Aristotelian realists believe in mathematics because they see it as describing abstract realities that exist independently of human thought and activity.
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Aristotelian realists believe in the reality of mathematics because they see it not as a human invention but as describing abstract realities that exist independently of human thought and activity. According to them, mathematical objects such as numbers, geometric shapes, and algebraic symbols have objective, mind-independent existence. This view is often summarized as the idea that mathematics is discovered rather than invented.
As philosopher Michael E. Marmura explains, “Mathematics, for the Aristotelian realist, describes and explains a world of abstract entities, a world of being that is independent of the human mind and created order.” This belief in the objective reality of mathematics is related to the Aristotelian notion of substance, which holds that objects have specific properties that are inherent to them.
Aristotelian realists also argue that mathematical objects and concepts have a certain level of necessity and universality. As Aristotle states, “As for the mathematical sciences, they studied not what occurs and passes away, but forms and separate substances, which were to be considered not only in the sphere of Being, but also in that of truth and opinion.”
Moreover, Aristotelian realists view mathematics as being intimately connected to the natural world. They see mathematics as providing a language and a set of tools for understanding and describing the patterns and regularities found in nature.
Some interesting facts about Aristotelian realism and mathematics include:
- The Aristotelian view of mathematics as discovered rather than invented is in contrast to the more modern view of mathematics as a human invention or construction.
- Aristotelian realism has a long history and can be traced back to classical Greek philosophy. Aristotle himself wrote extensively on the subject of mathematics and its relation to the natural world.
- Aristotelian realism has been influential in the development of many areas of mathematics, including geometry and calculus.
- The idea that mathematical objects have objective existence outside of the human mind has been the subject of much debate and controversy among mathematicians and philosophers over the centuries.
In conclusion, Aristotelian realists believe that mathematics is a real and objective part of the natural world that exists independently of human thought and activity. This belief is based on the idea that mathematical objects have a necessary and universal quality that is inherent to them. Ultimately, this view of mathematics as a discovery rather than an invention has important implications for our understanding of the relationship between mathematics and the physical world.
| Aristotelian Realism and Mathematics |
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| Believe in mathematics because they see it as describing abstract realities that exist independently of human thought and activity |
| Mathematical objects have objective, mind-independent existence |
| Mathematics is discovered rather than invented |
| Mathematical objects and concepts have a certain level of necessity and universality |
| Mathematics provides a language and a set of tools for understanding and describing the patterns and regularities found in nature |
See related 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.
I discovered more solutions online
Aristotelians believe that mathematics is just another science. Mathematics investigates the real world to learn about mathematical universals such as symmetry or ratio. These universals are mathematical properties and relations that are as real and physical as any other scientific universal.
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…