According to Georg Hegel, Encyclopedia of the Philosophical Sciences, medieval mysticism is the same as the speculative philosophy of the 19th century:
Enc§82d Hinsichtlich der Bedeutung des Spekulativen ist hier noch zu erwähnen, daß man darunter dasselbe zu verstehen hat, was früher, zumal in Beziehung auf das religiöse Bewußtsein und dessen Inhalt, als das Mystische bezeichnet zu werden pflegte. Wenn heutzutage vom Mystischen die Rede ist, so gilt dies in der Regel als gleichbedeutend mit dem Geheimnisvollen und Unbegreiflichen, und dies Geheimnisvolle und Unbegreifliche wird dann, je nach Verschiedenheit der sonstigen Bildung und Sinnesweise, von den einen als das Eigentliche und Wahrhafte, von den anderen aber als das dem Aberglauben und der Täuschung Angehörige betrachtet.
Hierüber ist zunächst zu bemerken, daß das Mystische allerdings ein Geheimnisvolles ist, jedoch nur für den Verstand, und zwar einfach um deswillen, weil die abstrakte Identität das Prinzip des Verstandes, das Mystische aber (als gleichbedeutend mit dem Spekulativen) die konkrete Einheit derjenigen Bestimmungen ist, welche dem Verstand nur in ihrer Trennung und Entgegensetzung für wahr gelten. Wenn dann diejenigen, welche das Mystische als das Wahrhafte anerkennen, es gleichfalls dabei bewenden lassen, daß dasselbe ein schlechthin Geheimnisvolles sei, so wird damit ihrerseits nur ausgesprochen, daß das Denken für sie gleichfalls nur die Bedeutung des abstrakten Identischsetzens hat und daß man um deswillen, um zur Wahrheit zu gelangen, auf das Denken verzichten oder, wie auch gesagt zu werden pflegt, daß man die Vernunft gefangennehmen müsse.
Enc§82e Nun aber ist, wie wir gesehen haben, das abstrakt verständige Denken so wenig ein Festes und Letztes, daß dasselbe sich vielmehr als das beständige Aufheben seiner selbst und als das Umschlagen in sein Entgegengesetztes erweist, wohingegen das Vernünftige als solches gerade darin besteht, die Entgegengesetzten als ideelle Momente in sich zu enthalten. Alles Vernünftige ist somit zugleich als mystisch zu bezeichnen, womit jedoch nur so viel gesagt ist, daß dasselbe über den Verstand hinausgeht, und keineswegs, daß dasselbe überhaupt als dem Denken unzugänglich und unbegreiflich zu betrachten sei.
It should also be mentioned here that the meaning of the speculative is to be understood as being the same as what used in earlier times to be called “mystical”, especially with regard to the religious consciousness and its content. When we speak of the “mystical” nowadays, it is taken as a rule to be synonymous with what is mysterious and incomprehensible; and, depending on the ways their culture and mentality vary in other respects, some people treat the mysterious and incomprehensible as what is authentic and genuine, whilst others regard it as belonging to the domain of superstition and deception. About this we must remark first that “the mystical” is certainly something mysterious, but only for the understanding, and then only because abstract identity is the principle of the understanding. But when it is regarded as synonymous with the speculative, the mystical is the concrete unity of just those determinations that count as true for the understanding only in their separation and opposition. So if those who recognise the mystical as what is genuine say that it is something utterly mysterious, and just leave it at that, they are only declaring that for them, too, thinking has only the Significance of an abstract positing of identity, and that in order to attain the truth we must renounce thinking, or, as they frequently put it, that we must “take reason captive.” As we have seen, however, the abstract thinking of the understanding is so far from being something firm and ultimate that it proves itself, on the contrary, to be a constant sublating of itself and an overturning into its opposite, whereas the rational as such is rational precisely because it contains both of the opposites as ideal moments within itself. Thus, everything rational can equally be called “mystical”; but this only amounts to saying that it transcends the understanding. It does not at all imply that what is so spoken of must be considered inaccessible to thinking and incomprehensible.
For Hegel’s relation to Meister Eckhart, see there.
The philosopher Bertrand Russell concerned himself with mysticism and was, according to himself, influenced by a “mathematical mysticism” in his younger years.
In his essay “Mysticism and Logic” (Russell 1918) Russell understands mysticism as “the attempt to conceive the world as a whole by means of thought”. He gives four characteristics of it:
Russell finds at least some of these characteristics in the philosophies of Heraclitus, Plato, Hegel, and Spinoza. Interestingly, Heraclitus is judged far better than Plato, the former being ascribed “a nature [in which] we see the true union of the mystic and the man of science”. In the remainder of the essay Russell discusses the relation of the four characteristics of mysticism to his own views.
In the autobiographical book Portraits from Memory the passage
Mathematics … deals with the world of ideas and has in consequence an exactness and perfection which is absent from the everyday world. This kind of mathematical mysticism, which Plato derived from Pythagoras, appealed to me.
may serve as an example how Russell was influenced in young age by the exposure to mathematics; more poetically it is expressed in the essay The Study of Mathematics published along with Mysticism and Logic (Russell 1918)
Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of sculpture, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as in poetry. … Real life is, to most men, a long second-best, a perpetual compromise between the ideal and the possible; but the world of pure reason knows no compromise, no practical limitations, no barrier to the creative activity embodying in splendid edifices the passionate aspiration after the perfect from which all great works springs.
… Of the austere virtues the love of truth is the chief, and in mathematics, more than elsewhere, the love of truth may find encouragement for waning faith.
According to Ray Monk’s biography on Russell, the search for a foundation of this “Pythagorean” mysticism governed the first half of his life: Being deeply disappointed by the mathematical curriculum at Cambridge, which did not address his philosophical needs, Russell turned to idealism, which was at that time propounded in Britain by J.M. Ellis McTaggart following the German tradition of Kant and Hegel. Under this influence Russell devised in An Essay on the Foundations of Geometry a variation of Kant’s transcendental aesthetics claiming that the curvature of the space of our perception, i.e. of any ‘’form of externality’’, is a priori constant.
Exposure to the work of the German mathematicians Dedekind, Weierstraß, and Cantor along with a personal meeting with Peano caused a turn in Russell’s views. He now hoped to find a rigorous foundation of mathematics based solely on logical axioms, thereby justifying a kind of Platonic view on mathematics. But over the years he was frustrated in this attempt by many problems he encountered, especially Russell's paradox. He gradually gave up all his Platonic beliefs; arriving in the end (after Wittgenstein's Tractatus) at formalism.
He remarks on the power of rigorous mathematical definition:
“Continuity” had been, until he [Cantor] defined it, a vague word, convenient for philosophers like Hegel, who wished to introduce metaphysical muddles into mathematics. Cantor gave a precise significance to the word, and showed that continuity, as he defined it, was the concept needed by mathematicians and physicist. By this means a great deal of mysticism, such as that of Bergson, was rendered antiquated. (Chapter XXXI of “A History of Western Philosophy” (1945))
L.E.J. Brouwer had a long standing interest in the theory and practice of mysticism, setting out his manifesto while still a graduate student in the essay Life, Art, and Mysticism (1905). The essay quotes the Bhagavad Gita, Meister Eckhart, and Jakob Böhme.
However, he did not consider his intuitionism to depend upon his mystical views as such. For Brouwer, mysticism is the domain of “consciousness’ deepest home”, wherein there is no subject-object distinction. The first act of intuitionism, the intuition of time, propels us away from this pure state of “wisdom”. It is in fact the precondition for any experience of an outer world or experience of the self as something distinct. Mathematics thus comes into being in this exodus from the home of consciousness, so on Brouwer’s view, “mathematics is in a sense the very opposite of mysticism” (Van Atten, 2004), and mystical insight is only possible quite apart from the machinations of the intellect.
Brouwer even went so far as to remark on the perniciousness of mathematics in its keeping us from the home of consciousness. Thus in the notebooks in which he conceived his 1907 thesis we find: “One could see as the goal of one’s life: abolition and delivery from all mathematics”.
Wikipedia, Mysticism
Bertrand Russell, Mysticism and Logic and Other Essays, 1918 (on Wikisource). Containing Mysticism and Logic and The Study of Mathematics. The former was published already in 1914 in the Hibbert Journal.
L. E. J. Brouwer, Consciousness, Philosophy, and Mathematics, 1948
Ray Monk, Bertrand Russell – Mathematics: Dreams and Nightmares, 1997
van Atten & Tragesser, Mysticism and mathematics: Brouwer, Gödel, and the common core thesis, 2003
van Atten, On Brouwer, 2004
Arthur Melzer, Philosophy between the lines – The lost history of esoteric writing, University of Chicago Press, 2014
Decent discussion of the history of mysticism (or not) in quantum physics, such as induced by the concept of “wavefunction collapse” (see also interpretation of quantum mechanics) is in