The realist view of mathematics is that mathematical objects exist independently of human minds and are discovered rather than invented.
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The realist view of mathematics is a philosophical stance that posits the existence of mathematical objects independent of human cognition. This view holds that mathematical entities are discovered rather than invented, and that they exist whether or not we are aware of them or have any use for them. Mathematicians who subscribe to this view believe that their discipline is not merely a tool for describing the physical world, but also a means of apprehending deep truths about reality itself.
One of the most famous proponents of mathematical realism was the philosopher and logician Bertrand Russell. In his seminal work “Principles of Mathematics,” he argued that mathematics discovers truths about a realm of abstract objects that are just as real as the physical world. According to Russell, “Mathematics, rightly viewed, possesses not only truth, but supreme beauty” because it reveals the fundamental structures of reality.
Another notable realist was Kurt Gödel, who is famous for his incompleteness theorems. He argued that mathematical truths exist independently of human thought, and that there are some truths that cannot be proven within any formal system. This idea challenged the prevailing view of mathematics as a purely logical system that could be fully understood and mastered by human beings.
Despite its intuitive appeal, mathematical realism has been challenged by a number of philosophers who argue that mathematical entities are simply products of human thought. These anti-realists argue that mathematical concepts are invented rather than discovered, and that they are useful for describing the world precisely because they are constructed by human minds.
Ultimately, the debate between realism and anti-realism in mathematics is far from settled, and the field continues to be energized by new insights and discoveries. Whether or not mathematical entities exist independently of our minds may be a philosophical question, but it is one that has serious implications for the way we understand the nature of reality itself.
Table comparing Realism and Anti-Realism in mathematics:
| Realism | Anti-Realism |
|---|---|
| Believes in the existence of mathematical entities independent of human thought. | Believes that mathematical concepts are invented rather than discovered. |
| Posits that mathematics is a means of apprehending deep truths about reality itself. | Argues that mathematical concepts are useful for describing the world only because they are constructed by human minds. |
| Champions the idea that mathematical objects have supreme beauty because they reveal fundamental structures of reality. | Challenges the view of mathematics as a purely logical system that can be fully understood and mastered by human beings. |
Answer in the video
This video discusses the debate between those who believe that mathematics is discovered, and those who believe that it is invented. The video provides examples of how mathematics has been used to solve problems in the real world.
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Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities.
Mathematical realism is the view that the truths of mathematics are objective and true independently of any human activities, beliefs or capacities. Mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create. In this view, mathematics is objective and independent of mathematicians. Mathematical realism is a necessary condition of science.
Mathematical realism is the view that the truths of mathematics are objective, which is to say that they are true independently of any human activities, beliefs or capacities. As the realist sees it, mathematics is the study of a body of necessary and unchanging facts, which it is the mathematician’s task to discover, not to create.
In mathematics, realism is a way of looking at the world from the outside. In this view, mathematics is objective and independent of mathematicians. Hence, mathematical realism is a necessary condition of science. It is impossible to create a mathematical object, without a world.
Yes there is. Maddy takes this position in Perception and Mathematical Intuition, and so do recent mathematical Aristotelians. The alternative of believing it all real usually comes with full blown Platonism (forms in a separate realm), and is not very popular, see however Brown’s Platonism, Naturalism, and Mathematical Knowledge for a modern defense. Let me add however that many modern realists would pragmatically admit fictional mathematical entities beyond merely constructible ones, like inaccessible cardinals or Lebesgue non-measurable sets, although Aristotle might have been more conservative.
Here is a long quote from Franklin’s Aristotelian Realism, which I give in full because it seems to address the questions directly:
“The thesis defended has been that some necessary mathematical statements
refer directly to reality. The stronger thesis that all mathematical truths refer to reality seems too strong… Statements about negative numbers can refer to reality in some way, since…
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