Changes

*'''Countablism''': countablists reject the existence of uncountable mathematical objects; only countable mathematical objects really exist. Aleph-null is the largest cardinal. Most commonly this is a consequence of a constructivist or intutistionist position

*'''Countablism''': countablists reject the existence of uncountable mathematical objects; only countable mathematical objects really exist. Aleph-null is the largest cardinal. Most commonly this is a consequence of a constructivist or intutistionist position

*'''Finitism''': finitists reject the existence of infinite mathematical objects; only finite mathematical objects really exist.

*'''Finitism''': finitists reject the existence of infinite mathematical objects; only finite mathematical objects really exist.

*'''Ultrafinitism''': ultrafinitists go beyond finitism, to reject the existence of mathematical objects which while finite are so large as to be impossible for any human being to think about. According to classical mathematics, there are infinitely many natural numbers, almost all of whom are so large that no mathematician will ever think of or refer to them individually. Almost all of them are so large that there is not enough space in the universe to even refer to them in any concievable notation. Ultrafinitists deny that such finite numbers exist.

*'''Ultrafinitism''': ultrafinitists go beyond finitism, to reject the existence of mathematical objects which while finite are so large as to be impossible for any human being to think about. According to classical mathematics, there are infinitely many natural numbers, almost all of whom are so large that no mathematician will ever think of or refer to them individually. Almost all of them are so large that there is not enough space in the universe to even refer to them in any conceivable notation. Ultrafinitists deny that such finite numbers exist.

*'''[[Paraconsistent logic|Paraconsistency]]''': paraconsistent logic and mathematics rejects the principle that every proposition must be either true or false, but not both -- it posits that there are propositions that are both true and false simultaneously. Thus, for example, there may be some set, which both does and does not contain a given element. (This is proposed as an alternative solution to paradoxes such as [[Russell's paradox]] and the [[liar paradox]].)

*'''[[Paraconsistent logic|Paraconsistency]]''': paraconsistent logic and mathematics rejects the principle that every proposition must be either true or false, but not both -- it posits that there are propositions that are both true and false simultaneously. Thus, for example, there may be some set, which both does and does not contain a given element. (This is proposed as an alternative solution to paradoxes such as [[Russell's paradox]] and the [[liar paradox]].)

Constructivism, finitism and ultrafinitism generally entail a rejection of much of classical mathematics; so, they tend not to be popular with practicising mathematicians. However, there are alternative systems of mathematics which have been worked out based on their assumptions -- in particular, constructivist or intuitionistic mathematics, is a significant and continuing area of mathematical research. Some of its formulations (e.g. intuitionistic logics) have been found useful by those who don't necessarily agree with the philosophical theories which motivated their development.

Constructivism, finitism and ultrafinitism generally entail a rejection of much of classical mathematics; so, they tend not to be popular with practicising mathematicians. However, there are alternative systems of mathematics which have been worked out based on their assumptions -- in particular, constructivist or intuitionistic mathematics, is a significant and continuing area of mathematical research. Some of its formulations (e.g. intuitionistic logics) have been found useful by those who don't necessarily agree with the philosophical theories which motivated their development.

[[Category:Philosophy]][[Category:Mathematics]]

[[Category:Philosophy]][[Category:Mathematics]]