All this actually happened MANY times in my actual real-life research;
I made a computer program for simulating the random sampling of a hypothetical and newly-defined new type of probability distribution.
After I inputting many thousands of combinations of input values into the program and running the simulations to see what the corresponding output would be, I observe a pattern between input and the output.
Question 1;
can that observation of that input-output pattern of those computer simulations be completely validly called "empirical evidence" in the strictly SCIENTIFIC sense of the term?
I then work out a maths equation E that fully describes that pattern and make the assumption that that equation E is always correct.
I then test E for yet many more combinations of input values in yet more simulations and find that the way those combinations of input values correspond to the resulting output also conforms to E.
I conclude via induction that E is PROBABLY always true BUT I don't, at least not yet, have a mathematical/deductive proof that it is true and don't currently know of any purely DEDUCTIVE-logic reason why it SHOULD always be true so cannot (yet) be absolutely rationally sure.
Question 2;
Can that assumption and conclusion that E is PROBABLY always true be completely validly called a "SCIENTIFIC theory"?
Is E a "theory" and/or a "SCIENTIFIC theory"?
If the answer to question 2 is "yes";
Question 3;
What if I suddenly were to accidentally stumble across a valid mathematical proof that E must always be true?
Can E still be called a "theory" and/or a "SCIENTIFIC theory" but now is a PROVEN one (and is now also call a 'theorem' but that isn't part of my question) ?
I personally see science as a study of the natural world (with study of humans still falling withing the 'natural world'😉. You are studying mathematics not nature. So you are using scientific methods to study maths. I don't think I would call your theories 'scientific theories' as it might cause confusion.
I would suggest sticking to normal maths terminology.
Originally posted by twhiteheadSo you imply that although I DID make a "theory BUT my "theory" is NOT a "scientific theory" but rather is ONLY a "mathematical theory".
I personally see science as a study of the natural world (with study of humans still falling withing the 'natural world'😉. You are studying mathematics not nature. So you are using scientific methods to study maths. I don't think I would call your theories 'scientific theories' as it might cause confusion.
I would suggest sticking to normal maths terminology.
But then I am a bit confused by your "you are using scientific methods to study maths" comment because I thought "scientific method" has a meaning that is such that it NECESSARILY involves formulating "scientific theory"?
BUT perhaps in the above context that shouldn't be said as "scientific method" but rather said something like "scientific-like method" or "quasi-scientific method"? So I didn't use 'true' "scientific method" but rather a kind of "quasi-scientific method"? -perhaps that's the answer?
We can apply the scientific method to many things. We could for example apply it to a fictional world, such as
'The Lord of the Rings' or 'Harry Potter'. We could ask questions, make hypotheses and theories etc. We would need to get information from the books or the authors to test those hypotheses and theories.
But I wouldn't call them scientific theories. Not because they are not based on the scientific method, but rather the subject matter.
Mathematics has a well established system of hypotheses and theories and the meanings of those words within mathematics (which is slightly different from the domain of science) and testing using examples, exhaustive searches and more are hardly new. There is no need to introduce new terminology or to use the word 'scientific' at all.
https://en.wikipedia.org/wiki/Theorem#Relation_with_scientific_theories
I believe the correct term for your not yet rigorously proved ideas is 'conjecture':
https://en.wikipedia.org/wiki/Conjecture
Originally posted by twhiteheadwhat you are saying seems to be hinted at with;
https://en.wikipedia.org/wiki/Theorem#Relation_with_scientific_theories
"...Nonetheless, there is some degree of empiricism and data collection involved in the discovery of mathematical theorems. By establishing a pattern, sometimes with the use of a powerful computer, ..
... Such evidence does not constitute proof.."
Hay! That is exactly what I was doing!
I believe the correct term for your not yet rigorously proved ideas is 'conjecture':
https://en.wikipedia.org/wiki/Conjecture
Yes, I think so. From that link;
"...In mathematics, a conjecture is a conclusion or proposition based on incomplete information, for which no proof has been found...."
And I made a LOT of them!
So, I will call them 'conjectures'.
So, in answer to my questions;
My observation of that input-output pattern of my computer simulations of random sampling of a hypothetical probability distribution CAN be completely validly called "empirical evidence" in the strictly SCIENTIFIC sense of the term.
HOWEVER, despite that, it is NOT true that the unproven conjectures I make from such observation can be validly called a "SCIENTIFIC theory"; they are "conjectures", NOT scientific theories.
And if then a sound mathematical proof is found for one such conjecture then it becomes a 'theorem' and a "fact" (but NOT a "scientific fact" ? )
ANYONE;
Just one more question just to make this absolutely clear;
Is what a completely valid (as in soundly proven) mathematical theorem says a "scientific fact" or just a "mathematical fact"?
(I am pretty sure it is just a "mathematical fact" but just checking)
Originally posted by humyI would say yes.
My observation of that input-output pattern of my computer simulations of random sampling of a hypothetical probability distribution CAN be completely validly called "empirical evidence" in the strictly SCIENTIFIC sense of the term.
HOWEVER, despite that, it is NOT true that the unproven conjectures I make from such observation can be validly called a "SCIENTIFIC theory";
Here I would say it is a matter of convention, not a true or false thing. We are, after all, talking about a definition here.
they are "conjectures", NOT scientific theories.
Conjecture is the standard term in the field of mathematics.
Is what a completely valid (as in soundly proven) mathematical theorem says a "scientific fact" or just a "mathematical fact"?
(I am pretty sure it is just a "mathematical fact" but just checking)
Yes. Mathematical proofs depend on axioms, not facts about the world. A mathematical theorem is a logical conclusion based on assuming certain axioms to be true.
Science uses maths and logic and empirical evidence to investigate reality. It is necessarily dependent on empirical evidence thus never progresses beyond the equivalent of conjectures. However, patterns may exist that can be explained by simpler sub systems and the relationship between them may be essentially entirely mathematical. So for example, if you take the claim that a gas is made of particles with certain forces, then various laws of fluid dynamics may follow directly from that through logic rather than observation. ie I am saying that many things we call 'science' may be derived entirely logically from prior scientific findings of simpler systems. Overall, there is much overlap between science and mathematics.
Originally posted by twhiteheadSo to update that, that should be;
I would say yes.
[b]HOWEVER, despite that, it is NOT true that the unproven conjectures I make from such observation can be validly called a "SCIENTIFIC theory";
Here I would say it is a matter of convention, not a true or false thing. We are, after all, talking about a definition here.
they are "conjectures", NOT scientific theories. ...[text shortened]... fic findings of simpler systems. Overall, there is much overlap between science and mathematics.[/b]
My observation of that input-output pattern of my computer simulations of random sampling of a hypothetical probability distribution CAN be completely validly called "empirical evidence" in the strictly SCIENTIFIC sense of the term.
HOWEVER, despite that, whether the unproven conjectures I make from such observation can really be validly called a "SCIENTIFIC theory" is a matter of convention and exact definition/meaning of "SCIENTIFIC theory" used, which varies, so it is had to say. But we can at least definitely say they are validly called "conjectures".
And if then a sound mathematical proof is found for one such conjecture then it becomes a 'theorem' and a "fact" but NOT a "scientific fact".
So what a completely valid mathematical theorem says is NOT a "scientific fact" but is still a "mathematical fact".
ANYONE;
Do you completely agree with all the above assertions (which me and twhitehead have come to think is the case ) ?
Originally posted by humyTo explore this a bit further, there is a whole domain of mathematics dealing with the Cartesian plane. Now one could investigate a Cartesian plane using a computer simulation, and come up with various conjectures and possibly prove some of them mathematically. But the Cartesian plane is a logical construct not a real world object, so I wouldn't call any of that science.
So what a completely valid mathematical theorem says is NOT a "scientific fact" but is still "mathematical fact".