I have a background in Mathematics, and am starting to wander into the complex realm of Philosophy. I'm interested in trying to understand what is the meaning of the scientific investigation in relation with natural "reality" (an undefined term for the time being). To properly address the issue, I think I need some serious introduction to phenomenology, so I ask you if you could provide some good introductory reads, preferably palatable to a scientific oriented mind.
I recommend Wittgenstein's Philosophical Investigations. About third way into the book he starts to deal more and more with phenomenology — he approaches the topic through our use of language and mixes in mathematics as well.
He is considered one of the greatest philosophers (of the 20th century if not ever) and yet contemporary philosophers of mind seem to ignore his critique with no apparent justification.
For example, while he argues very forcefully against there existing a criteria of identity that can be used to compare private experiences, philosophers of mind continue to discuss thought experiments of fading qualia and inversion of qualia as if they have never heard of him.
I think it is therefore valuable to read him, before you get carried away with all the rest.
He is also very readable and does not use cryptic jargon.
Edmund Husserl is one of the founders of phenomenology. Husserl has even studied mathematics, but afterwards switched to philosophy.
Husserl has published Philosophy as Rigorous Science besides many essays like Ideas Pertaining to a Pure Phenomenology and to a Phenomenological Philosophy with phenomenology already in the title.
It is up to you, to decide how helpful phenomenology is for you, serving as an introduction to philosophy.
In my opinion, possibly the writings of Karl Popper are more helpful to a reader with a mathematical or scientific background, e.g. Popper's book Objective Knowledge: An Evolutionary Approach. Popper does not deal with phenomenology. I think he did not estimate this type of philosophy.
Phenomenology has a narrow meaning in contemporary philosophy as a style of philosophical inquiry originated by Husserl, and I do think that it is particularly congenial to a mathematician. Husserl worked as Weierstrass's assistant in his youth, and later personally knew and corresponded with Cantor, Hilbert, Courant, Minkowski, and other major mathematicians of his time (he worked in in Göttingen until 1916), see On Husserl's Mathematical Apprenticeship and Philosophy of Mathematics.
Husserl's phenomenology is somewhat unusual in philosophy, understood pragmatically it can be seen as a system of mental techniques for exploring ideal objects, discovering structure in them and connections between them, and provides insight into the workings of mathematical intuition when finding new results and their proofs. His writing style can be surprisingly lucid to a mathematician, his early training shines through in striving for precision of language, he often coins new terms to make exposition more precise, and in the presentation of examples and arguments. I found Ideas for Pure Phenomenology (often abbreviated as Ideas I and available online) particularly insightful in this regard. Husserl's idea of ideal perception attracted Gödel, who saw it as the basis of mathematical intuitions, writing "our ideas referring to physical objects contain constituents qualitatively different from sensations or mere combinations of sensations, e.g. the idea of object itself".
But as a first reading I recommend Phenomenology, Logic, and the Philosophy of Mathematics by Tieszen, a contemporary phenomenologist of mathematics. He gives a general introduction and presents detailed examples of applying phenomenology to mathematical subjects. He also describes the work of mathematicians who credited Husserl as their inspiration, notably Weyl and Gödel, and connections between phenomenology and philosophies of mathematics, particularly Frege's logicism and Brouwer's and Weyl's intuitionism.
I agree that Husserl would be the obvious starting point. He is considered the founder of modern "phenomenology" and his interest was originally in working out the origins of mathematics by "returning to the things themselves," as he put it.
It is also worth noting that Husserl and Frege corresponded with one another and shared an interest in mathematics, yet set modern philosophy on two very different courses. From Husserl's phenomenology came Heidegger, existentialism, and much of what is roughly called the Continental side, while Frege gave rise to Russell, the logical positivists, and so-called Analytical side.The complete genealogy of "phenomenology" runs roughly from Hegel's "Phenomenology of Spirit" to the existentialists like Merleau-Ponty, Gestalt psychologists, and various modern Heideggerians.
I would suggest browsing a copy of Husserl's "The Crisis of European Sciences and Transcendental Phenomenology," which is not too mystifying, alongside one of the synoptic introductions to phenomenology, plentiful from publishers. An interesting accompaniment, though not phenomenology per se, might be Wilfred Sellar's essay "Philosophy and the Scientific Image of Man." Not easy, but an intriguing distinction between the "manifest" and "scientific" images of "man-in-the-world."
You might be interested in Gian-Carlo Rotas "Indiscrete Thoughts". Apart from his work in mathematics he also happened to publish in phenomenology. The book has some fascinating essays on 20th century mathematicians and a whole section devoted to phenomenology.
Introduction to Phenomenology by Robert Sokolowski is what I read. Very enjoyable read.