It appears to me that the secret of this matter is that it is the way of the Torah to do everything as 'a miracle within a miracle.' Thus [the Torah] removes injury by means of the cause of the injury, and heals illness by means of the cause of the sickness, (as the Rabbis have mentioned in connection with the verse, and the Eternal showed him a tree, and as occurred with the salt that Elisha cast into the water).
Now it is a well-known medical principle that all people bitten by poisonous creatures become dangerously ill when they see them, or even when they only see their likeness, so that if people who have been bitten by a mad dog or other mad animals look into water, they see there the image of the dog or the attacker, and this can cause them to die, as is written in medical books and mentioned in the Gemara of Tractate Yoma. Similarly doctors keep people from mentioning in their presence the name of the animal that bit them, because their minds cling to this thought and do not turn away from it altogether until it kills them.
These doctors have already noted that it is an empirical fact, amongst the wonders of the species, that if the urine of a person bitten by a mad dog is put in a glass receptacle after he has become rabidly sick, there will appear in that urine the image of puppies. And if you pass the liquid through a cloth and filter it, you will not find any trace of them at all; but if you return the liquid to the glass bottle, and it remains there for about an hour, you will again clearly see puppies in it. This is a true fact, and one of the wonders of the forces of life.
Now in view of all this, it would have been correct that the Israelites, who had been bitten by the fiery snakes, should not look upon a snake, and should not mention it or bring it to mind at all. But the Holy Blessed One, commanded Moses to make for them the likeness of a serpent, which was the thing that killed them.
And it is well-known that these fiery snakes have red eyes and wide heads, and their bodies at their necks are like copper. Therefore Moses could not fulfill God's command to make a fiery serpent except by making a snake of copper, which appears similar to a fiery snake. For had he made it of any other material, it would have had the likeness of a snake, but not that of a saraf, a fiery serpent.
And those who said [that Moses made a copper snake] in order to “let this language fall on that language,” meant that the mention of the name alone would usually have been harmful. But the principle here is that God commanded that they be healed with the very thing that causes harm. So they had to make it like that in both form and name, so that when a person gazed upon the copper snake which completely resembled the damaging agent, they lived. And this was all to make them realize that God alone gives death and life.