I found the following deep in my email inbox:
Dear Mr Sir,
I do apologise for not having written back earlier. I wrote to you last year concerning the nature of mathematics and where I might study it. Your advice was (I think) profound : solve problems, study at the appropriate level and form my own view.
I have now tried this with mixed results.
I began with 1+2=3 and most people have confirmed this as true, as was 1/10=0.1
But when I tried 0.1+0.2=0.3 (which I think is true), my computer claims it is false ( Python language ).
My computer helpfully suggested that 0.1+0.2 > 0.3 so is the sum greater than “the parts” ?
Yet also 0.1+0.7 < 0.8 , so now the sum is less than “the parts”…
I have visited several branches of PC-World(Large computer retailer in UK)and most of the computers they sell produced the same answer, even the Apples! The staff did not think this merited a refund of my purchase, and they were quite happy the machines were working(!), although they said there were software updates which might “fix” the problem.
Apparently this “error” is due to a mysterious process called “rounding” where one number is replaced by another to save space, or make it more or less accurate.
I have a friend who knows a lawyer, and this sounds very similar to the things that she does to convince people to hand over large amounts of money.
My neighbour runs a small green-grocer’s shop and he said it was complicated but not a problem.
My landlord said it was the sort of thing an engineer would do, and reflected the sorry state of the world…
We began our conversation when I said that my employer had asked me to improve my maths, this in turn to improve the accounting that I do for him. Yet it would now appear that after a year of study, I have only found that if I use my computer I am likely to get less accurate results than the way I do it by hand at the moment.
I fear that the “basic understanding” of mathematics which I have is too shallow for me to understand what it actually is and the books that I have purchased (and tried to read) on the subject are not in fact maths books but something else. I have been reading Lewis Caroll’s Alice-in-Wonderland texts ( I remember these from school) and whilst interesting maybe they are too advanced ? Would Harry Potter provide a better view or does it not really matter ?
It scares me to wonder what would happen if a large bank believed 1+2=3 , but sold stocks and bonds on a different basis…
This is a bit of a ramble, and I do apologise for that. But I am confused, and you have been helpful in the past. I do hope I have laid out and expressed my problems here.
Could it be that rather than ask “what is mathematics”, I should try a different, easier question instead such as “what is mathematics good for?” ?
I’m forming the view that mathematics is more about communication, and accuracy is an allusion or a false god. Seen in this light, the Potter/Wonderland distinction becomes clearer. Of course Star Wars would admit the interplay between the light (communication) and dark (accuracy) side of The Force.
What do you think ?
Andrew
The leadup to all this is interesting : It all started when somewhere around the age of 40 I came to the conclusion that I didn’t really know what mathematics was.
This was further confused that I had a degree in pure mathematics from a highly regarded academic institution. Three years down the drain…
After all, I’d passed the final exams and got a 2.1 honours degree and had successfully been applying it for years. So what was the problem ? The problem was that I couldn’t pin down exactly “what” mathematics was.
That frustrated me so I thought I’d ask around.
Wikipedia told me it was :
“Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline.”
All very vague and hand-wavy. But at least the final line gave me some solace in my ignorance.
The BBC told me it was :
“Maths. Mathematics is the study of numbers, calculations, shapes and equations, written using special notations and symbols.”
Jo Boaler wrote in her book :
“Mathematics is a human activity, a social phenomenon, a set of methods used to help illuminate the world, and it is part of our culture”
She also wrote :
“numbers” or “lots of rules” (children). Ask mathematicians what maths is and they will more typically tell you that it is “the study of patterns” or that it is a “set of connected ideas”. Students of other subjects, such as English and science, give similar descriptions of their subjects to experts in the same fields”
So now we have lots of different ideas, including the idea that the same description works for English and Science…
A least the kids were honest : “numbers” or “lots of rules”.
It was at this stage I realised I had not a science degree , but an arts degree. Surely maths would be classed as science ? What if my employer found out ?
So I shot off an email to various universities to see if there was a difference between arts degrees and science degrees pretending to be an absolute beginner :
Dear Mr Sir,
I apologise for the direct approach, but your maths website provided no other obvious email.
I am interested in studying mathematics ( I failed to pass CSE in 1983 but encouraged by the popular press feel there is opportunity yet) , but am confused by the options offered by the Universities.
Some offer me a BA, some a BSC and others a BMATH.
Are they the same or different ?
Can you help or suggest someone who can please ?
Kindly Yours
Andrew
The response was somewhat surprising :
Dear Andrew,
Thank you for your query which has been passed on to me as admissions tutor in Mathematics. The answer to your question is that the exact title of a mathematics degree, i.e., BA, BSc or BMath is unimportant. I don’t think any UK university offers a BMath. The decision whether a Mathematics degree is awarded as a BA or a BSc will depend upon internal bureaucracy at a given institution and does not reflect the material that is taught in any significant way.
Kind regards…
So it doesn’t matter then. I thought I’d press on and ask then exactly what mathematics is (or was — maybe it changes over time??!):
The reply to that was :
Dear Andrew,
My advice would be to solve problems — take a course at the appropriate level whether GCSE or OU — and actively form your own judgement on these matters. The best way to understand mathematics is to do some.
Kind regards…
Aha ! Some concrete advice at last. Form my own view and proceed. And that brings us to the first email I have shared. Well I had been doing some for years, but was no closer to the truth. Maybe my starting point of “1+2=3” had been too ambitious after all…
So fast-forward to today now, I decided to ask that great arbiter of truth chatGPT. I’d surely get an opinion!
“Mathematics is a broad and deep discipline that uses an abstract, logical language to describe the patterns and structures found in the natural and theoretical world. It involves the study of quantities (arithmetic and number theory), structure (algebra), space (geometry), and change (calculus and analysis), among other topics. Mathematics is not just about numbers and equations but also about reasoning, making logical arguments, and solving problems.
At its core, mathematics is a way of thinking and a toolset for understanding the universe. It enables us to model the physical world, predict outcomes, design structures and technologies, and analyze data. Mathematics is essential in science, engineering, technology, finance, economics, and many other fields, serving as the foundation upon which many theories and discoveries are built”
More “numbers” and “structures”. But also a “way of thinking”.
Now I knew from my university days that postulating and proving theorems was a key part of mathematics. In fact, maths really seemed to be concerned with “truth”. Making statements that are “true”. Whatever that means.
So maybe “1+2=3” could be a good place to start after all ?
I mean “1+2=3” is obviously true right ?
In Whitehead and Russell’s “Principia Mathematica” the proof of “2+2=4” only arrives on page 83.
Now that tremendous piece of work was definitive in trying to formalise mathematics. But if “2+2=4” is that complex, “1+2=3” is going to be jolly hard too.
Maybe we should start from “0=0” ? Now we’ve just got the number “zero” and the “infix-operator” “equals”.
Why is that statement true — and while we are on the topic, exactly what is “true” and “false” ?
I came to the conclusion that a “piece of mathematics” was a sequence of statements, each one “following” from the former ones. The “following” idea being that there would be a “proof” following some rules of deduction of each statement in turn. The initial statements in this sequence wouldn’t have proofs, but would be irrefutable truths.
More “truth” !
It was at this stage that I got very interested in computer proof verifiers : programs which (somehow) formalised mathematics to the extent that they could certify the validity of proofs.
The problem now was the fact that these programs relied on languages which themselves were very sophisticated and concealed large amounts of background in their implementation.
But I was getting somewhere. Then I found the https://us.metamath.org project. It creates a large corpus of mathematics from a simple substitution language and a text file.
So a “piece of mathematics” in the Metamath world is simply a text-file which runs to success.
So it’s a very large character string which “satisfies” the Metamath program.
Pure text.
Meaningless nonsense if you like ! Looks like Alice-in-Wonderland was on the right track after all.
Yet somehow it seems to work and all fit together. So I remain interested !
