Roger Penrose - Is Mathematics Invented or Discovered? (Short Version)

Mathematics describes the real world of atoms and acorns, stars and stairs, with remarkable precision. So is mathematics invented by humans just like chisels and hammers and pieces of music?

[ameerhamza]

I'm here in comment section to see what the geniuses have to say

[Mikey-mike]

"One reason why mathematics enjoys such special esteem above all the other sciences is that its laws are absolutely certain and indisputable, whereas the laws of all the other sciences are to some extent debatable and always lie in constant danger of being overthrown by newly discovered facts.
In spite of this, investigators in the other departments of science would not need to envy the mathematician were the laws of mathematics only applicable to our imaginations and not to objects of reality."
Albert Einstein

[EPSTomcat]

I love how closer to truth asks the same question to different people. This here makes the most sense to me.

[BUTechTips]

congrats Sir Roger Penrose. You are really a great mathematician.

[jonathansaraco]

Mathematics is only "invented" insofar as the set of axioms we use and the notation is "invented". All properties of the system that are a result of those axioms are discovered.

[kamranamir]

Penrose is right here. The greatest philosopher of mathematics, Kurt Gödel, too said that mathematics has been discovered, not invented.

[JasperXoR]

Before I even watch I'm going to say the answer has to be invented, I bet it sure feels like discovered though. My reasoning, Just as Mathematics feels like a more pure language to describe the universe when compared to the written word, it is still very human relative. Math is still just the best human created tool to help describe the universe around us.

[thevisitor]

I’m beginning to suspect it’s a little of both
That this platonic reality of mathematical structures exists and our math systems are what we use TO discover and make sense of these thigns which I believe now to be the very “codes” of the reality we experience

[williamotoole]

I could listen to Prof Penrose, all day, every day. 😁

[Ruisu]

The key point behind objective reality is that it exists whether we describe it or not (or does it? That's another can of worms...), and Penrose touched on that in this video. Case in point; the universe didn't need a language invented by something borne of that universe to describe that universe, assuming that in order to exist, it isn't reliant on, say, a collective, subjective conscious to build an objective reality. For all intents and purposes, then, it was going to exist whether we were going to be in the picture or not. Contradictorily, the properties and behaviours that 'our invention' describes then exists with or without us, and _so in that_ sense, mathematics is a discovery. It really depends on what you're defining as 'mathematics'. If you readily define it as a type of objective language, then you're readily sitting on the invention fence. If you readily define it as the underlying properties of the universe, then you're sitting on the discovery fence. If you're doing both, then get a god-damned hobby, already.

Either way, it seems preferential, whether mathematics was invented or discovered. You say it was invented as an accurate way to describe reality, and you say it was discovered as a possible means of accurately describing reality, I say it was a good waste of several years of my school life in preparation for something Iw as never going to do. It doesn't matter - we can use it to describe things that our senses aren't going to. Mind over matter and all that shit.

[jacksonkerins]

I am a Senior in college coming close to finishing my undergrad as an Economics major, although sometimes I wish I had studied Physics or possibly been an Econ/Physics double major. I have a profound love for mathematics and the way it explains our natural universe. I often contemplated ideas like this when going through high school. Although I have enjoyed my studies thus far, I wish that I was pushed towards math more by my parents and instructors.

[abhayjaiswal]

Mathematics never be invented . It's the science on which all natural phenomenon are based. And it's never wrong to say that Mathematics is the base language in which Supreme God written his Mysterical universe🙂

[templesein]

A lot of people seem not to know enough about numbers to say that mathematics is "only" a language. Numbers have autonomous properties which are not only arithmetical and algebraic but geometrical also. Numbers are shapes and shape generators. Square numbers, for instance, are actual squares (a basic example which an astounding number of people are unaware of). Or all natural numbers are the side of an equilateral triangle. According to the pythagoreans "all things are number" - this is to be taken literally, but the mistake most people make is to project their own understanding of "number" onto this thesis and then consider it absurd. So, before you do this you have to ask what in reality is number.

Even though there are invariant structural elements of language, language does not possess the same autonomy as mathematics does. There's also a difference between mathematical reality itself and mathematical description, which may depend on human constructs. Most people don't even know the difference between digits, numerals and numbers. But the interesting fact is that the ancients didn't seem to make a rigid distinction between digit, numeral and number. To say that mathematics is just a language is an unjustified opinion.

I won't say that numbers are eternal ideas and archetypes, as it seems the pythagoreans and platonists have claimed, but the fact is that reality is mathematical. The dualism between "logic" and "being" is an artificial one (I am aware that to equate mathematics and logic is problematic, but we may suppose that mathematics has a logic of its own). The best would be to say that in our human constructs (mathematical description) we try to approach mathematical reality, NOT that mathematics is a tentative description of (illogical, unmathematical) reality.

[kpsudhakar]

Very interesting 'fact' - the Abstract is being explained in Mathematical precision... Listening to this, Mathematics is always there existing and We are just 'discovering'..

[pshehan]

Mathematics can be used to describe the physical world, but sometimes only to a certain level of accuracy.

Mathematics can also be entirely self consistent, but not describe the world, or any other world.

For example it was assumed for millenia that Euclid's geometry was the only geometry and described physical space. Then at the end of the 19th century, non Euclidean geometries were discovered which had different axioms and gave different theories. For example the interior angles of a triangle could be greater or less than 180 degrees.

So the question became, does Euclid's geometry describe the space in our universe? It seems that Euclid's "flat" space with internal angles of a triangle adding to 180 degrees is only an approximation on a 'small scale' but on cosmic dimensions space is curved, according to relativity. whther the curvature is 'positive' or negative is another matter, not determined.

So the question of whether the maths is 'there' waiting to be discovered or is an invention of the human mind is really one of choice according to philosopical inclination. Not being much of a mystic, I find the idea of a disembodied mathematics existing somewhere waiting to be discovered problematic.

I view it as an invention of the human mind.

[Tonyrg]

Mathematics describes relationships. The question thats really being asked is wether the relationships that we observe exist with or without an observer. The answer is yes because how else would they give rise to an observer.

[lginc]

Thank you for these wonderful conversations, brilliant minds who are inspiring all of us.It's a pleasure to Listen and Learn.

[OnCharmLee]

[Mathematics and Physics - Ideal and Actual Worlds]: Mathematics is often called a language that describes nature or its changes or movements, and is also used as a tool to model logic and judge their right and wrong.

Is mathematics simply an artificial thing created by human beings? Even its representing symbols are artificial, is a concept or a logic inherent? If it is inherent, does it exist only together with nature, or does it exist more priory and originally than nature?

The various symbols and expressions used in mathematics must have been defined by humans. But it can not be said that the concepts and theories in it are also artificial things created by humans.

For example, when describing a natural law with an expression, the law can not be said to have been created by a human being, and when any right logic is expressed by a formula, the logic itself can not be said to have been created by man.

The law of nature - if the law is right, regardless of whether it is good or bad to human beings - is a description of mathematical modeling of the causal relationship that the same phenomenon always occurs in the same condition in the physical world. It is a logical problem that human beings can not intervene.

The main task of physicists is to find these laws or principles that exist in nature. And engineers - even though they find such laws or principles themselves - make them available as machines or products mediated through matter, so that the found laws or principles can be used in real life.

However, the individual cases that occur in nature or are applied in real life correspond to special cases of actual cases that - can be expressed as numerical values - are infinitely possible in these mathematically modeled natural laws and principles. Thus, realization in nature and everyday humans life can be said to be the realization of some cases of mathematics based on matter.

In this case, if the mathematical logic is wrong, the behavior of the manufactured product will not work consistently with the corresponding principle as it is predictable, and - especially for satellite or missile - remote observing, monitoring, control, adjusting, management and maintenance will not be possible.

The question is that if there is a perpetual logic that human beings have found or thought out through thinking, even not in nature, that is always judged to be right for anyone who thinks right, transcending the physical space and time, anywhere, anytime, then who made the logic and how can it always remain right?

Examples of such logic include the consequences of reasoning proven through various pure mathematical axioms or rational thought processes. As an example, the logic of the most basic arithmetic rules (addition, subtraction, multiplication and division) has been found and used separately in different parts of the world in the past when traffic and communication never developed, and the logic that the physical world is finite and always changes has never been denied and confirmed right in natural daily life.

Thus, mathematics, logic, and philosophy are more primitive than physics, and they are independent of the physical world - or have no relevance to physics - and have been able to study independently. Concept or right reason has been felt to exist before the physical world.

Plato felt mathematics as the domain of the Idea world and the God’s world. Mathematics deals with infinity, but physics rejects infinity. This is because the world of matter is fundamentally a finite world, and mathematics can handle any logical world away from the physical world.
For the fact that while the physical world is an imperfect, always changing, temporary, and relative world that includes tangibleness, finite, quantized discontinuity, and error rates, the world of reasons is intangible, infinite, continuous, complete, immutable, permanent and absolute, it is dealt with in ‘3. Physical World - incomplete, ever-changing and temporary Material World’ and ‘4. Truth's World - never-changing and ever-right God's World’.

For example, the theorem that the sum of two sides is greater than the other side in a triangle - corresponding to the pure concept of logic that the straight line distance is shorter than a certain curve distance - is right in the ideal world, but it is impossible to realize it perfectly in the actual world where refraction occurs even in light.

Repeating the process of bisecting each line of the oblique sides and attaching them to the base side on an equilateral triangle in an infinite number of turns, results in the final that the sum of the two oblique sides becomes equal to the base, and the above theorem becomes wrong. This means that it is not possible in the actual world to repeat the bisecting without any loss and infinitely.

The Pythagorean theorem that the square of the base of a right triangle plus the square of the height equals the square of the hypotenuse, and the mathematical formulas that determine the circumference or width of a circle are also ideal cases. In he actual world, we can not make perfectly right triangles or circles without error.

This is why some industrial products (eg, construction, machinery, electronics, etc.) used in the actual world can not be completely perfect, and – this is because the physical world is ceaselessly and interchangingly changing as a whole – have limited performances and life spans. And we may understand that the fact that the natural exponents and cosmological constants (eg, the ratio of the circumference of a circle to its diameter π, natural exponent e, light speed c, Planck constant h,


[Difference between mathematics and physics in 0 and infinite handling]

Physics deals with the physical world. And mathematics deals with the logic world. Physics deals with a world in which material existence exists, and mathematics deals with a logical world that may not require any material existence.

Therefore, in physics, zero (0) and infinity are not handled. The physical world treats the change mainly on the assumption that there exist somewhat physically in finite space-time, and it does not handle the case that existence is infinity. In other words, physics deals mainly with the existence of certain physical beings in a finite world.

However, mathematics can treat zero (0) without physical existence, and also treat infinity. The world without a physical being is a primordial world, and the world of infinity means to contain all possible cases.

Thus, physics corresponds to special cases of mathematics and can be explained based on mathematics. In other words, the laws of physics appear as mathematical expressions in special cases in various cases in mathematics. This shows that the physical world is a realization of a part of the logical world.

(Extracted from OnCharm's book "Humans & Truth")

[dr.satishsharma]

reality, it is already out there " ( means beautifully , sweetly, eloquantly, in plain language & in seven words, explained by distinguished Sir.ROGER PENROSE, once for all & thus extinguishing the ambiguity on mathematics being as discovered or invented ....and that. is it, no more discussions thanks 🙏.

[thetruthoutside]

It just his method in explaining these difficult ideas is in itself exceptionally convincing and concise and is enough to make wonder 🤔 and allows you to really think.