# Questions about mathematics as a deductive system

## Main Question or Discussion Point

Hello, I have a few questions about mathematics as a deductive system in which mathematical statements follow as a logical consequence from axioms and other statements. In particular, consider the real number system and Euclidean geometry.

Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?

Say the axioms are literally descriptions of our world. Then I guess the deductions of the axioms will also describe our world closely. What happens if we derive from the axioms a statement that is different from observations in our world? (Have this occurred in Euclidean geometry?)

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PeroK
Homework Helper
Gold Member
Hello, I have a few questions about mathematics as a deductive system in which mathematical statements follow as a logical consequence from axioms and other statements. In particular, consider the real number system and Euclidean geometry.

Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"? The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?
Your question is one that many people have asked. Try, for example:

https://en.wikipedia.org/wiki/The_Unreasonable_Effectiveness_of_Mathematics_in_the_Natural_Sciences

fresh_42
Mentor
I read the page in the past, but I didn't get much out of it. Can you help me with my questions instead? I really want to know!
Is doing arithmetic with numbers and working on theorems that follow deductively from axioms of Euclidean geometry simply a "game of logic"?
It was, until we needed e.g. encryption and encoding.
The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so?
This is not so. With a few exceptions, the ratio between circumference and diameter of a circle, or the growth rate of e.g. bacteria, real numbers are quite irrelevant in real life, rationals will do. Nobody actually needs $\sqrt{2}\in \mathbb{R}$. $1.41$ will be significant enough for your carpet.
One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?
Our world is flat like a board, hence the Euclidean geometry. Lengths have been introduced as ratios between two distances, hence the terms rational and irrational. The reals only came into play since people refused to stop cutting lengths into smaller parts and continued infinitely.
Say the axioms are literally descriptions of our world. Then I guess the deductions of the axioms will also describe our world closely. What happens if we derive from the axioms a statement that is different from observations in our world?
Then we learn something about the capability and the limits of our models to describe the real world, e.g. Banach-Tarski.
(Have this occurred in Euclidean geometry?)
Yes. When people like e.g. Gauß measured the real world in greater distances, Euclidean geometry was no longer valid and non Euclidean geometry has been established. The wording different from our observations doesn't mean anything. How can the notion of e.g. a long exact cohomological sequence be an observation? We enhanced our axioms as our observations became better and more detailed, not the other way around.

Stephen Tashi
The real numbers and Euclidean geometry are useful and accurate in real life. Why is that so? One idea I could think of is that axioms of real numbers and Euclidean geometry are not chosen without any meaning. They are set up with the aim of mathematizing natural observations. So how do the axioms eventually lead to mathematizing natural observations?
You should think of it as Nature and human beings creating mathematics by inventing useful axioms, not as pre-existing axioms leading to the mathematics and workings of Nature. A set of axioms that didn't describe some natural phenomena wouldn't be popular among applied mathematicians.

We could imagine different axiom systems competing for popularity among the community of mathematicians and scientists. Those that work best would kept alive and taught. Those that were ineffective and awkward would be discarded. Whether this Darwinian vision is correct is a topic for people who study intellectual history, in particular the history of math and science.

People who study mathematics intensively usually think of mathematical structures as existing independently of any natural phenomena that are examples of such structures. So the popularity of different axiom systems is not simply a question of which axiom systems describe Nature well. There is a feedback effect. People want mathematics that is useful for describing mathematical structures as abstractions.

You should think of it as Nature and human beings creating mathematics by inventing useful axioms, not as pre-existing axioms leading to the mathematics and workings of Nature. A set of axioms that didn't describe some natural phenomena wouldn't be popular among applied mathematicians.
I totally agree, this is how I view Mathematics now (but not so in the past), which leads to a few questions in this thread that I've recently posted.

Unfortunately, I haven't gotten a reply that I'm looking for. Can you kindly take a look at the thread and help me? Thank you!

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Stephen Tashi
I totally agree, this is how I view Mathematics now (but not so in the past), which leads to a few questions in this thread that I've recently posted.

Unfortunately, I haven't gotten a reply that I'm looking for. Can you kindly take a look at the thread and help me? Thank you!
I've looked at that thread. As I understand it, you ask for physical examples that illustrate certain mathematical statements. Your introductory remarks indicate that you think the proper way to study mathematics is find such examples for theorems, axioms, and defnitions before studying the logical connection between them. Of course, if you are studying mathematics as a hobby, you can adopt whatever style of studying you want. However, if you are studying mathematics with some career goals in mind, you are making a mistake if you expect this approach to always work.

Looking to Nature as the cause for mathematical definitions and theorems develops intuition, but Mathematics is essentially legalistic, not an exercise in intuition.

For 1) and 2), if you do much mechanical drawing, these statements, these statements will seem obvious. However, whether there are situations in Nature where these statements are (exactly) true is unclear, because the geometry of the physical world is not known to be Euclidean.

As to 3), you would have to associate some quantifiable Natural phenomena with the sequence $\{1/n\}$, such as volumes of water and imagine a sequence of events taking place in time, like a basin emptying. However, this type of intuition is actually a hinderence to understanding the mathematical definition for the limit of a sequence. The mathematical definition does not refer to a process taking place in time or in a series of "steps". The formal definition of the limit of a sequence uses the quantifiers "For each ......there exists...." It resembles a legal contract, not a description of basin emptying.

fresh_42
Mentor
Looking to Nature as the cause for mathematical definitions and theorems develops intuition, but Mathematics is essentially legalistic, not an exercise in intuition.
There is a fundamental difference between mathematics and physics, indeed. Physics is a descriptive science, math a Geisteswissenschaft, normative and deductive.

Many mathematical developments originated from physical problems, but by far not all. Sometimes it was even the other way around: physicists looked whether mathematicians already had an appropriate tool.

I like to say it this way: Physics cannot prove that the apple will always fall towards the ground, just because it wasn't observed otherwise. Mathematics, however, can prove that it will always fly up in the sky, by assuming an according framework.

HallsofIvy