I asked this question about Scientism and the answers there quite brilliantly explained to me why Scientism is philosophically inconsistent.
But I just want to know: What is the relationship between Philosophy and Science?
Personally, I think the scientific method must be assumed as an axiom/premise in Philosophy. My argument is that there is no way to deny the sure success of the sciences (like Physics, Chemistry, Medicine, etc).
So, what is the relationship between Philosophy and Science?
I find it useful to approach this question from the bottom up. You made a philosophical argument:
My argument is that there is no way to deny the sure success of the sciences (like Physics, Chemistry, Medicine, etc).
One of the things I love about philosophy is once you take a stand like this, there's always a line of questioning which can unsettle your position. I find it's useful to unsettle it first, and then use that uncertainty to explore the other interesting parts of the question. So that being said, I will not deny the sure success of the sciences.
Instead, I will question it.
I'm not going to say that the sciences haven't had sure success, I'm simply not going to immediately accept that the sciences have been a sure success. I'm going to ask you to convince me. How sure are we? You will almost certainly point to the wide acceptance of science. Science is everywhere in our society! To which I will reply that bacteria are everywhere as well. Does that mean bacteria are a success? Is it really good if something succeeds, simply because "succeeds" is the word we used?
Are we happy because of science?
Do we have meaning, because of science?
Are we better people because of science?
These are harder questions. There are a lot of very strong arguments suggesting that science and technology have actually made us less happy, and further from meaning. After all, that's the general pattern which drives people to become monks, and monasteries are alive and well today. They're "succeeding" too.
Now I, myself, tend to find the science arguments compelling. I'm choosing to be the person questioning science here, but I understand where you're coming from. What I'm really picking at is
I think the scientific method must be assumed as an axiom/premise in Philosophy
(Emphasis mine). Such wording is a very powerful claim when speaking in the philosophical world. And, generally speaking, powerful claims are very hard to justify in philosophy. And, in fact, you will find that the scientific method actually does not always appear in every philosophy. I once attended a great lecture on what Traditional Chinese Medicine is, and the speaker explained a difference in how their tradition is developed as compared to Western medicine:
There's clearly a difference in methodology. Many would claim that TCM does not follow the scientific method. But TCM is deeply steeped in Chinese philosophy, so a claim that philosophy must assume the scientific method as a premise invalidates 4000 years of Chinese philosophy, declaring it to "not be philosophy because it doesn't assume the scientific method." You can understand why such a claim is not popular.
Or you might claim that TCM has the scientific method of hiding in it. As it turns out, there is no one "The Scientific Method." We talk about it as though there is only one, but there isn't. It's actually a rather large class of approaches ranging from the very precise definitions that are used at the LHC when making observations of subatomic particles all the way out to a very broad "guess and check" mentality. You could choose to define the scientific method to be broad enough to admit Chinese philosophy, thus making it easier for others to accept your claim.
But there's a catch. The weaker you elect to define the scientific method, the harder it is to have absolute confidence that it must be a good thing and a successful thing. It's actually a really fun exploration, which digs at the heart of what it means to be "science."
This is the Philosophy of Science and the point where I can segue to answer your original question. The philosophy of science is considered to be a subdiscipline of empiricism. Empiricism is the study of that which we can "know" with our senses, i.e. empirical observation. There are other subdisciplines of empiricism as well besides science. Empiricism, itself, is a subdiscipline of epistemology, which is the study of what we can "know." Knowing is an incredibly complex and nuanced thing in philosophy because it has to actually address questions along the lines of "How do we know that we know something? How do we know that we know that we know something?" and so on and so forth.
Epistemology, itself, can be contrasted to ontology. Ontology is the study of what is real. Most of us assume that we know what "reality" is, but ontologists really question it. It turns out to be a tremendously interesting topic, but it's almost completely divorced from science.
Which is interesting. We often have the opinion that science is a tool which leads us to the truth of reality, but when you dig far enough down into philosophy, you start to see that that isn't quite what it does. In fact, some even argue that it does the opposite: instead of leading us to the truth, it tries to lead the truth to us.
But that's just one argument. The point being, that philosophy explores a class of questions and their answers which reach far broader than science does. And that's a good thing. Diversity is good. Science is great at what it does (that is to say, science is great at being scientific). Philosophy is great at what it does (that is to say.... whatever philosophy does. It's actually a rather funny exercise to try to pin down precisely what philosophy does). Both are valuable in this world, and neither should really replace the other.
Now I have pointed out how much separation there is between science and philosophy. But there is a very important connection between them: philosophers are people. Their ideas mingle with the ideas of the population around them. While there is no explicit connection between science and ontology, you will find that many ontological philosophers strive to develop their research to fit nicely with the ontological claims that science sometimes makes (when they technically can't make such claims, being empirical). Other philosophers will explicitly strive to make their work conflict with science's claims, just to explore that "what if the world isn't the way we think it is" sort of question. Their philosophy will naturally reject the claims that the scientific method must be assumed because the point of their work is to see what happens when you reject it.
So science absolutely influences all of philosophy, though often in a very indirect way. Likewise, philosophy forms the cornerstone of what science, itself, is. But many find it's digging and questioning to be a bit of a bore, so we often ignore this connection to philosophy.
Starting from Aristotle, science is a branch of philosophy, the branch that attempts to explain actual outcomes. 'The Physics' is as much a physics as any later one, but it failed. So it evolved and was replaced by things that worked better.
It is common now to think of things like Alchemy as simply never having been science. But Newton took Alchemy quite seriously, and it did have actual productive discoveries.
For us to retroactively redefine 'science' by looking backward from a place where we have centuries more experience with what does and does not work is intellectually dishonest. All of this stuff was science.
But nothing has changed the fact that science relates to logic, uses learning, has human processes, etc. It needs an ontology, an epistemology, and to some degree even a prescriptive politics (with a moral obligation to honest disputation, a proper structure for peer review institutions, etc.). So all science takes place within a broader philosophy, and can only ever be part of a philosophical system.
Philosophy is known as the "mother of the sciences," and in practice, when we understand a subject well enough that we can design reliable, replicable experiments around it, we start considering it as a science, not a branch of philosophy. But if science is a set of boxes containing tools, philosophy is what is outside the boxes --arguably including the design and definition of the boxes themselves.
The reason philosophy continues to exist is that there continue to be important questions we want to think about that we cannot yet approach scientifically --and things we might never be able to. There is no science of morality, nor of art, nor of religion, although people have made attempts at all of those. Also, when debates about the role and nature and ethics of science arise, philosophy must adjudicate them. Science was not designed to study itself --it is not its own proper subject.
As far as whether philosophers must accept the scientific method as a primary axiom, the best we can say is that some do, and some don't. Since philosophy lacks universal acceptance criteria there's no way of mandating something like this. It's worth noting, however, that philosophical inquiry far predates the development of the modern scientific method --before that point, what we now know as the sciences were instead considered "natural philosophy."
But I just want to know: What is the relationship between Philosophy and Science?
That's not a simple matter. F.e. nowadays some metaphysicians, like Ladyman and Ross in Every Thing Must Go, argue that metaphysics can and should be informed by physics (to some degree).
Personally, I think the scientific method must be assumed as an axiom/premise in Philosophy.
There's no such thing as the scientific method. In detail, the various conceptions of what the scientific method is supposed to be do vary.
My argument is that there is no way to deny the sure success of the sciences (like Physics, Chemistry, Medicine, etc).
Success isn't the same as truth. That is, we can hold scientific anti-realism and think that the results of our scientific processes are empirically adequate (see Van Fraassen) or functional while agreeing that the enterprise of science is important and successful.
So, what is the relationship between Philosophy and Science?
Like others mentioned, many philosophers think that science rests on a backdrop of epistemology etc. But some philosophers think that it can go the other way around. Some moderately so - I've given an example above -, fewer formulate it quite radically. Take for example Quine who thinks that epistemology and metaphysics can be "naturalized", that is, explained by natural science. For epistemology he thinks that the explanation can be done with psychology etc. Although he sort of argues against a distinction between science and philosophy in general. Quine is heavily criticized (and imo rightly so), so keep that in mind.
edit: typo
One important and clarifying distinction is that, science has no mechanism for making value judgments. That is, the scientific method can explain problems such as "how does a kidney function?" but not questions such as "how should I live my life?". It's very difficult to make a scientific argument as to what one should do in the Trolley Problem, for instance, but there are plenty of other philosophical arguments that can attempt to answer that question. It can describe which method of ammonia synthesis is optimal under certain conditions, but isn't well-equipped to tell us whether or not it's acceptable to lie.
Science is a facet of philosophy, and like any philosophical mode, it has certain uses and certain other areas that it is not useful for.
The answer should follow automatically from actionable definitions for philosophy and science. For that to work, we assume that the correctness of a definition is much less important that its predicativity.
== Definitions ==
statement: a claim, an expression fact: a change in the real world
Philosophy: statements about statements Science: statements about repeatable facts (for which it is possible to look for counterexample facts)
== Consequences ==
Hence, according to these definitions, statements about science are always philosophical: statements about statements (about repeatable facts).
Science, however, can impossibly make claims about philosophy (=not facts). Science cannot even make claims about itself. It is also obvious that the rules governing science can only be philosophical.
Your question has a simple answer. Long answers on matters as wide as this just add confusion and incoherence.
Philosophy is the root discipline dealing with all knowledge. Philosophy deals about the final truth.
Science is just one type of knowledge, featured by the scientific method. Science deals about empirical truth.
Example: thermodynamics describe gases as formed by little objects. And it works empirically, you use Avogadro's formula, you get a very approximate description of any experience. But we can say that, philosophically, gases are not formed by objects: they are instead formed by much complex manifestations created in part by with our subjective perception (see empiricism, Kant, etc.). Although we don't have a final truth regarding thermodynamics, much of the meaningful work of the last three centuries was dedicated to understand perception.
So, what is the relationship between Philosophy and Science?
Philosophy, as the study of experiential reality, is highly theoretical in nature. The main struggle with philosophy is to find a framework that can easily and consistently be applied on a practical day-to-day basis across a range of individual realities and subjectivities.
Science, as the study of the behaviors and characteristics of observable reality, is concerned with the quantifiable and objective. The main struggle with science is to find a framework that reconciles scientifically observable truths with our individual yearnings for meaning and direction.
In some ways, you could say philosophy is a "top-down" approach to understanding existence, taking theoretical frameworks and seeing how far down they can be applied until they break or become internally inconsistent, whereas science is "bottoms-up", starting with the minutiae of observation and continually extrapolating and building upon that into increasingly sophisticated and complex models of the natural and observable world until they also start to lose their predictive value.
Philosophy and Science are thus both related but also diametrically opposed in their approach to understanding reality.
Interestingly, they also both share a common failing with respect to their practical application. In the most basic decisions we need to make on any given day, both fail us consistently. Which school should I send my child to? How do I improve the quality of my life? Why is it important for me to be healthier?
Philosophy would approach these questions again from a highly generalized perspective that, if lucky, may apply perfectly for one individual, but seldom applies consistently to a range of individuals. What is "right"? What is "good" or "better"? These questions have confounded philosophers for centuries. Wars have been waged based on differing answers to those basic questions.
Similarly, Science has given us a measurably better standard of living. The key there is measurable, what we can accurately measure in the physical world, we can improve by way of technology. But what about our psyches? While technology and science give us increased functionality, they fail to bring us any closer to a consistent understanding of what is "better". How else can different tribes of humans now come to such different conclusions despite having the same tools, knowledge and technology available to them? Their subjective natures color even previously indisputable facts. Science struggles to explain this phenomenon. Subjectivity is a consistent blind spot for the scientific world and the scientific method.
All this said, this doesn't have to be the state of things forever. If there was common ground by which different individual subjectivities could be married with the objective facts revealed by the scientific method and applied consistently to daily human life, then we would effectively have bridged the gap between Philosophy and Science. We would have created a grand unified theory of the subjective and the objective.
I have a thought for such a framework which I am happy to discuss if there's interest. I'm also happy to kick its tires and try to bend and break it with others help.
