My question is whether a lack of knowledge about formal mathematics or theoretical science in general would have an impact on a philosopher's ability to think and make judgments.
Why should a philosopher acquire a deeper understanding of natural sciences or develop mathematical and scientific ways of thinking?
This depends very much on the area of philosophy. If you're interested in philosophy of quantum mechanics, for instance, you need at least undergraduate level training in physics (and the mathematics that entails). If you're doing ethics or political philosophy, then maybe the need for that sort of knowledge is lessened (although knowledge of some basic logic and some economics would have vastly improved several talks I've had to sit through...). Philosophers of mind need a good understanding of neuroscience and possibly some psychology.
Now these are just examples of the maths/science that philosophers need in order to usefully contribute to an area. There is the broader question of what understanding could be recommended even if it's not a prerequisite for doing the philosophical work. A basic knowledge of mathematics and science is always a good thing. Having studied maths gives you a particular way of thinking through problems that Intro to Logic just doesn't. That extra facility with thinking logically is always useful.
Then there's the even broader question of what any right thinking person should know. And as a consequence what every philosopher should know. This includes, I think, some basic physics: an understanding of electricity; mechanics and kinematics; conservation of energy... Some basic maths: what a function is; how to calculate a percentage; how to read statistics that crop up in newspapers and adverts (the difference between a relative and an absolute increase...)
I would expect a philosopher to know about propositional and first order logic, and proof techniques. It would be worth your while to read Herbert B. Enderton's book A Mathematical Introduction to Logic. That may sound a bit harsh, however I encounter the use of them in my everyday life.
Countless times, I have seen people use phrases like "this is the proof of that" or "it is clear that every X stems from Y". However, when I look at what they actually are talking about I see neither a proof nor the clarity they are talking about even though the words like proof or every (∀) are well defined under mathematics and mathematical logic.
Also, when one listens to a person good at public speaking (like a preacher or a politician) one can easily think "Wow, he is damn sure right!" However, when you dissect each of his sentences and express it in first order logic (when you can), you can clearly see the fallacies, mistakes, and blabber.
I respectfully disagree with Seamus, mostly because of his rather audacious use of the word "need". I don't think a philosopher needs to know anything in the sciences—in some cases they ought to in some regards, but they don't need to. The first (ancient) philosophers of course started in exactly this way, and they got along fine. Even centuries later, when the sciences were moving full steam ahead, philosophers like Immanuel Kant* and Hume and many others proposed many great theories without the aid of science that are still accepted today, because they were founded on logic and reason; phenomenological experience being the only thing they needed to start.
Furthermore, in some ways scientific theories can be perniciously misleading. Science is, after all, a collection of constantly revised theories that go through rather large changes from time to time, called paradigm shifts. Basing your arguments on potentially false theories in science could have obvious potential drawbacks. There were points in history with concepts like the flat Earth, the Sun revolving around the Earth, Atlas holding up the sky, souls being located in the pineal gland, etc... To have based a philosophical theory on such things would have been disastrous.
That said, I think it is the duty of every philosopher to never stop learning more, and those who continue to seek knowledge will always be more well-rounded and more prepared than those who do not. As Seamus and others have suggested, if you are interested in a particular field of philosophy that relates directly to a science (say, the philosophy of quantum mechanics), obviously it behooves you to learn about said field from as many perspectives as possible. Outside of that, the bad philosophers and the good philosophers will always be divided simply by their ability to form rational conclusions**.
*I'm aware the Kant was a scientist and taught many classes in the sciences. But he also created many theories regarding the scope of human knowledge and the limits of the human mind when there was no formal understanding of psychology in his time.
**Note that I don't think one's ability to use first-order and predicate logic makes a difference in their ability to reason; that is, the ability to form logical conclusions and to tell when an argument is invalid is independent of being able to use the formal notation.
I absolutely vigorously advocate that all self-respecting philosophers know at least the basics of Grothendieck's categorical view-point of moduli stacks. It is both fundamental, and has implications about many important questions in philosophy. Some would argue with me, but I think it dwarfs Einstein's relativity as the most important philosophical advancement of the 20th century.
Grothendieck generalized geometry to the point of it coinciding with number theory (see here for example, but there are many others). The point of moduli stacks is the treatment of classification problems geometrically, in the proper generalization of geometry (think, as a baby example, an actual shape where each point represents an object you classify -- however the idea of stacks uses both a generalization of "classification" and of "geometry"). As a first step towards understanding this, I would look at, say, moduli of curves.
These generalizations completely transformed how people think about numbers, and how geometry is perceived (meaning that they are both projections of the same abstraction). Einstein's relativity, in comparison, is just the realization that spacetime is a 4 dimensional pseudo-Riemannian manifold with a specific metric - an enlightening idea to be sure, but still in a mathematical language that is standard to the way we think of geometry (there's nothing new about pseudo-Riemannian manifold, there's just something new about space being one).
Unless one is intending to actually do research in the sciences, I would hesitate to say that a philosopher needs a great deal of formal mathematics. Mathematicians themselves do very little formal mathematics (despite appearances); most of the work is in actually understanding the concepts & how they relate amongst each other, and their relative importance, which questions are important and why. Formal mathematics is a tool which helps & hinders this - rather than the thing itself.
After all, the philosophers first came up with the idea of atoms (which has been applied to matter, then to energy - the 'quanta' and perhaps in future to space-time).
Another example would be the Liar's Paradox that seeded Gödel's discovery of the his incompleteness theorems. He was astute enough to understand its ramifications and turn into formal mathematics.
A philosopher should really, really study cognitive science / neuroscience. There are all sorts of insights we have now about how perception works, how we learn, and so on.
For example, after studying neuroscience, one realizes that introspection is an absolutely horrible guide to how perception actually works. The system is designed to present a seamless view of the world without alerting us to its inbuilt flaws and nature. This is very helpful when you're trying to dodge a lion long enough for the rest of your tribe to arrive and save you, but not so helpful when you're trying to understand the nature of perception sans neuroscience.
For anyone interested in the philosophy of science, I think Eugene Wigner's The Unreasonable Effectiveness of Mathematics in the Natural Sciences is an important piece, but it may be hard to truly appreciate his observations if you don't actually know physics on a technical level.
Here's philosopher Eric Steinhart's answer to the question (the math part anyways), his book More Precisely: The Math You Need to Do Philosophy. From the publisher's website:
More Precisely is a rigorous and engaging introduction to the mathematics necessary to do philosophy. Eric Steinhart provides lucid explanations of many basic mathematical concepts and sets out the most commonly used notational conventions. He also demonstrates how mathematics applies to fundamental issues in various branches of philosophy, including metaphysics, philosophy of language, epistemology, and ethics. This second edition adds a substantial section on decision and game theory, as well as a chapter on information theory and the efficient coding of information.
Here's the table of contents. And here it is in collapsed form, in case the link goes dead:
The short answer is "Enough not to be easily misled". And to me what that means is that one needs to be immersed in an academic culture that will allow others to care about your work and to contribute perspectives from outside your discipline.
I do not think that the philosophy of mind, for instance requires any understanding of neurophysiology. But it requires the ability to capture a metaphor that will not be derailed by idiosyncratic or purposefully misleading interpretations of neurophysiology. You need a grasp, at a man-on-the-street level of someone who might watch Horizon or Nova.
Most importantly, you need a good sense of who is making sense. To some degree, Kuhn can save you here: when it is most important, that is a sociological question, and need not be addressed by actual understanding of the field's internals.
As a biologist, I tend to refer to biology as the "central science". It builds upon physics and chemistry, and studies how we are related to other animals, including our physiology, psychology and behavior. Lack of a minimum knowledge of biology usually turns wanna-be-philosophers into endless readers of endless texts, from which they have no idea where ends the known facts and begins the hypothetical tales and religious prejudices.
Beyond that, the relation of biological sciences with a good knowledge of health could avoid, for instance, the alarming rate of obesity in many "developed" countries, or the way we are destroying all Earth's biodiversity to buy an unsustainable lifestyle. The list of important points is huge, and it astonishes me how the word "physics" appear much more in such discussion than the word "biology".
Absolute reality = truth = fact is that which exists independent from observation and experience.
Absolute reality is that with exists objectively whether or not we are aware of it and believe it. Absolute reality exists totally seperate from anyone's capacity to observe, measure, define and/or believe it. It exists independent from any source or origin.
Philosophy is a methodology developed to contemplate the nature of absolute truth and incrementally improve our understanding thereof by first making hypotheses and validating those hypotheses with thought experiments applied to our subjective experience.
The scientific method is a refinement of philosophical methodology replacing thought experiments with empirical data and eliminating logical fallacies, making the validation mechanism far more reliable and therewith significantly reducing error.
The scientific method is thus a methodology that gradually reveals objective reality with increasing detail and accuracy and therewith also allows us see the interconnection between the different elements and attributes of objective reality that in turn helps us manipulating objective reality to our advantage. It allows us to understand how we can find happiness and safety and build the tools to fulfill them.
As a refinement of philosophical methodology, it's also worth pointing out that the scientific method is capable of adressing every issue philosophy deals with with greater accuracy and detail.
Questions the scientific method cannot answer with any degree of certainty or accuracy whatsoever are therefor also questions philosophy cannot answer with any degree of certainty or accuracy whatsoever. Abandoning the scientific method to answer questions it cannot answer at this point in time and looking for an answer in philosophy instead is therefore utterly pointless.
The vast majority of things that were mysteries to our ancestors have already conslusively been answered by science and science alone. Wherever mysteries remain, science is indefinitely the most reliable methodology to answer it. The belief that any mystery left unsolved by science can be solved or has been solved by any other means is nothing but wishful thinking and a naive response to fear of the unknown.
What about ethics?
Science both teach us a lot about how our actions influence our happiness and stability. Herein, neuropsychology typically focuses on individual behavior, whereas sociology focuses on the collective components components and biology focuses on genetic components of our behavior.
Combined, neuropsychology, sociology and biology give a rather complete picture of human behavior and human consciousness. It allows us to model human nature in a consistent way, which in turn allows us to develop a rational moral foundation based this model.
Adding to that, mathematical studies like game theory can help us determine the impact of our actions and assess that impact with greater clarity.