Obviously there are multiple ways to explain this, so the correct answer in this case would be the most reasonable and detailed answer with as many explanations as possible.
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Rather than explaining how magic can be explained scientifically since there are far too many examples in how that can be done I'll explain how you can make magic logical by asking yourself some questions.
How is Magic used?
- Does a chant/aria/mantra need to be recited?
- If so, how is it structured?
- Do the words have meaning in relation to the effects?
- Do special tools need to be used?
- If so, are they man-made?
- How is the tool made?
How is Magic powered?
- Is the power visible?
- Is it natural?
- If so, where in nature does it come from?
- Is a conceptional source where it comes from (IE. Crystal, Vortex)?
Who can use Magic?
- Can anyone just use it?
- Are they born with it?
- If so, is it inherited?
- Is there something in their bodies which is different from someone who can't use magic?
- Do they have to partake in a ritual in order in order to gain the power?
And of course most importantly;
Do we really need to know how exactly how Magic works?
Sure it's good if you can explain it, but leave some mystery in there. Mystery is what separates magic from highly advanced technology.
Here's an example:
Shadow Magic involves the manipulation of matter by forming it into a nebulous cloud in which it's magnetic polarity is always in flux. The cloud itself is formed from microscopic minerals found in the earth which seem to have existed since the formation of the planet which resonate with the caster allowing the change in magnetic polarity.
One can't be taught to use Shadow Magic, they must be born with it. This is because of a growth within the body which in the past was mistaken to be a cancerous tumor, which is formed when the fetus is exposed to Anti-Matter while in the womb.
Shadow Magic is cast by the user using a special paint, made from the blood of the caster, on the caster's fingers which allows them to "draw" in the air. The first symbol is the location where the magic will form, the second is the shape of the magic, the third is the effects and the final symbol is where it is to be bound to. This can be omitted if the user intends to "shoot" the magic but is required when forming veils/cloaks on the body or commanding tendrils/ribbons with one's limbs.
As you can see above we answer How is Magic Used, How is Magic Powered andWho Can Use Magic while also keeping some mystery in there. If we're always asking "why?", it allows further explanation.
From there you can then think of how one could go though the process of discovery. Using the example above you could say that the paint was found to contain blood which matched that of the user, after several positive matches between users and their paint they had they bodies examined which all had the same form of tumorous cancer. Closer inspection revealed it wasn't a cancer at all and not only that, but all users were exposed to Anti-Matter at birth.
I imagine science would do this via controlled, repeatable experiments that measure its effects, as it does with every new object or phenomena (and indeed many old phenomena) that has been observed. Science is not actually defined by a certain technological state, but is rather a systematic way of asking questions, known as the scientific method.
How successful these scientists would be depends a great deal on the nature of the "magic" involved, spesifically whether or not is has rules or limits or is internally consistent; basically on whether or not it's possible for experiments in magic to be repeated or peer reviewed. If it can, I suspect they will treat it much as they do any other promising new branch of research: with a great deal of enthusiasm!