Oxford University Press UK Authors University of Stirling Abstract Bob Hale and Crispin Wright draw together here the key writings in which they have worked out their distinctive neo-Fregean approach to the philosophy of mathematics.
Mathematical objects are independent of intelligent agents and their language, thought, and practices. The first two claims are tolerably clear for present purposes. Abstractness says that every mathematical object is abstract, where an object is said to be abstract just in case it is non-spatiotemporal and therefore causally inefficacious.
For further discussion, see the entry on abstract objects. Independence is less clear than the other two claims. What does it mean to ascribe this sort of independence to an object?
The most obvious gloss is probably the counterfactual conditional that, had there not been any intelligent agents, or had their language, thought, or practices been different, there would still have been mathematical objects. However, it is doubtful that this gloss will do all the work that Independence is supposed to do see Section 4.
For now, Independence will be left somewhat schematic. Many older characterizations of platonism add strong epistemological claims to the effect that we have some immediate grasp of, or insight into, the realm of abstract objects. Many philosophers who defend platonism in this purely metaphysical sense would reject the additional epistemological claims.
Examples include Quine and other philosophers attracted to the so-called indispensability argument, which seeks to give a broadly empirical defense of mathematical platonism. See the entry on indispensability arguments in the philosophy of mathematics.
Again, this exclusion is justified by the fact that some philosophers who are generally regarded as platonists for instance, Quine and some adherents of the aforementioned indispensability argument reject this additional modal claim.
If the view is true, it will put great pressure on the physicalist idea that reality is exhausted by the physical. For there is little doubt that we possess mathematical knowledge.
The truth of mathematical platonism would therefore establish that we have knowledge of abstract and thus causally inefficacious objects. This would be an important discovery, which many naturalistic theories of knowledge would struggle to accommodate.
Although these philosophical consequences are not unique to mathematical platonism, this particular form of platonism is unusually well suited to support such consequences. For mathematics is a remarkably successful discipline, both in its own right and as a tool for other sciences.
So if philosophical analysis revealed mathematics to have some strange and surprising consequences, it would be unattractive simply to reject mathematics. For instance, when theology turns out to have some strange and surprising philosophical consequences, many philosophers do not hesitate to reject the relevant parts of theology.
Object realism is thus just the conjunction of Existence and Abstractness. Because object realism leaves out Independence, this view is logically weaker than mathematical platonism. The philosophical consequences of object realism are thus not as strong as those of platonism.
Many physicalists would accept non-physical objects provided that these are dependent on or reducible to physical objects. They may for instance accept objects such as corporations, laws, and poems, provided that these are suitably dependent or reducible to physical objects.
Some views in the philosophy of mathematics are object realist without being platonist. One example are traditional intuitionist views, which affirm the existence of mathematical objects but maintain that these objects depend on or are constituted by mathematicians and their activities.
The view also holds that most mathematical statements that are deemed to be true are in fact true. So truth-value realism is clearly a metaphysical view.
But unlike platonism it is not an ontological view. For although truth-value realism claims that mathematical statements have unique and objective truth-values, it is not committed to the distinctively platonist idea that these truth-values are to be explained in terms of an ontology of mathematical objects.
Mathematical platonism clearly motivates truth-value realism by providing an account of how mathematical statements get their truth-values.
But the former view does not entail the latter unless further premises are added. For even if there are mathematical objects, referential and quantificational indeterminacy may deprive mathematical statements of a unique and objective truth-value.In particular, he published three major books: Abstract Objects (Blackwells ), The Reason's Proper Study: Essays Towards a Neo-Fregean Philosophy of Mathematics (OUP , jointly written with Crispin Wright) and Necessary Beings: An Essay on Ontology, Modality, and The Relations Between Them (Oxford ).
|mathematics and statistics online||The challenge takes the following form. There exist infinitely many ways of identifying the natural numbers with pure sets.|
These works – . Download ebook The Reasons Proper Study Essays Towards A Neo Fregean Philosophy Of Mathematics, Where to get access file The Reasons Proper Study Essays Towards A Neo Fregean Philosophy Of Mathematics Online, Library of book - The Reasons Proper Study Essays Towards A Neo Fregean Philosophy Of Mathematics Pdf, Easy get access pdf The Reasons.
Bob Hale, emeritus professor of philosophy at the University of Sheffield, died last webkandii.com was 72 years old. Professor Hale was known for his work in philosophy of mathematics, logic, and metaphysics.
Prior to Sheffield, he taught at the University of Glasgow, the University of St. Andrews, and the University of Lancaster. Towards a multi- & meta-cultural Postmodern Philosophy.
Prelude. Muslim scholarship Regarding the many historical influences determining the outbreak of the Renaissance, the earliest phase of humanistic modernism, at the end of Medieval Europe, one should not (as was & is usually done by Europacentrists to suggest the originality of modernism) underestimate the major role played by the masters.
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Oxford: Oxford University Press. () "Necessary Beings: An Essay on Ontology, Modality, and the .