You often hear people saying that numbers aren’t real, but in fact there is a fairly simple proof of numbers that anyone with a normally-digited human body can perform for themselves.
Mathematics of course begun with the ability of human beings to count up to ten on their fingers and thumbs. We thus have a very apparent proof of the existence of numbers up to ten staring at us all the time from our palms. The philosopher G.E. Moore thought that refuting skepticism was as easy as claiming that he had one hand, then another. Certainly he would have been right if he was talking about absolute skepticism about the existence of numbers.
But could he have proved that there were numbers higher than ten? The easiest way of doing this is to switch from fingers to toesies. So then, assuming that one has five toes on each foot, we have a proof of the existence of numbers up to twenty.
It is at this point that a lot of societies throughout history typically stopped. Before around 1000 BC, there are few documented instances of numbers being referred to that are larger than twenty: the closest is a vague sense, intimated by some Indian and Babylonian sages, that such numbers might exist, but they would be incomprehensible to the human imagination. But then at roughly the same time, the great early philosophers of China and Greece discovered numbers higher than twenty, by realising that they could count on their knuckles.
Thus followed a great dynamic period of innovation in the science of mathematics, as the amount of numbers available to the human mind appeared to increase exponentially. Just counting on finger and thumb-knuckles increases the numbers to thirty. Then someone usually noticed (in Greece this was Anaximander) that the toes have their own vestigial foot-knuckles, giving us forty. The collections of ridges on the fingers and thumbs are then counted, two on each finger and one on each thumb, giving us the ability to count to 58.
Now, in China it was at this point that the accumulation of numbers simply stopped, but the Greeks, who were always more interested in numbers (not least because many of them e.g. the Pythagoreans, worshipped them) produced two further innovations that allowed them to count as far as 100. The first was to notice that the toes have their own sort of ridges to them, which gave then an additional ten numbers which they could count. The second was to count their teeth. This gave them anywhere from an additional zero to 32 numbers, depending on how many teeth they had remaining.
Thus it only stands to reason for us to claim that: given a normal human body with all its digits and teeth intact, we have a performative proof of numbers up to 100.
Are there numbers higher than 100? Of course, the relentless march of intellectual history has suggested that there is: certainly some very great thinkers thought so. As early as the first century AD, heterodox number systems based on nail-counting and suchlike gained purchase, particularly through Gnostic and mystery religions. During the Renaissance, Giordano Bruno posited an infinity of numbers based on counting the stars (also held to be infinite). Even Aristotle thought that in addition to the standard bodily proof of numbers there was an additional metaphysical proof of numbers up to 101, given that the 100 countable digits and teeth required a single unifying principle in which they all might be contained.
Such proofs are obviously deeply controversial. And, to my mind, we would be better off avoiding such controversy altogether, since even if we could prove the existence of numbers larger than 100, it seems that we have little reason to think that doing so would be at all necessary or desirable. Being able to count to 100 gives us all the numbers we need: who, for instance, would want to build a house over 100 metres tall, or to live to an age greater than 100? Would you want to spend more than 100 pounds on an item of clothing, or eat more than 100 bananas in one sitting? Are books ever made any more enjoyable to read by being greater than 100 pages? But numbers greater than 100 are not simply superfluous: they are actively dangerous. All the greatest atrocities in history have involved the deaths of more than 100 people. The existence of distances greater than 100 miles has historically made travel and communication over them very difficult if not impossible. I could go on.
Therefore, it seems to me that the best answer to the question of the reality of numbers is to say that: yes they are real, but the largest possible number is 100, the largest number that admits of any sort of direct proof.