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In *The Foundations of Arithmetic: A Logico-Mathematical Enquiry Into the Concept of Number" by G. Frege pages XV and XVi we read:

A typical crudity confronts me, when I find calculation described as "aggregative mechanical thought". I doubt whether there exists any thought whatsoever answering to this description. An aggregative imagination, even, might sooner be let pass; but that has no relevance to calculation.

The present work will make it clear that even an inference like that from n to n + i. which on the face of it is peculiar to mathematics, is based on the general laws of logic, and that there is no need of special laws for aggregative thought.

What is the exact meaning of "aggregative"? I searched Google Books and the internet but there is not any obvious explanation of it.

Frege says in the footnote that Kuno Fischer in his System der Logik und Metaphysik: oder Wissenschaftslehre section 94 describe calculation as "aggregative mechanical thought" but I dont know German and cannot read the source. Would you please help me grasp it?

  • Dedekind in "The nature and meaning of numbers"(in Essays on the theory of numbers) mentioned that: If we scrutinise closely what is done in counting an aggregate or number of things, we are led to consider the ability of the mind to relate things to things, to let a thing correspond to a thing, or to represent a thing by a thing, an ability without which no thinking is possible. – Root Lopht Apr 2 '13 at 9:55
  • Is there any relevance to the Context Principle and the way Frege developed the Classical logic against traditional i.e. first considering the meaning of whole and then find, extract the use and meaning and so on of every parts from the whole against the method of thinking and discussing about e.g. subjects and predicates separately and then put them together to form a proposition or an thought? I also ask this question on AskPhilosophers [link] and hope to see some comments from Prof. Heck (askphilosophers.org/question/5107) – Root Lopht Apr 9 '13 at 4:18
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an opportunity to get out my battered old copy of Grundlagen, woo.

So the introduction is where Frege is establishing the point of the book: when you're writing an entire book to establish what people think they already know, you need to make a good case that they're wrong. So that's what he does; he takes on the current state of philosophical thinking about mathematics and bitchily tears it up.

"Aggregative" means "put together", basically. "Aggregative mechanical thought" is Fischer's idea that numbers are what happen when we start from 1 and just keep adding things together in our head. Frege points out that this is nonsense; thoughts aren't things that can be put together, and numbers don't work by some special kind of reasoning of their own but are just features of the same reasoning we use for everything else.

If you're reading to learn about Frege's ideas, though, I wouldn't worry too much about the specifics of this; he moves on to jucier targets (gets really bitchy about Mill on page 9) and actually starts to talk about his own ideas eventually. I had a look at the Fischer (it had never occurred to me to just Google it before), but my German isn't quite up to translating it properly; basically he's talking about calculation just being counting number in a mechanical way.

  • thanks a lot, nowadays I see similar arguments like Fischer's when we ask about numbers in mathematics community and I want to know more about the ideas Frege were about to answer which I can see around even after a century. – Root Lopht Apr 30 '13 at 4:10

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