Let's assume that I have lead music notes for a song. I do not have a musical instrument with me to actually play it. But I do have an understanding of sound frequencies in western music, understanding of all musical notations (how each note differs from other, their length of play, etc). Just by looking at the notes and with nil understanding of how a chord would sound, is it scientifically possible to determine the chords for the song by mathematically examining the notes in each bar without hearing.
Yes, to an extent... There are always alternate possibilities for chord progressions. So while it is possible to determine a chord progression that works, it will not necessarily be a unique chord progression, or even the best chord progression. You could always use chord substitutions to create a different, but still valid solution.
To a large extent, this is how I play piano accompaniment. I read the melody line and fill what the chord that I know should probably be there... And there are times when I guess wrong.
As a concrete example, lets say you see a note that is the fourth scale degree of whatever key you are in.
First, there is always the possibility that this note is a "non-harmonic tone" -- i.e. not part of the active chord. Whether this is the case or not can often be guessed from the context of the melody. For our purposes, we'll assume that it is a chord tone.
If you limit yourself to triads, you have three possible chords it could harmonize with: ii, IV, or vii-dim. Statistically, diminished chords are pretty rare, so you can probably ignore that option (except when you can't). The remaining two chords (ii and IV) both act as a predominant, and are often somewhat interchangeable so you can probably pick either one.
However, if you include seventh chords, than this could also be part of a dominant seventh chord (V7). Since dominant has a different function to the preceding predominant chords, the choice between these options will create a greater effect on the final sound. The dominant chord will generally have a greater tension, and a need to resolve. This is where an feel for the underlying direction of the overall progression can come in handy (or better, and knowledge of the actual song) because you can infer whether a dominant chord is desired.
If you consider chords outside of the scale, some additional possible chords for the 4th scale degree include iv (functions as a predominant, similar to IV, but with a flat 6 to make a more melancholy minor sound) or ♭II, which can function either as a predominant (a Neapolitan chord) or as a dominant (via tritone substitution).
Because there are multiple solutions that will work, some better than others depending on context and desired sound, if you have "nil understanding of how a chord would sound", as posed in the question, then you will only be able to guess randomly between these options. Thus you run the risk of making a choice that, while not dissonant, may end up sounding slightly awkward within context.
I would add to the above answers that it depends on the musical style of the song. For instance, in many folk traditions you are more likely to find a more modal kind of harmony than in more modern pop. Thus, it might be more fitting to use a ii chord (d minor) in the key of C to accompany the melodic note D than the more classical/pop V chord (G major).
There are computer programs that attempt to automatically harmonise a melody, so yes, it's possible. In the context of a simple folk song, there might even be agreement on what the "right" harmony is! But it's more likely there will be several mainstream choices, let alone all the possibilities in left field! And, of course, what is "mainstream", what is "left field" all depends on context.