If we were merely to begin with a word [ ta mathemata ] “the mathematical [things]” we find “that which is learned” or “things which are learned” or some translation near to that, like an un-English “the learnables” or some such.
Later we find ourselves wondering whether the “learnables” correspond to the “teachables” — and end up muttering to ourselves quietly: “probably not”.
One perspective: Mathematician Leopold Kronecker said
God made the integers, all else is the work of man.
I lean to “invented” since to me math is a form of language. But like “free will” vs “determinism”, the “discovered” vs “invented” are still tied in the fourth quarter. For anything I’ve ever need to do with math it has not made any difference.
well they had to discovered, by way of reason,
physicsguy, if Harrigan wants to do it, I’ll contribute to his campaign. He just entered Congress; I don’t know if he’d want to jump to the Senate immediately. He looks like an outstanding candidate. North Carolina Republicans are sick and tired of Tillis, and I don’t think he’s that attractive to Dems either. He’s got the charisma of a soggy carpet.
My old college roommate went on to teach applied math at MIT for 5 years. This subject came up with some frequency because we met at philosophy and politics student groups at the university in our home state.
One sympathetic and intuitively gratifying answer is provided by the great philosopher of science, Karl Popper.
In a post of only 10 or 12 paragraphs, this non math-gened blogger explains his stance concisely its title, “Invented and Discovered: Mathematics in Popper’s World 3”.
The author cities two or three recommended authors on math and philosophy, culminating in philosopher and mathematician Reuben Hersch’s restatement of Popper thusly:
“Start from two facts”, Hersh wrote. “(1) mathematics is a human creation, about ideas in human minds; (2) mathematics is an objective reality, in the sense that mathematical objects objects have definite properties, which we may or may not be able to discover. Platonism is incompatible with Fact 1, since it asserts that maths is independent of humans; constructionism is incompatible with Fact 2, since there no properties until they are proved constructively; formalism with both, since it denies the existence of mathematical objects… Mathematics is an objective reality that is neither subjective nor physical. It is an ideal (ie non-physical) reality that is objective (external to the consciousness of any one person). In fact, the example of mathematics is the strongest, most convincing proof of such an ideal reality.
This is our conclusion, not to truncate mathematics to fit a philosophy too small to accommodate it – rather, to demand that the philosophical categories be enlarged to accept the reality of our mathematical experience…. The recent work of Karl Popper provides a context in which mathematical experience fits without distortion. He has introduced the terms World 1, 2, and 3, to distinguish three major levels of distinct reality. World 1 is the physical world, the world of mass and energy, of stars and rocks, blood and bone. The world of consciousness emerges from the material world in the course of biological evolution. Thoughts, emotions, awareness are nonphysical realities. Their existence is inseparable from that of the living organism, but they are different in kind from the phenomena of physiology and anatomy; they have to be understood on a different level. They belong to World 2.
________
The inveterate seekers and curious here like our friend huxley are invited to read on with profit!
I only vaguely remember Michael Ledeen from the Reagan administration. I was pretty young. But I remember thinking he seemed really smart.
If I walk in a field and see a boulder, the boulder exists and has substance no matter what language I speak or what sounds I utter to reference it when communicating with others. Leibniz and Newton used different terms to refer to their discovery of The Calculus but their methods were identical because they had discovered the same thing.
CONTINUED BLOG POST FROM ABOVE:
In the further course of evolution, there appear social consciousness, traditions, language, theories, social institutions, all the nonmaterial culture of mankind. Their existence is inseparable from the individual consciousness of the members of the society. But they are different from in kind from the phenomena of individual consciousness. They have to be understood on a different level. They belong to World 3. Of course, this is the world where mathematics is located.”
“Mathematical statements”, Hersh concluded, ” are meaningful, and their meaning is found in the shared understanding of human beings, not in an external nonhuman reality. So mathematics deals with human meanings in the context of culture, it is one of the humanities; but it has a science-like quality, since its findings are conclusive, not matters of opinion…. As mathematicians, we know that we invent ideal objects, and then try to discover the facts about them.”
@Rufus T. Firefly:If I walk in a field and see a boulder, the boulder exists and has substance no matter what language I speak or what sounds I utter to reference it when communicating with others.
Let me know when you see 7, or pi, or the square root of 17, or the proof that e is not the root of any polynomial with integer coefficients, while walking in field. Please take a photo.
Leibniz and Newton used different terms to refer to their discovery of The Calculus but their methods were identical because they had discovered the same thing.
I would say they had carried out exercises in mathematical logic and come to the same conclusions. Their methods were totally different.
well numbers are visible, in the listing of objects, pi is a harder thing to describe, how to measure a circle, the sides of objects for geometry, now calculus would be a harder sell as to it’s practical application,
I’m with Penrose in that I believe that mathematics is discovered. It is too deeply embedded in the nature of reality for it to have been a human invention.
I often find that, in arguments over whether math is invented or discovered, the two sides use the word “mathematics” to refer to two distinct things. The “discovered” side is often referring to the underlying mathematical relationships, what Niketas Choniates might call mathematical logic. The “invented” side is often referring to the symbolic representations of the relationships, or to particular techniques for exploiting the relationships.
That’s an excellent column by David Strom, Mike Plaiss.
Leftism is a religious movement–a substitution of ideology for religion, and like religion–especially, in the 20th and 21st century, Islamism–it provides meaning to people with a hole in their souls that was left behind when Christianity was rejected. For liberals, technocracy and scientism fill that gap; for leftists, it is Marxism and decolonization.
I would add that this leftism is also filling the holes in the souls of Jews who reject their religious heritage in favor of secularism.
I began seeing articles about the confluence of Marxists and Islamists years ago, at Legal Insurrection. At first I had a hard time understanding how this could be, but now I see it’s the destruction of civilization which attracts them both. A pagan and barbarian future is a chilling thought.
Here’s my argument about mathmatics being discovered or invented:
What we call “mathmatics” is a language made up of a logically consistent series of procedures using objects and operaters that we’ve developed over thousands of years to make predictions that can be used to answer questions and solve all sorts of problems. But the fundimental underlying truths that mathmatics can be used to illuminate are sort of another matter. Expressions like 1 + 1 = 2 represent something fundimental to the Universe and perhaps beyond (multiverse? reality itself?). For example, the concept that the number “1” represents is immutable… as far as we can tell anyway. So in a sense, mathmatics is “invented”, but what it represents is “discovered” by using mathmatics.
Mathematics is a tool. It’s used in any number of ways. To count objects, such as money, aircraft, bushels of wheat, etc. It also describes various things we experience – wind velocity, temperature, altitude, etc. It also is used to predict things – seasons, building design strength parameters, weather, etc. It’s also used to describe things we can’t see sch as atoms, electrons, gravity, etc.
In many ways it’s like the tools humans have developed over time. Was the lever discovered or invented? How about the hammer? How about our control of fire? I say discovered.
Each step along the path of improvement required thought and analysis, but the basics of each step already existed – waiting for someone to figure out how to put them together into something useful. And the discovery of mathematics was a major tool in that progress.
“God made the integers, all else is the work of man.”
Counterexample: pi
The fact that men are still disputing about such things is indicative of the fact that there are matters that, despite the intellectual firepower trained on them, are simply beyond our ken. The protestations of materialists to the contrary notwithstanding, “science” is inadequate to provide answers to the deepest questions. For example, does it taste great or is it less filling? Mankind awaits the answer.
@miguel cervantes:well numbers are visible, in the listing of objects
You give it away right there. You can’t count objects with numbers until you’ve made a decision about how to distinguish objects.
Suppose there’s pieces of fruit on a table. If you count them, and say (for example) “seven”, you have made a decision about what properties of fruit can be abstracted away enough in order to lump them together. You can say “seven pieces of fruit” but if three are oranges and four are apples, and that’s important, then you have to say “three oranges and four apples”. Whether it’s important was a decision you made, not the universe, and it didn’t affect what was on the table, but it did affect whether you could say “three and four” or “seven”.
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I vote for discovered.
For Kate:
Kurt Schlichter proposes this morning that Patrick Harrigan could be a good primary candidate against Tillis. Your thoughts??
https://townhall.com/columnists/kurtschlichter/2025/05/22/lets-roll-the-dice-on-primarying-these-rinos-n2657394
New post: Real or Mythical? America’s Past Golden Age
https://chicagoboyz.net/archives/74106.html
If we were merely to begin with a word [ ta mathemata ] “the mathematical [things]” we find “that which is learned” or “things which are learned” or some translation near to that, like an un-English “the learnables” or some such.
Later we find ourselves wondering whether the “learnables” correspond to the “teachables” — and end up muttering to ourselves quietly: “probably not”.
One perspective: Mathematician Leopold Kronecker said
I lean to “invented” since to me math is a form of language. But like “free will” vs “determinism”, the “discovered” vs “invented” are still tied in the fourth quarter. For anything I’ve ever need to do with math it has not made any difference.
well they had to discovered, by way of reason,
physicsguy, if Harrigan wants to do it, I’ll contribute to his campaign. He just entered Congress; I don’t know if he’d want to jump to the Senate immediately. He looks like an outstanding candidate. North Carolina Republicans are sick and tired of Tillis, and I don’t think he’s that attractive to Dems either. He’s got the charisma of a soggy carpet.
My old college roommate went on to teach applied math at MIT for 5 years. This subject came up with some frequency because we met at philosophy and politics student groups at the university in our home state.
One sympathetic and intuitively gratifying answer is provided by the great philosopher of science, Karl Popper.
In a post of only 10 or 12 paragraphs, this non math-gened blogger explains his stance concisely its title, “Invented and Discovered: Mathematics in Popper’s World 3”.
https://theoccasionalinformationist.com/2017/01/29/invented-and-discovered-mathematics-in-poppers-world-3/
The author cities two or three recommended authors on math and philosophy, culminating in philosopher and mathematician Reuben Hersch’s restatement of Popper thusly:
“Start from two facts”, Hersh wrote. “(1) mathematics is a human creation, about ideas in human minds; (2) mathematics is an objective reality, in the sense that mathematical objects objects have definite properties, which we may or may not be able to discover. Platonism is incompatible with Fact 1, since it asserts that maths is independent of humans; constructionism is incompatible with Fact 2, since there no properties until they are proved constructively; formalism with both, since it denies the existence of mathematical objects… Mathematics is an objective reality that is neither subjective nor physical. It is an ideal (ie non-physical) reality that is objective (external to the consciousness of any one person). In fact, the example of mathematics is the strongest, most convincing proof of such an ideal reality.
This is our conclusion, not to truncate mathematics to fit a philosophy too small to accommodate it – rather, to demand that the philosophical categories be enlarged to accept the reality of our mathematical experience…. The recent work of Karl Popper provides a context in which mathematical experience fits without distortion. He has introduced the terms World 1, 2, and 3, to distinguish three major levels of distinct reality. World 1 is the physical world, the world of mass and energy, of stars and rocks, blood and bone. The world of consciousness emerges from the material world in the course of biological evolution. Thoughts, emotions, awareness are nonphysical realities. Their existence is inseparable from that of the living organism, but they are different in kind from the phenomena of physiology and anatomy; they have to be understood on a different level. They belong to World 2.
________
The inveterate seekers and curious here like our friend huxley are invited to read on with profit!
I only vaguely remember Michael Ledeen from the Reagan administration. I was pretty young. But I remember thinking he seemed really smart.
https://asiatimes.com/2025/05/michael-ledeen-a-reagan-revolutionary-passes-at-83/#
Everything true is discovered.
Niketas @ 11:19am,
If I walk in a field and see a boulder, the boulder exists and has substance no matter what language I speak or what sounds I utter to reference it when communicating with others. Leibniz and Newton used different terms to refer to their discovery of The Calculus but their methods were identical because they had discovered the same thing.
CONTINUED BLOG POST FROM ABOVE:
In the further course of evolution, there appear social consciousness, traditions, language, theories, social institutions, all the nonmaterial culture of mankind. Their existence is inseparable from the individual consciousness of the members of the society. But they are different from in kind from the phenomena of individual consciousness. They have to be understood on a different level. They belong to World 3. Of course, this is the world where mathematics is located.”
“Mathematical statements”, Hersh concluded, ” are meaningful, and their meaning is found in the shared understanding of human beings, not in an external nonhuman reality. So mathematics deals with human meanings in the context of culture, it is one of the humanities; but it has a science-like quality, since its findings are conclusive, not matters of opinion…. As mathematicians, we know that we invent ideal objects, and then try to discover the facts about them.”
And it appears that the guy who shot the two Israeli diplomats is a socialist moron (sorry for the redundancy) from Chicago who lived a few blocks from a coworker of mine. David Strom has a wonderful essay this morning:
https://hotair.com/david-strom/2025/05/22/the-left-inevitably-embraces-terror-n3803025
@Rufus T. Firefly:If I walk in a field and see a boulder, the boulder exists and has substance no matter what language I speak or what sounds I utter to reference it when communicating with others.
Let me know when you see 7, or pi, or the square root of 17, or the proof that e is not the root of any polynomial with integer coefficients, while walking in field. Please take a photo.
Leibniz and Newton used different terms to refer to their discovery of The Calculus but their methods were identical because they had discovered the same thing.
I would say they had carried out exercises in mathematical logic and come to the same conclusions. Their methods were totally different.
well numbers are visible, in the listing of objects, pi is a harder thing to describe, how to measure a circle, the sides of objects for geometry, now calculus would be a harder sell as to it’s practical application,
I’m with Penrose in that I believe that mathematics is discovered. It is too deeply embedded in the nature of reality for it to have been a human invention.
I often find that, in arguments over whether math is invented or discovered, the two sides use the word “mathematics” to refer to two distinct things. The “discovered” side is often referring to the underlying mathematical relationships, what Niketas Choniates might call mathematical logic. The “invented” side is often referring to the symbolic representations of the relationships, or to particular techniques for exploiting the relationships.
the inexplicable chasing the indefensible,
https://twitchy.com/samj/2025/05/22/oh-look-se-cupp-posted-something-dumb-again-tapper-book-n2413181
https://twitchy.com/samj/2025/05/22/harmeet-dhillon-nukes-jasmine-crockett-n2413155
when there is some measure of accountability, well squirrels are let loose,
i know in serious times, why focus on such small fry, because they allow
the most agregious frauds to be perpetrated, and free of responsibiility,
https://x.com/SenatorHousakos/status/1925318933537493020
That’s an excellent column by David Strom, Mike Plaiss.
I would add that this leftism is also filling the holes in the souls of Jews who reject their religious heritage in favor of secularism.
I began seeing articles about the confluence of Marxists and Islamists years ago, at Legal Insurrection. At first I had a hard time understanding how this could be, but now I see it’s the destruction of civilization which attracts them both. A pagan and barbarian future is a chilling thought.
Here’s my argument about mathmatics being discovered or invented:
What we call “mathmatics” is a language made up of a logically consistent series of procedures using objects and operaters that we’ve developed over thousands of years to make predictions that can be used to answer questions and solve all sorts of problems. But the fundimental underlying truths that mathmatics can be used to illuminate are sort of another matter. Expressions like 1 + 1 = 2 represent something fundimental to the Universe and perhaps beyond (multiverse? reality itself?). For example, the concept that the number “1” represents is immutable… as far as we can tell anyway. So in a sense, mathmatics is “invented”, but what it represents is “discovered” by using mathmatics.
“…free of responsibility…”
Let’s hope that will be changing.
Fast.
https://www.zerohedge.com/political/top-doge-official-details-discovery-shocking-voter-fraud-tip-iceberg
Mathematics is a tool. It’s used in any number of ways. To count objects, such as money, aircraft, bushels of wheat, etc. It also describes various things we experience – wind velocity, temperature, altitude, etc. It also is used to predict things – seasons, building design strength parameters, weather, etc. It’s also used to describe things we can’t see sch as atoms, electrons, gravity, etc.
In many ways it’s like the tools humans have developed over time. Was the lever discovered or invented? How about the hammer? How about our control of fire? I say discovered.
Each step along the path of improvement required thought and analysis, but the basics of each step already existed – waiting for someone to figure out how to put them together into something useful. And the discovery of mathematics was a major tool in that progress.
“God made the integers, all else is the work of man.”
Counterexample: pi
The fact that men are still disputing about such things is indicative of the fact that there are matters that, despite the intellectual firepower trained on them, are simply beyond our ken. The protestations of materialists to the contrary notwithstanding, “science” is inadequate to provide answers to the deepest questions. For example, does it taste great or is it less filling? Mankind awaits the answer.
@miguel cervantes:well numbers are visible, in the listing of objects
You give it away right there. You can’t count objects with numbers until you’ve made a decision about how to distinguish objects.
Suppose there’s pieces of fruit on a table. If you count them, and say (for example) “seven”, you have made a decision about what properties of fruit can be abstracted away enough in order to lump them together. You can say “seven pieces of fruit” but if three are oranges and four are apples, and that’s important, then you have to say “three oranges and four apples”. Whether it’s important was a decision you made, not the universe, and it didn’t affect what was on the table, but it did affect whether you could say “three and four” or “seven”.