Mathematics does not need philosophy, since pure mathematics is the science of drawing necessary conclusions about strictly hypothetical states of affairs. However, philosophy needs mathematics, since philosophy depends on such reasoning - appropriately adapted to address human experience (phenomenology), the identification and pursuit of ends (esthetics/ethics/logic), and the nature of reality (metaphysics).

I can't remember when I needed or thought I might need mathematics for philosophy.

Mathematics are just some symbols that humans created to assist in develop equations that approximately predict the behavior under certain circumstances of certain non-living matter. Quite simply, mathematics is a tool created by consciousness. I am far more interested in that what created mathematics than the tool itself. Mathematics may be of interest to other professions that create and use it. I don't see any relevance to philosophy especially as it concerns life.

Strictly speaking, you don't need mathematics as we have it in order to do philosophy. However, in its essence, mathematics is a systematic and disciplined approach to producing and thinking about structures and relationships, and as such its relevance to any reasoning is obvious. Moreover, the centuries of mathematical work have already produced a rich toolbox of structures, methods and insights, and it would be foolish not to use it.

The nature of number is a fascinating philosophical question, best summed up as: are numbers and mathematical functions invented or discovered? Are mathematical 'objects', such as natural numbers, already in existence, awaiting discovery by minds capable of grasping them? Or are they strictly internal to the workings of the brain, something which humans utilise to manage and predict things around us?

Generally speaking, the idea that number is real is in keeping with Platonism. Exactly what is meant by that statement is subject to considerable debate and conjecture. But mathematical Platonism is the idea that numbers are just as real as the hypothetical tables, apples and stars of philosophical discourse.

Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it's descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception. In his essay "What Is Cantor's Continuum Hypothesis?", Gödel wrote that we're not seeing things that just happen to be true, we're seeing things that must be true. The world of abstract entities is a necessary world—that's why we can deduce our descriptions of it through pure reason. ...

Platonism has always had a great appeal for mathematicians, because it grounds their sense that they're discovering rather than inventing truths. When Gödel fell in love with Platonism, it became, I think, the core of his life. He happened to have been married, but the real love of his life was Platonism, and he fell in love, like so many of us, when he was an undergraduate.

Rebecca Goldstein, Godel and the Nature of Mathematical Truth

Frege seemed to assume a Platonist view:

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents - "just as a planet, even before anyone saw it, was in interaction with other planets."

Frege on Knowing the Third Realm, Tyler Burge

A famous essay on the power of mathematics to make scientific predictions was Eugene Wigner's The Unreasonable Efficiency of Mathematics in the Natural Sciences. It ought to be recalled that Wigner's Nobel was awarded on the basis of the discovery of mathematical symmetries in sub-atomic particles.

Another well-known example of the power of predictive mathematics was Dirac's prediction of anti-matter:

His research marked the first time something never before seen in nature was “predicted” – that is, postulated to exist based on theoretical rather than experimental evidence. His discovery was guided by the human imagination, and arcane mathematics.

For his achievement Dirac was awarded the Nobel prize for physics in 1933 at the age of 31.

New Scientist.

I think many of these kinds of discoveries were made on the basis of what Kant would have called 'synthetic a priori statements' - 'synthetic judgments are genuinely informative but require justification by reference to some outside principle'. In other words, they enable accurate predictions about things hitherto unknown, on the basis of logic. In this case, experimental science provided later confirmation of what the maths told Dirac 'must be there'. (There were many similar examples in respect of Einstein's predictions, also.)

Despite this, mathematical Platonism is a minority view with today's philosophers. Why? Because numbers are obviously not material objects. So if they're real, but not material, then that is a real fly in the ointment for materialism or physicalism. This is mentioned in an article on 'the indispensability of mathematics':

Some philosophers...claim that we have a special, non-sensory capacity for understanding mathematical truths, a rational insight arising from pure thought. But, the rationalist’s claims appear incompatible with an understanding of human beings as physical creatures whose capacities for learning are exhausted by our physical bodies.

Myself, I have no such qualms. I agree with Heisenberg's assessment, that the discoveries of 20th century physics seem to indicate that maybe numbers are in some sense more real than the phenomena that are described by mathematical analysis:

I think that on this point modern physics has definitely decided for Plato. For the smallest units of matter are, in fact, not physical objects in the ordinary sense of the word; they are forms, structures or—in Plato's sense—Ideas, which can be unambiguously spoken of only in the language of mathematics.

The Debate between Plato and Democritus

The reason numbers enable us to grasp principles and regularities is because there is a deep connection between the nature of number and nature herself - much more than current philosophical materialism would like to acknowledge.

Arithmetic and geometry
are the formalisation of
the most basic intuitions
we use in out interactions
with the material world.

Mathematics in its more
advanced forms are further
abstractions and generalisations
of that initial formalisation.

When applied to domains where our
basic intuitions break down
e.g. relativity and the quantum
mechanical world, mathematics is
a tool used to construct
isomorphisms (loosely speaking)
betwen said domains and our basic
intuitions.

Suffice to say that symbolic multiplicities can never be anything more than a rough approximation of flowing, ever changing continuity. Mathematics is not only just a tool, it is actually an hindrance. Far better tools and metaphors can be found in any of the arts.

Mathematics is not only just a tool, it is actually an hindrance. — Rich

You wouldn't have the instrument on which you recorded that idea wiithout it.

yes, which is why my fundamental interest of philosophical inquiry is in the human consciousness that created the instrument and less so in the instruments they create. Now, when I was a computer consultant, my interest was in the tools of the industry. It all depends upon the intent of my inquiry.

Do they need one another, are they intertwined that deeply? — River

We need to number pages of philosophical books.

I do believe that there are another player in this connection. I do agree that Mathematics is a science used to deduce conclusions and axioms which does not "need' philosophy. However, my objection is that philosophy also, is a discipline that does not necessarily need mathematics. I think that both mathematics and philosophy is a discipline in their own right.
The connection between them lies on third discipline that uses devices from both mathematics and philosophy, Bertrand Russell coined it as "Mathematical Philosophy" in his book 1919 book "Introduction to Mathematical Philosophy". Russell distinguishes "Mathematics", or "ordinary mathematics", from "Mathematical Philosophy". He said that both are dealing with the same objects but with a different mindset. (The details could be read from the book)
I also agree that philosophy borrows devices from mathematics, but the claim that philosophy necessarily needs mathematics is still broad, or it is maybe that the case it is not true at all. I also suggest that we should inquire first the nature of mathematics before proceeding to the claim that philosophy necessarily need mathematics through reason/reasoning.

We need to number pages of philosophical books.

-@TheMadFool

Makes sense.

Do they need one another, are they intertwined that deeply? — River

They are intertwined deeply.

The difference in the mathematical schools of thought cannot be done away with some mathematical proof. The differences are indeed philosophical. And the schools themselves are important, even if the average mathematician will say he has no school of thought or philosophy when doing math.

Just like the scientist who is ignorant of the philosophical schools in science will likely unintentionally adhere to one. And usually to a very classical view from the 18th Century.