How did Euclid go about forming his propositions?

[Manny46]

What I basically want to ask here is, about the process of forming mathematical truth/theorem. This seems like a bit broad question, but I have this specific query. We all know that Euclid started with his basic postulates or what we may call axioms, and common notions. Now did he form those propositions in a gradual manner, something like a follow up step which naturally proceed the previous proposition or postulate with the support of common notions? Or were those propositions intuitive in nature, which he reached through intelligent speculation, believed in its truth, and then deductively, with the help of other propositions/postulates proved its truth, forming a proposition finally?
This can be generalized to more broader question as to how mathematicians go about forming their theorems, as far as starting a new theory with completely new axioms are concerned, or even extending the previous work. Is it gradual or is it something where mathematicians form different statements, and guesses and with more careful observation convince themselves of the truth of the statement, and thereafter work deductively to prove the statement/conjecture to convince others and themselves of its truth, thereby forming a theorem/proposition from a mere statement?

 

[fresh_42]

The question what Eulicd did or how he thought is on thin ice. Neither do we know him nor are we able to anticipate the thinking of his times. All that can be said is deduced from the Elements. I suggest to read a book about them.

This can be generalized for more broader question as to how mathematicians go about forming their theorems as far as starting a new theory with completely new axioms are concerned or even extending the work.

I have seen both, although I wouldn't go as far as 'a new theory with completely new axioms'. It's always a construction work which founds on previous results. Even what you may call a new theory is often the product of parallel developments and based on previous work, even if we only associate it with the one who finished it. Lie used results form Engel, Noether from Lie etc. Cases like Galois' field theory are the exception, and even this took others to reformulate it.

Developing a new idea is a creative process. What do you except from a painter to hear, if you asked him how he came up with the idea of a certain painting?

 

[Manny46]

The question what Eulicd did or how he thought is on thin ice. Neither do we know him nor are we able to anticipate the thinking of his times. All that can be said is deduced from the Elements. I suggest to read a book about them.

I have seen both, although I wouldn't go as far as 'a new theory with completely new axioms'. It's always a construction work which founds on previous results. Even what you may call a new theory is often the product of parallel developments and based on previous work, even if we only associate it with the one who finished it. Lie used results form Engel, Noether from Lie etc. Cases like Galois' field theory are the exception, and even this took others to reformulate it.

Developing a new idea is a creative process. What do you except from a painter to hear, if you asked him how he came up with the idea of a certain painting?

Is it something which proceeds like a follow-up/deduced from one proposition to another? Or is it something where one forms a conjecture, convinces himself of its truth, and thereafter tries to prove it deductively?
Do you have any recommendation for a book on Euclid with creative process as its concern?

 

[fresh_42]

Is it something which proceeds like a follow-up/deduced from one proposition to another?

As said: It is usually a point of convergence of work by different scientists. But I also have seen results which were developed from scratch. Strassen had a paper about algorithmic bilinear complexity which started with a strange geometric concept and ended with a new upper bound for matrix multiplication. This has been an astonishing thing to watch. He didn't reinvent tensors or the framework as such, he just added a new idea. That's why I said that a production process normally doesn't start at the level of axioms, but often with a new idea to consider certain objects.

Or is it something where one forms a conjecture, convinces himself of its truth, and thereafter tries to prove it deductively?

This can happen as well. Mochizuki, Perelman, and Wiles come to mind. I think these are exceptions. Far more have failed than succeeded with such an approach.

Do you have any recommendation for a book on Euclid with creative process as its concern?

Unfortunately not. Myself is interested in the history of science. IIRC van der Waerden dealt with the subject (history) at the end of his life and he wrote a few things. But I don't know whether geometry was among his papers or books, but I would have a look at his work.

 

[Manny46]

As said: It is usually a point of convergence of work by different scientists. But I also have seen results which were developed from scratch. Strassen had a paper about algorithmic bilinear complexity which started with a strange geometric concept and ended with a new upper bound for matrix multiplication. This has been an astonishing thing to watch. He didn't reinvent tensors or the framework as such, he just added a new idea. That's why I said that a production process normally doesn't start at the level of axioms, but often with a new idea to consider certain objects.

This can happen as well. Mochizuki, Perelman, and Wiles come to mind. I think these are exceptions. Far more have failed than succeeded with such an approach.

Unfortunately not. Myself is interested in the history of science. IIRC van der Waerden dealt with the subject (history) at the end of his life and he wrote a few things. But I don't know whether geometry was among his papers or books, but I would have a look at his work.

Thanks for clearing that up. So the key point is, that we understand the concept, the idea behind the theory rather than cooking up new theorems by just maneuvering axioms/theorems. I mean it does involve the latter but at It's core It's a creative process where one needs to fully grasp the concept to produce something meaningful. Is that a correct assessment?

 

[fresh_42]

Thanks for clearing that up. So the key point is, that we understand the concept, the idea behind the theory rather than cooking up new theorems by just maneuvering axioms/theorems. I mean it does involve the latter but at It's core It's a creative process where one needs to fully grasp the concept to produce something meaningful. Is that a correct assessment?

More or less, yes. 'To fully grasp the concept' is the ideal case, but I suppose that many results are just small improvements which don't necessarily require the full concept. I have a book about the mathematical developments in the 18th and 19th century (roughly). It shows that many results were in fact a collection of several developments and achievements. Many results came from the necessity to solve physical or mathematical problems, and there are usually many results before someone made a theory out of them. E.g. Riemann wasn't the first who considered manifolds, but as far as I know, was the first who recognized, that those surfaces don't need to be embedded in some surrounding space. And if you start to analyze similar results, you will see that the vast majority has been found this way.

 

[Manny46]

and there are usually many results before someone made a theory out of them. E.g. Riemann wasn't the first who considered manifolds, but as far as I know, was the first who recognized, that those surfaces don't need to be embedded in some surrounding space. And if you start to analyze similar results, you will see that the vast majority has been found this way.

I'm missing something here, would you elaborate on this point? What I get from here is that mathematicians build up from the previous works and results of other mathematicians/their own work.

 

[fresh_42]

Yes. Sometimes there are examples like Wiles and Fermat's last theorem, in which case the work itself is a seed of new mathematics, but even Wiles proof is founded on theories which were already known, and thus not completely new. This is as with everything in life: things are not black and white. The majority is in between. I'm sure Euclid didn't invent his Elements. He was the one who brought a systematic and wrote it down. Plus he certainly added new insights and theorems. Newton and Leibniz developed the concept of infinitesimals in parallel, simply because they needed it. Even Einstein's work wasn't out of the blue.

Mathematics has indeed the property that you can "invent" objects and functions with certain properties. To be useful, however, it's more than likely that many others dealt with the same subject and already found partial results.

 

[Manny46]

Thank you so much for your insight.I guess, the only way to get hang of this process is to read and do more math. I have just started doing real analysis, what do I know yet.

 

[sysprog]

Especially in the modern era, new mathematics may be developed in response to something suggested by someone else, in an ongoing set of dialectic, individual and collaborative efforts. For example, much has been contributed to discipline in response http://mathworld.wolfram.com/HilbertsProblems.htm. As for how Euclid arrived at his methods and insights, I think that's rather inscrutable, beyond observing what is implicit and explicit in the extant work Itself.

 

[Manny46]

Especially in the modern era, new mathematics may be developed in response to something suggested by someone else, in an ongoing set of dialectic, individual and collaborative efforts. For example, much has been contributed to discipline in response http://mathworld.wolfram.com/HilbertsProblems.htm. As for how Euclid arrived at his methods and insights, I think that's rather inscrutable, beyond observing what is implicit and explicit in the extant work Itself.

I see. Thanks

 

[Manny46]

Special thanks to fresh_42 and also sysprog for their valuable insights. While it's not entirely possible to know the exact process, but I think this extract from Foundations and Fundamental Concepts of Mathematics by Howard Eves does a great job of clarifying it.
This is not to say that the Greeks shunned preliminary empirical and experimental methods in mathematics, for it is probably quite true that few, if any, significant mathematical facts have ever been found without some preliminary empirical work of one form or another. Before a mathematical statement can be proved or disproved by deduction, it must be thought of, or conjectured, and a conjecture is nothing but a guess made more or less plausible by intuition, observation, analogy, experimentation, or some other form of empirical procedure. Deduction is a convincing formal mode of exposition, but it is hardly a means of discovery.
(pp.9, section 1.3)https://books.google.co.in/books?id=J9QcmFHj8EwC&lpg=PP1&pg=PA9#v=onepage&q&f=false

 

[sysprog]

I agree substantially that deduction is not the main driver of discovery, but it is probably the main thing in the process of proof. Figuring out which rule to apply is a creative part of the process of deduction. Informally, induction may be thought of as the formation of a general rule from observation of specific instances of it, while deduction is the process of systematically applying of a rule already discovered. The two often go hand-in-hand in the overall processes of mathematical discovery and invention, neither of which makes great strides without formidable feats of observation, insight, and imagination.

Certainly one can identify predecessorial influences on Euclid -- mathematicians such as Thales, Pythagoras, and Zeno, and philosophers, from the pre-Socratics through Socrates, Plato, and Aristotle, all contributed significantly to mathematics and to the refining of concepts of proof.

Hippocrates of Chios was the earliest Greek geometer to have written a first-principles-based geometry exposition entitled (Στοιχεῖα) Stoicheia -- in English, Elements. He lived about a century before the days of Euclid, and Euclid is known to have been familiar with his work. Even so, ironically enough, the geometry in that work is of the kind now known as Euclidian.

 

[Manny46]

I agree completely. Both of these processes have their own place. While intuition, induction and empirical evidences are useful for forming conjectures, but it can't really be established as a theorem without deduction.
Thanks for that
trivia. Makes me more interested in the history of Mathematics.