Frege and Husserl on numbers

So a couple of weeks ago I bailed out of my series of posts on Derrida's Speech and Phenomena because I got too lost in Chapter 4. I found a secondary source, Vernon Cisney's Derrida's 'Voice and Phenomenon', which has been very helpful. The first thing I learned is that I should probably be calling the book Voice and Phenomenon instead of Speech and Phenomena, as that seems to be the favoured translation now, so I'll do that in future. More importantly I learned some useful backstory from the 'Historical context' section.

One bit I found interesting was an argument between Frege and Husserl on the concept of number. Frege had this concept of 'equinumerosity' where two set of things had the same number of elements if there was a one-to-one map from elements of one to elements of the other. So five pennies can be mapped to five cats can be mapped to five buckets. Then the number 'five' is a 'second-order concept' that generalises all these examples of 'five cats', 'five buckets' etc.

Husserl says that fine, this explains how we know there are the same number of things but not how we know that that number is five and not six, and this other part is the core of the concept of number. In Philosophy of Arithmetic he looks at the intuitive processes behind this more fundamental process. For a start, you need to be able to identify a bunch of objects as 'the same' in some way, so that you count all the cats together separately from the pennies. There's also some sort of concept of aggregation, where five pennies is one penny and another penny and another penny and another penny and another penny.

Husserl then has this distinction between 'authentic presentations', which are immediately present to the subject, and 'inauthentic presentations', which are given only by way of indirect symbolic meaning. According to Husserl we can have an authentic presentation of 'five' because we can directly see five pennies as five pennies. We don't have an authentic presentation of 'eighty-seven', because we can't directly see 87 pennies as 87 pennies. So for numbers like 87 we'd have to rely on symbolic reasoning. This looks like it'll be important for thinking about expressive vs. indicative meaning.

Where's the cut-off point between five and eighty-seven? Twelve, apparently. This is mentioned completely in passing by Cisney but apparently twelve is authentic and thirteen is inauthentic. So now we know.

One nice feature of this theory is that explains why thirteen is unlucky: it's the first inauthentic number. I'm getting facetious so I'll stop here.