After following the White Rabbit down the rabbit-hole and meeting all kinds of strange and wondrous animals, Alice finds herself at the mad tea party. Her three companions – the Hatter, the Dormouse and the March Hare – keep asking Alice difficult questions, which make her feel very annoyed.
The Hatter leaned over the table in the direction of Alice. “Tell us, my dear girl,”, he began, almost shoving the Dormouse, which had fallen asleep again, with its nose in the sugar-pot, “what is the largest number that you know?”
The March Hare chuckled. He knew that the Hatter’s question was a trick-question. Whatever number Alice would come up with, the Hatter would always be able to come up with a larger number. He’d just take Alice’s number and add one to it.
‘This is no way to treat your guests’ Alice thought. ‘All these questions…it’s almost as if I’m back at school.’ She decided that she would no longer play along. She would not answer the Hatter’s question.
“I don’t know any numbers.” Alice answered after a lengthy silence, during which she rested her chin on the palm of her hand, “Of course I’ve heard of them,” she continued while smiling to herself, “but to say that I know them… I think I don’t really like numbers.” Alice felt very proud of herself for having countered the Hatter’s question without actually answering it. She tried hard not to be impolite, but she decided to take her boldness one step further. Frowning deeply – as if she was struggling with her thoughts – Alice asked: “What is a number, anyway?”
“Well, that’s simple” the dormouse said, without raising its head, “just imagine a number of things and take away the things”.
“Yes…yes!” the Hatter shouted enthusiastically, “just take a number of apples and throw away the apples.”
“Well, I still don’t like them” Alice said grumpily. The hysterical laughing of the Hatter had made her even more annoyed than before. “Do we really need numbers?”
What is a number?
The philosopher and mathematician Bertrand Russell (1872-1970) wrote extensively on the philosophy of mathematics. In his 1919 Introduction to mathematical philosophy Russell defines ‘number’ as follows:
“A number is something that characterises certain collections, namely, those that have that number” (p 12)
Among philosophers of mathematics there is no agreement on what is the definition of number. Russell’s definition above is very much like the definition the Dormouse gave Alice (or is it?).
Modern philosophers do not unanimously adopt Russell’s definition of number. There is, for example, the idea that mathematical entities like numbers exist in the same sense as do atoms and electrons. This alternative view rests on the Quine-Putnam indispensability argument for mathematical realism. This argument goes something like this. If we believe in atoms because the idea of atoms is very useful for describing the world around us, then we should – for the same reason – believe that numbers exist. Numbers are very useful (if not indispensable) for a description of the type that we use in science.
The indispensability argument hinges on the idea that scientific descriptions without numbers are less useful than scientific descriptions which do involve numbers (or the idea that science without numbers is impossible). The question we must first answer, therefore, is what makes a scientific description ‘useful’? We want scientific descriptions to help us predict what will happen in future experiments. For the indispensability argument to work, therefore, it must be the case that science without numbers isn’t very helpful for making predictions. But is that true? To get back to Alice’s question at the end of our story – do we really need numbers?