#1) What If ?
What if, for the sake of discussion, for the sake of argument, we start from some extreme position, & follow it to some resulting "down–stream" conclusions, just to see where it gets us ? I can't promise that we stick to any given criteria of strict Logic or Proof, but perhaps some "ring" of truth remains after the speculative exercise. The other promise might have to be relinquished, for instance, if the validity of that same Logic or Proof only "existed" at a far lesser "down–stream" level in the chain of speculation. To anchor these abstract ground-rules a bit, let's begin with our Protagonist deeply asleep. A very Wise One that he knows shakes his shoulder, awakens him rapidly, asks quickly:
          Do you exist ?
The former sleeper responds right away or later relates the immediate response in his Mind:
          Yes, certainly I exist ! I just know that's true, spontaneously & undoubtedly.
The Wise questioner counters with an observation & a 2^nd confirming question:
          Oh I see, you "know" you exist like that. So can I agree that you "exist" & that you also "know" – at least that much.
Nodding his head in assent, the freshly awakened asks once for clarification & then listens for a while.
            Granted what you say, what is the significance ?
Asking the awakened to suspend judgment for bit, much as expressed at our beginning, the Wise One ventures on with is own "What if ?" scenario, with minimal detailed explanation or proof, getting no further argument or interrupting question from the listener. He speculates as follows, or with words to that effect.
That "inner place" in which you felt that certainty of your Existence, & also that certainty of your knowing so, your Consciousness, is actually the same "place". Moreover that "place" is not a place, having size or shape, has no location in Space, or in Time for that matter. The very same "place" where you are certain of your Consciousness is actually your Consciousness itself. As also the very same "place" where you are certain of your Existence, it truly is your Existence, it is what you are, your Self.

And not only your Self, but the same is my Self, & the Self of everyone & all that exist, have ever existed, or will ever exist. That singular Existence or Being is the literal Truth proclaimed by Parmenides & Shankara, by Huang Po, Meister Eckhart, Ib'n al Arabi, Brahmagna Ma, & Ananda Ma. The point being that no culture, gender, creed, or period of history has unique claim to this recognition, this Knowledge known as the Perennial Philosophy.

Further observations include the fact that very same not-a-place "place" or Brahman or the Self is also experienced routinely as the Source of Happiness, Love, Bliss, Joy, Peace, Satisfaction, Freedom & the like. These are all one with Consciousness & Being, though Happiness in full is more often the complete "reference value" we all intuit, perfect Happiness, the goal, whether we currently believe it to be attained or not. That Source of Happiness is the same "place" where true Happiness (not mere pleasure or triumph) is actually felt, routinely. Even the sense of its lack could be said to be felt there along with all negative emotions that seem so [negative] by coloration with conceptual self-delusion.
Nodding his head again in awe more than yet assent, the freshly awakened asks once again in the same words:
          Granted what you say, what is the significance ?
Much can has been said & written about all the ramification & also restoring true Self-Identity through the meditation of inward Self-Inquiry. Any sincere seeker can find all that, especially now with the Internet to get started with. Given some sifting through apparently conflicting alternatives (& some are conflicting) & perhaps a few false turns, the one who truly desires Freedom (Liberation from ignorance) will have it.
This much can be said in closing for now. That begins above as a "What if ?" certainly is found to be factual, though this cannot be proven or even fruitfully argued from "this side" of the issue. Experimentally proceeding in a neutral, judgment-suspended "What if ?" approach, the following could be said to be the significance you ask about.

You & I & all are that same Self, though we may seem to be "caught" in a Waking Dream world of Time & Space (at night, other dream words supervene, while in deep sleep, none at all). While seeming so "caught", we best pursue Self-Inquiry, with authentic texts that support that & qualified live guidance when possible. Other wise we progressively shine in our World (even if obscurely) fulfill responsibilities, relationships, work, etc. in Peace. Any help offered to what can appear as a stricken World & inevitable death in that World, comes from the inner basis, & less-so, the practical basis just described. This quest of Enlightenment, Liberation, is the most satisfying & significant venture possible.

 selection from Sri Ramana Maharshi

#2) How can an absolute Non-dualist appreciate Science ?
In simplest terms (the tone of this post throughout) Absolute Non-duality denies that there is a RWOT Real World Out There. Yet even the Illusion, a projection of Mind, is a an ephemeral reflection on the one Consciousness. In the seeming Waking State (while not in Dream or Deep Sleep) Science can be said to be a "high" view, an insightful perspective on the Illusion (the Universe) from within that Illusion. If there is no World for Science to actually speak to, Science does in relative (non-absolute) terms speak to the Mind itself. Science models much of the Mind in a fairly clear manner. Depths of Mathematical Physics are awesome (but beyond our scope here), diving into Fractals can be beautiful, & so on, all through Neurology, crystal structure, what have you. But here, with no need for illustration, because the reader can so easily visualize her own, we conclude with a truly elementary example that is yet a platform for much insight. The following could even play some small part in self-examination that paves the way toward Liberation. We glance briefly at what makes middle school math interesting.

Rene Descartes on holiday in the mountains of Bohemia was lazy late in bed on a morning when his daydreaming bore fruit. Tracing a fly walking across a structured Inn ceiling, he found himself track the fly's meandering with what he later called the 2-D Coordinate system patterned on the ceiling. Rene himself had been groping for such a device in recent study of quadratic curves in Conic Sections, math he'd credit Greeks of an earlier millennium. Ancient fishermen had tide knots in anchor ropes & later in nets. Egyptian architects borrowed the net & rope-ruler schemes for pyramid construction. Greeks had further pioneered the geodesy of Latitude & Longitude. Arab philosophers & artists preserved these Greek & Egyptian "technologies" during Europe's Dark Ages & these ideas entered Andalusian Spain during their occupation. The same prompted perspective drawing & architecture so fundamental to the Renaissance, but only Descartes organized 2-D "graphs" soon after that Bohemian holiday morning. What does  urban legend say about the Universe projected by Mind & Mind itself?

The middle schooler might easily overlook the organized 2-D Space on her graph but in many way this model evolved into the computer screen, & along the way a common sense of the 2-D "screen of the visual Mind."  The similar 3-D "theater of the Mind" is a close relative to our sense, & Newton's sense of Space itself. The older knotted rope might similarly be found tracing back from the sequence of Thought & the flow of Time, sound, & so on.

Overlooking the widespread concept of the Coordinate Space in all its forms might have bee harmless for the middle schooler but the underpins the Structural Realists adamant committed to a "real" Space or Space-Time. What could trip up the middle schooler eventually, however, would be to overlook the essential reference role of the Graph's Origin, point (0, 0) in 2-D. In the 3-D World of Space, the same is called Here. Along the 1-D Space of Time, the Origin is called Now. In the Space of Mind, the constellation of Thoughts are similarly all measure or referenced to the Origin call Me, or the Ego–"I". This is not the grand Brahman–Self "I"–less "I", the True Self. That Origin of Mind is the Ego, the imaginary obstacle to Enlightenment, gives every thought the meaning & seeming "reality" each has.

region in 3-D Space shifted to put Origin in its Center – could be a Mini-Container-Space in itself, & Macro-Origin for a Vicinity

To run around with those Thoughts & run around in the World they create is the opposite of Enlightenment. Appearing to function in a World is fine, but Freedom requires recognition of that Ego-Origin, its falseness, its seem "place" or arising in Consciousness & final identification with that Nondual Consciousness.

Non-dualist appreciating Science (short epilogue):

Simple points appended to mention of Cartesian Coordinate Space as a model for:

1) Mind – where thoughts are the points & the Ego-"I" Me is there reference Origin (0, 0) etc.

Mind Space seems:      
1-D for the sequence of thought

2-D for the visual screen of the Mind

3-D for the theater of the Mind

4-D for lifetime identity Personality, born, growing, dying in Waking State Time

2) Time – the 1-D Space of Time itself overlaps somewhat with the Mind's 1-D for the sequence of thought, with Moments for the points on the Timeline & Now for the Origin (0).

3) Space – physical, is 3-D with Here as the Origin, or as 4-D Space-Time with Here & Now at (0,0,0,0). 

4) World – fills Space with objects, etc. as points with an extended as the Body. While Hearing naturally parallels Time in Sound & Music, all being mostly 1-D, Vision naturally scans 3-D Space. These sense perceptions along with Tactile, etc. senses concretize a World. While the Body is an extended or Macro-Origin reference Origin for the coordinate space of the World, that same Body can be a Mini-Space in itself, with it own smaller Macro-Origin as perhaps the Brain. The Body-skin marks the inside–outside boundary in the World.

Similar Macro-Origins / Mini-Coordinate Space concepts could be inserted around each of the other Origins mentioned above, thus also defining inside–outside boundaries. For Time, this could be My Lifetime. For Space it might be my Vicinity, a changing reference by which locations or positions in Space are is variously appraised. Around the core Ego Origin of Mind, some extended Personality could be imagined to separate thoughts in some inside–outside way:

The point of these simplistic models is to suggest that Mind, Time, Space, & the World are not objective "givens" of fixed qualities. Different people experience these differently, & each of us does also at different times. However modeled, or not, these are mental projections, not fixed realities. The Coordinate Space models suggest that the same old "space with a center" Idea is repeated over & over again for Mind, Time, Space, & the World. More importantly, these represent the net of Illusion that distracts us from recognition of the core Identity, the Nondual Brahman, the Self of absolute Existence, pure nondual Consciousness, & complete Happiness, Love, Bliss, Peace, & Liberation.

Self-Inquiry traces back along short or long routes from any of the above, especially from Thoughts, questing "Whose thought?"  "Mine."  "Who am I ?"  "Whence am I ?" which is to say finding the "place" in Consciousness  where the Ego-"I" seems to arise. Find it to never arise there, remain as that Consciousness, the Non–Dual Self.