Cash Flow. A DCF compares Benefit - Cost over the lifetime of a proposed investment with the return that would be obtained by putting the money in a safe Building Society and just taking the interest. (Today's ultra-low interest rates justify UK government's distinctly un-Conservative spending policy, and a cynic buying a house today, might worry the politicians might be planning to push the value of the debt down by allowing a risky burst of high-inflation, hoping they can control it.)
• Engineering is closely related to money in that 'Any fool can do for a Guinea, what an Engineer does for a Pound'. Much more to engineering than bolting stuff together to get a result, especially when more than one option is available. But it too has levels:
  • Given walls of 'n' square metres, how many litres of paint to I need to buy? Or. how much electric cable is needed to run a spur from the Consumer Unit to a garden workshop, after noticing Manhattan Routing the wire around the outside of the house is least work and intrusion? Fairly simple - extracting information from data .
  • If 50,000 litres of water per hour are to be pumped from a 50 metre deep well, how powerful does the pump need to be, and how much pressure will the pipes have to withstand? Moderately complicated, especially if the cost of high-pressure pipework is reduced by pumping in two or more stages, but still extracting information from data and using it to select off-the-shelf solutions..
  • If a nuclear reactor produces 100MW of waste heat, what combination of pipework, coolant, and radiator provides the cheapest fail-safe way of managing it, ideally in a profitable way? This example is much more complicated: the coolant could be air, water, Carbon Dioxide or molten Sodium, each with it's own peculiar set of advantages and disadvantages. There are multiple answers, and I think the only practical way to compare them is with mathematical models, Aerospace and many other design activities call for a similar approach.

Engineering maths can be and often is highly advanced. For example, it was once fairly easy for an amateur to hot up a car engine by haphazardly opening up anything that impeded the flow of fuel into the engine, and exhaust out of it. Bigger carburettor(s), supercharger, chamfered ports, wide bore exhaust, silencer removed, pipes polished internally etc. Such no-sums-necessary techniques get significant improvements, but doing better requires deep thought and analysis. For example, early IC's engine designers worked hard to implement the maximum efficiency Carnot Cycle. Clever stuff, but modern engines are polytropic, balanced to optimise several other desirable features such as acceleration and reduced pollution. In short, getting the best out of an IC engine requires a solid understanding of mathematical thermo-dynamics beyond me and most petrol-heads. The modern motor car has been optimised in many other ways too, from road-holding suspensions to absorbing crash energy by folding up gracefully. The day of the individual inventor developing a successful mechanical product in a shed has pretty much gone.

Practical engineering is vital it's own right - it's how things get done, demanding skill and experience academic types rarely have. I admire Alec Issigonis and the chap who serviced and MOT'd my daughters car yesterday. They both do technical stuff I can't!

Scientists go a step further: the maths that led to E=mc squared, Black Holes  and Dark Matter are exploratory, rather than Information from data.

Dave

Edited By SillyOldDuffer on 01/06/2022 11:06:34

An interesting discussion about something I don't understand very well. Anything higher than differential calculus was too much for me back in my schooldays, long ago...
However, for unknown reasons rooting back in time I found in my stack of unread/partially read books the following one:
The Mathematical Experience, by Philip J Davis and Reuben Hersh. Wikipedia says a lot about these two mathematicians. The book has an interesting comment by the New Scientist on the front page: "An instant classic, it deserves to be read by everyone with an interest in the future of the human race". Oh well, maybe I should read more than just a few pages, but it is quite hard work for me. And my "interest in the future of the human race" is maybe not so high now.

But what bears reference to this thread is the beginning of the first chapter, which has the title "What is Mathematics?" This is followed by what the authors call a naive definition: "Mathematics is the science of quantity and space".

Thinking hard about that,
Hans

I would agree with that definition, but push it a bit more to say it is also about motion, and related things such as velocity, acceleration, etc.

One part of mathematics that is useful is the "bean counting" aspects of it.

Another part of mathematics is how it can be used as a predictive tool, assuming your model is sufficiently accurate. With the correct mathematical model, on can predict the behavior of things that one has never seen, which is quite magical in some respects, and very profitiable if you run an engineering firm.

.

And another great man said : "In theory, theory and practice are the same. In practice, they ain't". (In theory said by Albert Einstein. In practice, attributed to various, ranging from Richard Feynman to baseball coach Yogi Berra.)

If you really want to know what maths is, a good place to look is the philosophy of mathematics. Wiki has a succinct (not!) summary here **LINK**

The greatest human minds have wrestled with this question since the days of the ancient Greeks and come up with multiple theories of what maths is (are?? Hmm??) In practice, it seems nobody can really say completely or for sure what it all is.

Now, one scientist once said that scientists need the philosophy of science about like birds need ornithology. But one would hope that scientists, including mathematicians, have a greater level of self-awareness than the average bird. Although, sometimes I am not sure...

The 'scientific method' is to derive a theory from first principles, but to then test it by experiment to make sure it properly describes the real world. If it doesn't then either the theory is wrong or the experiment is wrong. Usually the former, but sometimes the latter. Working out for example how much a beam will bend is based on theory, but it is well supported by long years of use.

Is looking up a formula and plugging in the numbers mathematics? Arguably it's arithmetic.

Why are some people, including those in positions of power, proud of being inumerate? Let's set the cat amongst the pigeons, accountants don't do maths, they do arithmetic. Same applies to the majority of the population. Why then do we complicate the issue by teaching schoolkids the clever stuff rather than life skills like not taking out loans at usurious interest rates. Some of the stuff my kids were taught was simply wrong, for example one question in an exam paper had one equation with 2 unknowns and they were asked to find both unknowns. The maths teacher would not have it that there were an infinite number of answers

Duncan is pulling my leg with 'inumerate', he knows I'm innumerate!

He's right though, the number of people I meet who are positively proud of not knowing things is scary. Worse, lots of people think ignorance of technology means they are superior beings, describing those in the know as 'mere technicians'. Not just their ignorance that's alarming, it's the faulty logic and self-delusion that goes with it! How many Prime Ministers would you trust to return home with the right stuff after being sent shopping?

I argue 'Arithmetic; is a member of the Set 'Mathematics' and is therefore respectable!

Bertrand Russell showed maths had limits with paradoxes such as 'Is the set of all sets a member of itself or not?' In Doctor Who, asking this question reversed the polarity of the neutron flow and the resulting infinite recursion caused the Big Bang. This is why Health and Safety is paramount in my workshop - who knows what this Clown will do next...

Dave

Edited By SillyOldDuffer on 03/06/2022 13:50:58

Being innumerate is even worse than being inumerate, but it shows you can spell, which I plainly can't!