958; italics in the original.) (7) Consequently, mathematical statements
have to be interpreted in a non-realist way:
Equations, statistics, and other kinds of mathematical statements
are fundamentally inapposite to human experience and cannot in any way assert the truths of human experience.
The Gordon hypothesis is that complex mathematical statements
are less likely to be operational relative to other economic statements: We offer evidence on this proposition.
Vector graphics is the creation of digital images through a sequence of draw commands or mathematical statements
that place lines and shapes in a given two-dimensional or three-dimensional space.
He devised a way to construct self referencing mathematical statements
on the order of: This [insert a Godel type statement] cannot be proven to be true within this mathematical system.
It is certainly feasible to replicate this experiment with similar types of open-ended problems that call for establishing the truth or falsity of mathematical statements
For example, compared to all of the other countries, a greater percentage of mathematics problems per lesson in Japan involved proving or verifying mathematical statements
, and a smaller percentage of mathematics problems per lesson were repetitions of previous problems.
Is it possible to measure the intuitive acceptance of mathematical statements
? Educational Studies in Mathematics, 12, 491-512.
Benacerraf (1973/1989) poses the following dilemma: On the one hand, realism provides the most straightforward interpretation of mathematical statements
; on the other hand, a realist interpretation of mathematics seems to preclude the possibility of mathematical knowledge.
There is a great deal of similarity between Resnik's and Shapiro's views, so many of the above questions are answered along similar lines: both Resnik and Shapiro are mathematical realists - they both believe that mathematical entities exist independently of our knowledge of them and that mathematical statements
are mind-independently true or false; they both agree that structuralism and realism are independent of one another; and they both provide essentially the same account of sameness of structure.
I framed my introductory remarks in terms of the existence of mathematical entities; but it is sometimes convenient to restate the issues in terms of the truth of mathematical statements
. As long as certain presuppositions are made, the formulation in terms of mathematical truth (which I will give below) will be equivalent to the one in terms of mathematical existence.
The argument states that mathematical statements
are indispensable components of our best scientific theories, that our best scientific theories are highly confirmed by observations, and hence that the mathematical statements
are worthy of our beliefs.