"Evidence of mathematical reasoning" really throws this one open. That definitely predates modern humans by an exceedingly long time. Off the top of my head, I might suggest the Oldowan style of stone tool, or hand axe, which dates back 1.5 - 2.5 million years. Suggested to be the first evidence of cultural activity, it would also be the first evidence of math. Everything about stone tool production, from the understanding of how the rocks striking each other will interact, to the reasoning behind using a blade instead of a smooth surface, requires at least some primitive grasp of a mathematical principle or two.

Oldowan tools

If you're referring to formal written mathematics, not just evidence that our ancestors possessed mathematical ability, I daresay that is more restricted by the development of writing than by the development of mathematical knowledge. Some of the earliest examples of writing we have are things like inventories and accounting data, inherently mathematical activities. It would appear that in some cases, writing was adopted to facilitate the math already being done.

Examples:

The oldest Egyptian hieroglyphics, used for inventory

The oldest Greek writing, used for accounting