Logicism

Introduction:

Logicism is a philosophical theory about the status of mathematical truths, to wit, that they are logically necessary or analytic.
What it is
Just the claim that the theorems of mathematics are logically necessary or analytic.
Some Original Readings
The primary sources of the logicist doctrine are Frege and Russell.
What it isn't
Not a claim about formalisability, either in principle or in any specific logical system. Not a claim about how mathematical truths are discovered or applied. Not concerned with any aspect of mathematics other than the theorems that mathematicians seek to demonstrate.
Arguments Against Logicism
It is generally understood that set theory is required to do modern mathematics. The existence of sets is, however, not logically necessary. The truths of mathematics are therefore contingent.
Arguments For Logicism
The truths of mathematics are the same in all possible worlds, since they do not depend upon the existence of sets, just upon the consistency of the supposition that the required sets exist. Since true in every possible world, mathematics must be logically necessary.

What it is:

Logicism is just the claim that the theorems of mathematics are logically necessary or analytic.
Logicism is the view that Mathematics is analytic and/or that Mathematics is reducible to or identical with Logic. The doctrine is most closely associated with Gottlöb Frege and Bertrand Russell. Though they may neither of them have used the term logicism, it has become firmly associated with their shared view of the status of mathematical truths.
Gottlob Frege
"I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgements and consequently a priori."
Bertrand Russell
"The fundamental thesis of the following pages, that mathematics and logic are identical, is one which I have never since seen any reason to modify."
William Hatcher
"Logicism is generally defined as the philosophical thesis that the truths of mathematics are all analytic and thus derivable from pure logic..."

Readings:

The primary sources of the logicist doctrine are Gottlob Frege and Bertrand Russell. However, similar doctrines go back a long way. For example, David Hume held that the truths of mathematics are "truths of reason", and susceptible of demonstrative proof.
Gottlob Frege
For Frege's less formal writings on this topic (in English translation) see his The Foundations of Arithmetic.
Willard van Orman Quine
A rewarding additional source of insight into some of the technical aspects of this work may be found in Quine's Set Theory and its Logic.
William Hatcher
For a broad study see also The Logical Foundations of Mathematics by William Hatcher.
Bertrand Russell
For a lucid account intended for a wide readership see Russell's Introduction to Mathematical Philosophy. For greater detail on Russell's position see The Principles of Mathematics, of which the first edition was published while Russell was still struggling for a satisfactory way of avoiding inconsistency in his subsequent masterwork with Whitehead Principia Mathematica. The traces are not covered over in the second edition, which therefore gives a sense of history in the making.

What it isn't:

  • Not about formalisability.
  • Not about mathematical discovery.
  • Only about mathematical theory.
  • Arguments Against Logicism:

    Modern mathematics is founded in set theory. The existence of sets is not logically necessary. The truths of mathematics are therefore contingent.
    Logical Systems
    Specific criticisms of the logical systems devised by Frege and Russell are now more of historical interest than in arguing against Logicism as a contemporary doctrine.

    Putting aside the defects of these particular systems the most telling argument which remains against logicism is probably that the ontological premises necessary to carry through the development of mathematics cannot be shown to be principles of logic.

    Ontology
    Modern mathematics is conducted in a rich ontological framework (such as is found in classical first order set theory), yet no convincing arguments have ever been presented to show that the existence of this elaborate hierarchy of sets (or any other adequate ontology) is logically necessary.
    Consistency of Finite Universe
    It is consistent to add to first order logic an axiom asserting that there is only one individual, and it cannot therefore be logically necessary that there are more than one.
    Gödel's results
    Gödel's incompleteness results have also impacted on the credibility of Logicism. Gödel's first incompleteness theorem shows that, however well patched up, neither Russell's Theory of Types nor any other formal logical system is capable of proving all true facts of elementary arithmetic.

    Arguments for Logicism:

    The truths of mathematics are the same in all possible worlds, since they do not depend upon the existence of sets, just upon the consistency of the supposition that the required sets exists. Since true in every possible world, mathematics must be logically necessary.
    alternatives
    Notwithstanding the arguments against logicism, no alternative account of the status of mathematical propositions is convincing.
    tradition
    Philosophers throughout history have claimed that propositions such as:
    1+1=2
    are necessarily true.
    This necessity flows from the meaning of the terms employed in the proposition.
    ontology presupposed, not asserted
    The proposition:
    Pegasus has wings
    might be considered analytic on the grounds that it presupposes (at worst) rather than asserts the existence of Pegasus, and Pegasus is by definition a winged horse.

    In a similar way we may claim that the existence of natural numbers is presupposed rather than asserted by the theory of arithmetic. Arithmetic conjectures are about the natural numbers and we are able to determine their truth or falsity in many cases, on the basis of knowledge of the intended structure of the number system which is independent of settling in any absolute sense whether the natural numbers "exist".

    mathematical methods a priori
    To the present day Mathematicians continue to work with methods which are appropriate only to the establishment of a priori, necessary truths. They regard an appeal to experimental evidence as irrelevant and inadmissible (notwithstanding their willingness to permit digital computers to contribute to a mathematical proof).
    methods not disputed
    Those who dispute the logicist thesis do not propose that mathematicians should change to experimental methods. At worst the concession that mathematical truths should be considered hypothetical might be forced. This is the position adopted by Russell when he concluded that the axiom of infinity was contingent. It seems however more plausible to describe the necessary premises as presuppositions rather than as hypotheses.
    scope of necessity
    A defence of Logicism can be founded on an argument about the extent of logical necessity, which is to some extent a matter of arguing the merits of a broader conception of logical necessity than is at present widely accepted. An alternative defence can be based on a formulation of the thesis in terms of analyticity, which need not depend on any "reduction to logic". The logical positivist A.J.Ayer adopted this latter approach when speaking of the analyticity of mathematics in Language Truth and Logic.


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