What all the FE believers in this thread do not want to understand is that maths, philosophy and science are three separate disciplines, and they cannot be intermixed any way we want.
Philosophical concepts like "oneness" have no place in either science or mathematics.
The abstract concept of number, used in maths' Group Theory, has no direct relationship with the apples used to explain to a child the equation 1+1=2, and therefore has nothing to do with the philosophical idea of "when an apple ceases to be an apple", or in this case, "when a raindrop ceases to be a raindrop".
The concept of experimental error has no place in either philosophy or maths.
The concept of truth is clearly defined in maths, heavily studied in philosophy and totally irrelevant and extraneous to science.
And so we can continue forever explaining the differences between the three disciplines. If you want to mix them you should be prepared to explain why the mix is valid or else be shown how ignorant you are, like the OP's author, for example.
Science is a philosophy.
And we only need your word for that claim, of course. Or, maybe you have any kind of justification for your claim, other than it sounds nice?
You can say science is an offshoot of philosophy, just as chemistry is an offshoot of alchemy. Having a common origin does not mean anything. The target of philosophy and the target of science are clear and almost distinct, with just a very small intersection.
Empiricism, realism, instrumentalism, epistlemology, how we should interpret data and analyze it, Ockham's razor, reductionism, induction, the methodology, assumptions and implications of science is all philosophy and is what constitutes science as what it is. At every single point there is a philosophical choice that has been made already. Be careful to realize them and not take them for granted. Ignoring that they do both have the same "targets" or aims. There is a paradigm behind and driving science, and this paradigm is philosophical.
As you say their interests intersect. This is because it is science is a subset of Philosophy. Squares are rectangles because they share traits with rectangles, but all rectangles are not squares. All science is philosopy, but not all philosophy is science.
Math is a formal science. Should I take it you disagree that Math is a science or that you agree that it is in the realm of philosophy?
There are several hundred (if not thousand) books on this.
Heidegger is a good starting point. Read a few of his books. A memorable quote off the top of my head is "science is philosophy, whether it knows and wills it- or not."
A source I could dig up online with minimal effort: Fuller, "Insight into value" - "Ever science is a philosophy, resting on basic presppositions."
Another: "Science is philosophy applied to a particular problem ; philosophy is science
made universal and complete" Methods of knowledge: an essay in epistemology? , Walter Smith
So no. You don't have to take my word for it. If one doesn't think science is a philosophy then one either doesn't know what science is, one doesn't know what philosophy is, or one is too arrogant to admit either.
“Philosophy of science is about as useful to scientists as ornithology is to birds,” according to physicist Richard Feynman.
Philosophers have indeed tried to find the relationship between the two disciplines, and try to find the philosophical relevance of science. But most scientists haven't even heard about realism, instrumentalism, epistemology, oneness, immanence, and have only a cursory knowledge that empiricism has to do with the Scientific Method. And guess what? Science has not slowed down because of it.
Lots of knowledge started being theological, then became philosophical and finally ended up being studied by science, just as some knowledge started being alchemy and ended being studied by chemistry. That does not mean that all of science is a part of philosophy, as you try to claim.
Feynman says nothing in that quote of the validity of the philosophy of science. He talks of its use to scientists. And you couldn't get a more self admitted biased source against philosophy.
"I fully agree with you about the significance and educational value of methodology as well as history and philosophy of science. So many people today -- and even professional scientists -- seem to me like somebody who has seen thousands of trees but has never seen a forest. A knowledge of the historic and philosophical background gives that kind of independence from prejudices of his generation from which most scientists are suffering. This independence created by philosophical insight is -- in my opinion -- the mark of distinction between a mere artisan or specialist and a real seeker after truth." Einstein
"When I think about the ablest students whom I have encountered in my teaching, that is, those who distinguish themselves by their independence of judgment and not merely their quick-wittedness, I can affirm that they had a vigorous interest in epistemology. They happily began discussions about the goals and methods of science, and they showed unequivocally, through their tenacity in defending their views, that the subject seemed important to them. Indeed, one should not be surprised at this." Einstein
And then, of course theres Newton too. And Bohr. And any number of other scientists that have made far greater contributions than Feynman. Perhaps you meant "Scientists have indeed tried to find the relationship between the two disciplines." Also, are we to take you at your word that "Science has not slowed down because of it", a seemingly unsupported idea that came to you at a whim?
What most scientists have heard about is irrelevant. Whether they know about what makes up their philosophical belief system or not has no relevance on whether they are using said belief system. The usefulness of the study of this belief system is also completely irrelevant. So, Ok, Most scientists are ignorant; fine. The study of the belief system they hold is perhaps less useful to them. However, before every experiment, hypothesis, during said experiment, and after scientists make philosophical decisions (whether ignorant of them or not.) A scientist cannot practice science without philosophy at every single turn. Like I said, if one doesn't think science is a philosophy then one either doesn't know what science is, one doesn't know what philosophy is, or one is too arrogant to admit either. I imagine the majority of scientists, if I take you at your word, are in the first two categories.
Also, its not a matter of the roots of science, its a matter of what the aims of science are and what science is at the heart. These aims fall under philosophy.
Let me ask you, at what point (since you claim it came from philosophy) did science stop being a subset of philosophy, and why?
Should I take it you disagree that Math is a science or that you agree that it is in the realm of philosophy?