What do logicists mean when they try to "reduce mathematics to logic"?

Question

I've read a lot about Russell and other Logicism advocates and their trial to reduce math to logic. But what does that mean? We know that all known mathematics can be reduced to Set theory, is that their goal? If not, what was their goal precisely?

I was told that their goal is to build mathematics using first order logic without any use of axioms. This seems weird, how can this be achieved?

I've studied mathematics and logic very well, so feel free to use whatever level of sophistication without an appeal to over-simplifying things for me to understand.

The assumption of Logicism was that in order to derive math from logic, some additional concepts and axioms of "pure logical" nature, like class and the Comprehension Principle would be enough. This was not so; in order to complete the program, W&R in Principia Mathematica needed the Axiom of Infinity the Reducibility Axiom, ... 1/2 — Mauro ALLEGRANZA, Jul 23 '16 at 07:00
as well as the Axiom of Choice, whose "logcal nature" was debated and not at all evident. — Mauro ALLEGRANZA, Jul 23 '16 at 07:00
Logicism dates back to Frege works (1884 and 1902), followed by Russell's Principles (1902) and W&R's Principia (190-13), while Zermelo's first axiomatization was 1908 and then 1930. Thus, the (today) "known reduction of mathematics to set theory" was not so clear at that times. — Mauro ALLEGRANZA, Jul 23 '16 at 08:20
History and the philosophical context of logicism are discussed here http://philosophy.stackexchange.com/questions/34074/what-is-the-philosophical-ground-for-distinguishing-logic-and-mathematics/34077#34077 — Conifold, Jul 25 '16 at 21:16

score 2 · Answer 1 · edited Jun 17 '20 at 08:34

From Wikipedia:

The overt intent of Logicism is to reduce all of philosophy to symbolic logic (Russell), and/or to reduce all of mathematics to symbolic logic (Frege, Dedekind, Peano, Russell). As contrasted with algebraic logic (Boolean logic) that employs arithmetic concepts, symbolic logic begins with a very reduced set of marks (non-arithmetic symbols), a (very-)few "logical" axioms that embody the three "laws of thought," and a couple of construction rules that dictate how the marks are to be assembled and manipulated—substitution and modus ponens (inference of the true from the true). Logicism also adopts from Frege's groundwork the reduction of natural language statements from "subject|predicate" into either propositional "atoms" or the "argument|function" of "generalization"—the notions "all," "some," "class" (collection, aggregate) and "relation."

As perhaps its core tenet, logicism forbids any "intuition" of number to sneak in either as an axiom or by accident. The goal is to derive all of mathematics, starting with the counting numbers and then the irrational numbers, from the "laws of thought" alone, without any tacit (hidden) assumptions of "before" and "after" or "less" and "more" or to the point: "successor" and "predecessor." Gödel 1944 summarized Russell's logicistic "constructions," when compared to "constructions" in the foundational systems of Intuitionism and Formalism ("the Hilbert School") as follows: "Both of these schools base their constructions on a mathematical intuition whose avoidance is exactly one of the principal aims of Russell's constructivism" (Gödel 1944 in Collected Works 1990:119).

I think whoever told you that the goal was to not use axioms meant that the goal was to avoid "numerical" or "algebraic" axioms.

What do logicists mean when they try to "reduce mathematics to logic"?

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