Likeness of Chameitz to Notar
The mishna (Pesachim 2:1) records a dispute between Rebbe Yehuda and the Sages regarding the disposal of chametz as Pesach approaches. According to Rebbe Yehuda, this should be done only by burning; the reasoning is that chametz has a likeness to notar, left-over sacrifices which must be eliminated by burning. But the Sages state that other methods of eliminating the chametz, such as crumbling it and scattering it in the wind, are also proper. While we rule according the lenient ruling of the Sages, the custom is to burn the chametz because of its likeness to notar (SA OC 445:1 and Mishna Berura).
The likeness to notar, as elaborated in the gemara (Pesachim 27b), is indeed noteworthy. Both sacrifices and chametz are permissible up to a certain moment of time, but then become forbidden to derive any benefit from, with a punishment of excision (karet). (Of course, there is a key difference: notar is burned only after the "deadline" has passed; it is forbidden to burn a sacrifice which is still edible. By contrast, chametz must be burnt before the "deadline"; the whole idea is to eliminate it before the time of prohibition arrives.)
Let us see if we can extend the likeness between these two prohibitions.
Sefer Hachinukh gives two reasons for eliminating notar. The first reason is that since the sacrifice is holy, it is unseemly to let it become putrid, even after it is no longer permissible. Thus it is necessary to eliminate it. This doesn't seem to be relevant to chametz; during Pesach we eliminate all chametz, not just old chametz, and anyway the whole idea of chametz is that it is something that is improved by delay and ripening. We don't want to honor the chametz, on the contrary we seek to eliminate it.
The Chinukh's second reason is that by eliminating the notar we show our trust in Hashem's providence; saving it even after it is forbidden would be like showing that we are worried that we may be so desperately in need that we may be forced to eat it despite the prohibition. This approach seems appropriate to the elimination of chametz. Normally, we are strictly forbidden to destroy food. Bread in particular must not be wasted; it is forbidden to throw bread away, and even discarding bread crumbs is considered wasteful (SA OC 180:4). By having a special mitzva to eliminate one kind of bread, we show our trust that God will provide for us. But why specifically chametz?
There are a number of approaches to the specific symbolism of chametz. One explains that chametz represents pride, since it causes the bread to rise; another that it represents sloth since the bread rises only when we leave it alone. According to the approach suggested by the likeness to notar as explained by the Chinukh, we can say that the unique characteristic of chametz is that it is by nature "old", since it takes time to produce. We eliminate from our domain any bread which is demonstratively old in order to show that we are starting a new life in which we will be dependent on the grace of Hashem.
On the first Pesach, on the eve of the Exodus, we particularly wanted to show our dependence on Hashem. The Exodus from the splendor of Egypt to the dearth of the desert was the ultimate demonstration of our trust in God: "I have remembered the loving- kindness of your youth, the love of your betrothal; your going after Me in the desert, in an unsowed land" (Yirmiyahu 2:2). One way we demonstrate this trust is by eliminating chametz. Eliminating chametz, the bread of luxury, in favor of matza, the bread of affliction, shows that we eschew the luxury of Egypt and favor even spare rations with dependence on Hashem. And eliminating the bread of the past for that of the present shows that we eschew the security of Egypt and trust in God's providence to renew our sustenance through reliance on Him.