Mathematical Platonism is the view which holds that mathematical objects exist independently of us and the things we do; how we think, how we speak, how we behave. It is one of the oldest and most influential attempts to give an account of the metaphysics of mathematics.
In this article, a basic definition of Mathematical Platonism is given and explained. We will explore Platonism – not just concerning mathematics, but in general – and the relationship between Platonism and the work of Plato will be clarified. Then we will take a look at Frege’s argument for Mathematical Platonism, which is widely understood to be the most influential such argument. Finally, we will also consider several objections to Mathematical Platonism.
The Basic Claim of Mathematical Platonism
Mathematical Platonism holds that mathematical objects exist independent of human activity, thought and language. Therefore, we might say that it is the view that mathematical objects are discovered or found, rather than constructed or made by human beings.
It is worth clarifying that ‘mathematical object’ doesn’t refer to anything technical or complicated; it refers to anything that can be defined in mathematical terms. This sounds like quite a simple view, but that is deceptive. For one thing, what it is to discover something, rather than create it, is far from straightforward.
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There may not be a strict distinction between discovering something and creating it ourselves. This is especially true when dealing with conceptual objects, such as the ones mathematicians are largely concerned with. Usually we are only confident in saying that a physical object would exist without human intervention (although even that is highly contentious).
The Metaphysics of Mathematics
Mathematical Platonism is also complicated by the fact that it is a metaphysical doctrine. What exactly counts as metaphysics is certainly contentious, but broadly speaking metaphysics is concerned with how things really are rather than, for instance, what the conditions of knowledge are; that would be the concern of epistemology instead.
So Mathematical Platonism is purportedly concerned with what mathematical objects really are, rather than how we come to know them. Yet it seems strange to divorce the question of what mathematical objects are from how we know them, and so historically the metaphysical view expressed by Mathematical Platonism has been attached to claims about direct or immediate knowledge of mathematical objects.
This isn’t the only epistemological move available; W.V.O Quine, one of the most influential analytic philosophers of the 20th century and an ardent Mathematical Platonist, held an empirical view of mathematics, meaning that our knowledge of it is accumulated through experience rather than directly. Clearly, any full account of what we are doing when we are doing mathematics will have to go beyond the purely metaphysical view, but there isn’t space here to do so here.
Consequences of Mathematical Platonism
There are a range of consequences of adopting Mathematical Platonism to keep in mind, but foremost amongst them are the implications for a physicalist view of the world. Physicalism, at its most basic, holds that the world can be explained solely in terms of physical facts about it. Mathematical Platonism is the view that mathematical objects are real, and given that mathematical objects are conceptual rather than physical, this would seem to imply that non-physical facts are a part of a full explanation of reality.
Mathematical Platonism is particularly powerful, because mathematics has a strong claim to be the most consistent, most scientific and the most secure field of knowledge production. Lastly, it’s worth clarifying that, although it bears his name, Mathematical Platonism doesn’t have much to do with what Plato actually said and thought about mathematics. Rather, ‘Platonism’ – which can apply to things other than mathematics – is simply the view that a certain thing exists independent of us. Plato was a ‘Platonist’ in this sense, but ‘Platonism’ is applied in ways and to things which Plato would not necessarily have done so himself.
Frege’s Argument for Mathematical Platonism
Perhaps the most discussed argument for Mathematical Platonism comes not from Plato himself, but from Gottlob Frege. Frege remains one of the most influential mathematicians, logicians and philosophers of language today, and his theory of Mathematical Platonism is similarly influential.
Frege’s argument hinges on two premises. First, a defense of what is known as ‘Classical Semantics’: namely, that “the singular terms of the language of mathematics purport to refer to mathematical objects, and its first-order quantifiers purport to range over such objects.” Second, a claim about the truth of mathematics: “Most sentences accepted as mathematical theorems are true (regardless of their syntactic and semantic structure).”
The former claim requires a working understanding of what mathematics actually does, which Frege clearly possessed, and amounts to the claim that the languages of mathematics really are languages, and the components of these languages make sense in roughly the same way as natural languages.
All that can be said in defense of this claim for now is that many mathematicians find this view of mathematics plausible; it wasn’t just a quirk of Frege’s to draw this similarly between mathematical languages and natural languages (meaning the languages we normally speak, like English or Italian). There are, of course, a range of arguments for the second premise, but for our purposes we will assume it; most philosophers do, although they disagree about why.
Ontological Commitment
The Fregean argument is normally understood to move from these two premises to the truth of Mathematical Platonism via a concept called ‘Ontological Commitment’. There are a number of ways of understanding ‘Ontological Commitment’, but the crux of the matter is that a sentence is ontologically committed to those objects which must be assumed in order for the sentence to be true.
When we claim that mathematical sentences are committed to independently existing mathematical objects in this way, we seem to be presented with a choice. Either we deny the premise about the truth of mathematical sentences (which few philosophers want to do), or we accept the existence of mathematical objects and therefore the truth of Mathematical Platonism.
The Independence of Mathematics
We have not yet touched on what it would mean for mathematical objects to be independent. Philosophers commonly understand this claim in a counterfactual way; that is, they understand it by way of asking what would be different about mathematics if intelligent creatures had never existed.
It isn’t hard to see why, when posed in this counterfactual way, it’s difficult to deny the independence condition. For one thing, there is a basic implausibility to the view that the truths of mathematics would be totally different if human beings happened not to evolve the intelligence required to discern them. Equally, mathematicians often reason towards the hypothetical (that is, not actual). E.N Zalta puts the point this way: “Since the truths of pure mathematics can freely be appealed to throughout our counterfactual reasoning, it follows that these truths are counterfactually independent of us humans, and all other intelligent life for that matter.”
Objections to Mathematical Platonism
As it has been presented so far, Mathematical Platonism might seem quite intuitive. To the non-mathematician, the idea that mathematical objects would cease to exist if human beings had never existed seems strange. However, there are many objections to Mathematical Platonism, some of which are merely attempts to soften it and some of which constitute an outright rejection.
Some of these objections are quite technical, and serve to complicate the problems presented by Mathematical Platonism for non-mathematicians. However, one objection concerns the thoroughly philosophical issue which was touched on at the beginning of this article: the problem of epistemology and its relation to metaphysics. The question becomes: how can we come to know mathematical objects?
One such objection comes from Paul Benacerraf. A brief version of this objection is as follows. If the conclusions drawn by mathematicians are reliable, and we should therefore be able to explain that reliability, Mathematical Platonism cannot hold.
This last premise seems to appear out of nowhere, but it is ordinarily ascribed to the fact that Mathematical Platonism posits mathematical entities as existing outside of space and time, and so they should be causally isolated from us. However, it is not clear that reliability must be defined in a causal way. In whatever sense we take the results of mathematicians to be reliable will be determined by our definition of what it is we take mathematicians to do in the first place. Our definition of reliability should follow from our definition of mathematics, not the other way around.
FAQs
Is math invented or discovered Plato? ›
Thirdly, since platonism ensures that mathematics is discovered rather than invented, there would be no need for mathematicians to restrict themselves to constructive methods and axioms, which establishes (iii).
Is mathematical platonism plausible? ›The central core of Frege's argument for arithmetic-object platonism continues to be taken to be plausible, if not correct, by most contemporary philosophers.
Did we create math or discover it? ›If the universe disappeared, there would be no mathematics in the same way that there would be no football, tennis, chess or any other set of rules with relational structures that we contrived. Mathematics is not discovered, it is invented.
What did Plato say about mathematics? ›Plato believes that the truths of mathematics are absolute, necessary truths. He believes that, in studying them, we shall be in a better position to know the absolute, necessary truths about what is good and right, and thus be in a better position to become good ourselves.
Who believed that math is discovered? ›The oldest written texts on mathematics are Egyptian papyruses. Since these are some of the oldest societies on Earth, it makes sense that they would have been the first to discover the basics of mathematics. More advanced mathematics can be traced to ancient Greece over 2,500 years ago.
Is Math just made up? ›2) Math is a human construct.
The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes.
Aristotle rejected Plato's theory of Forms but not the notion of form itself. For Aristotle, forms do not exist independently of things—every form is the form of some thing.
What is the goal of platonism? ›Platonist ethics is based on the Form of the Good. Virtue is knowledge, the recognition of the supreme form of the good. And, since in this cognition, the three parts of the soul, which are reason, spirit, and appetite, all have their share, we get the three virtues, Wisdom, Courage, and Moderation.
What is the concept of platonism? ›Platonism is the view that there exist such things as abstract objects — where an abstract object is an object that does not exist in space or time and which is therefore entirely non-physical and non-mental. Platonism in this sense is a contemporary view.
Who invented math first and why? ›The Sumerians were the first civilisation to have developed a counting system. It is a common belief amongst many scientists that some of the oldest and most basic mathematical functions, such as addition, subtraction, multiplication, and division have been used for over 4,000 years.
Is mathematics an invention or a discovery essay? ›
It is up to us to discover mathematics and its workings. Math is a human construct. The only reason mathematics is admirably suited describing the physical world is that we invented it to do just that. It is a product of the human mind and we make mathematics up as we go along to suit our purposes.
Who is Plato and what was his beliefs about math? ›Plato wrote The Republic in around 375 BC, so about 75 years before Euclid wrote The Elements. In this work Plato sets out his ideas about education. For this, he believes, one must study the five mathematical disciplines, namely arithmetic, plane geometry, solid geometry, astronomy, and harmonics.
What is the famous line of Plato? ›Here are some of Plato's most famous quotes: “Love is a serious mental disease.” “When the mind is thinking it is talking to itself.” “Human behavior flows from three main sources: desire, emotion and knowledge.”
What was Plato's main theory? ›The theory of Forms or theory of Ideas is a philosophical theory, concept, or world-view, attributed to Plato, that the physical world is not as real or true as timeless, absolute, unchangeable ideas.
Who proved math was incomplete? ›In 1931, the Austrian logician Kurt Gödel published his incompleteness theorem, a result widely considered one of the greatest intellectual achievements of modern times. The theorem states that in any reasonable mathematical system there will always be true statements that cannot be proved.
How was math first discovered? ›...
Table of Contents.
| 1. | Who is the Father of Mathematics? |
|---|---|
| 6. | Conclusion |
| 7. | FAQs |
| 8. | External References |
The oldest mathematical texts from Mesopotamia and Egypt are from 2000 to 1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical concept after basic arithmetic and geometry.
Is math a man made or natural? ›To put it more bluntly, mathematics exists independent of humans -- that it was here before we evolved and will continue on long after we're extinct. The opposing argument, therefore, is that math is a man-made tool -- an abstraction free of time and space that merely corresponds with the universe.
Are mathematicians made or born? ›Researchers have said that if one wants to be good at all types of math, they need to practice them all, and can't trust their innate natural talent to do most of the job for them.
Is math learned or inherited? ›Maths ability is known to be heritable. Several genes that play a role in brain development influence the ability to do maths also. A study published in the PLOS Biology journal identified genetic variations and brain regions that affect maths ability.
Who is the most misunderstood philosopher? ›
Nietzsche, the most misunderstood philosopher of them all, was born today. On his birthday, a list of others whose ideas and words have been spectacularly misappropriated. Friedrich Nietzsche was born today in 1844.
What was Plato afraid of? ›Plato was afraid of the poet's ability to evoke passion in audiences, and afraid that passion can overrun reason, even in trained minds. Plato was afraid of the impact of representational force. Reality, according to Platonic theory, is comprehensible through a logical process.
What was Plato accused of? ›Plato's The Apology is an account of the speech Socrates makes at the trial in which he is charged with not recognizing the gods recognized by the state, inventing new deities, and corrupting the youth of Athens.
What are some examples of Platonism? ›Theory of Forms
These forms are immutable, timeless and changeless. Plato views these forms as the foundation of reality. For example, all mountains have a shared form but manifest in great variety in the physical world. According to Plato, to understand all mountains one simply needs to understand their one form.
Aristotelian realism
It contrasts with Platonism in holding that the objects of mathematics, such as numbers, do not exist in an "abstract" world but can be physically realized.
The first calculation of π was done by Archimedes of Syracuse (287–212 BC), one of the greatest mathematicians of the ancient world.
Who is called the father of mathematics? ›A Greek mathematician Archimedes is known as the Father of Mathematics. He is considered the Father of Mathematics for his significant contribution to the development of mathematics. Notable inventions of Archimedes are: The calculation of measurement of a circle.
Where is the father of math? ›He is believed to have lived around 570 BC – 495 BC. After him, we have Archimedes. Many claim him to be the 'father of mathematics'. His native place was Syracuse, a Greek territory.
What is the biggest invention in mathematics? ›- Trigonometry. ...
- Mathematical Discoveries in India – Quadratic Formula. ...
- Euler's Identity. ...
- Amazing Discovery of Mathematics – Fibonacci Numbers. ...
- Math Discoveries for Modern Computers – Binary Numbers. ...
- Greatest Discoveries in Mathematics – Euclid's Elements.
Albert Einstein said that compound interest is "the greatest mathematical discovery of all time." The concept of compounding is at the heart of investing and is especially vital when it comes to value investing.
Is mathematics an invention of human brain or is it innate in nature? ›
Mathematics is the language of science and its structures are innate to nature. Even if the universe were to disappear tomorrow, the eternal mathematical truths would still exist.
What are 3 important facts of Plato? ›He was a student of Socrates and later taught Aristotle. He founded the Academy, an academic program which many consider to be the first Western university. Plato wrote many philosophical texts—at least 25. He dedicated his life to learning and teaching and is hailed as one of the founders of Western philosophy.
What are the two famous quotes in Plato's Apology? ›- “For to fear death, my friends, is only to think ourselves wise without really being wise, for it is to think that we know what we do not know. ...
- “The difficulty, my friends, is not in avoiding death, but in avoiding unrighteousness; for that runs faster than death.”
Plato preferred truth as the highest value, stating that it could be found through reason and logic in discussion. He called this dialectic.
What did Plato invent? ›Plato invented a theory of vision involving three streams of light: one from the what is being seen, one from the eyes, and one from the illuminating source.
What did Plato discover? ›Plato's Theory of Forms: Plato believed that there exists an immaterial Universe of `forms', perfect aspects of everyday things such as a table, bird, and ideas/emotions, joy, action, etc. The objects and ideas in our material world are `shadows' of the forms (see Plato's Allegory of the Cave).
Who is the creator of first in math? ›Inventor of the First In Math Online program. Robert Sun, chairman, president and chief executive of Suntex International Inc., is an inventor, engineer and entrepreneur who holds numerous U.S. patents and several copyrights in the field of educational games.
What was Plato's biggest theory? ›He is best known for his theories of Forms, known as Platonism. In this philosophy, Plato rejected the materialism common to ancient philosophy in favor of metaphysics. He believed in the existence of an immaterial world of perfect objects and Forms (ideas).
What is Plato's most theory? ›It is most of all from Plato that we get the theory of Forms, according to which the world we know through the senses is only an imitation of the pure, eternal, and unchanging world of the Forms.
What is Plato's main theory? ›The theory of Forms or theory of Ideas is a philosophical theory, concept, or world-view, attributed to Plato, that the physical world is not as real or true as timeless, absolute, unchangeable ideas.
What is Plato's famous quote? ›
“The first and greatest victory is to conquer yourself; to be conquered by yourself is of all things most shameful and vile.”
What is Plato's most important contribution? ›His greatest work, The Republic, developed an insightful analogy between harmony in the state and harmony in the individual, and it is often considered one of the greatest works ever written. Plato wrote dialogues that considered the nature of virtue itself, as well as the nature of particular virtues.
What is the oldest form of math? ›The earliest form of mathematics that we know is counting, as our ancestors worked to keep track of how many of various things they had. The earliest evidence of counting we have is a prehistoric bone on which have been marked some tallies, which sometimes appear to be in groups of five.
What was math first called? ›The study of mathematics as a "demonstrative discipline" began in the 6th century BC with the Pythagoreans, who coined the term "mathematics" from the ancient Greek μάθημα (mathema), meaning "subject of instruction".
How was math born? ›Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started. In Babylonia mathematics developed from 2000 BC.
When was mathematics discovered? ›Evidence for more complex mathematics does not appear until around 3000 BC , when the Babylonians and Egyptians began using arithmetic, algebra, and geometry for taxation and other financial calculations, for building and construction, and for astronomy.