One talks of mathematical discoveries. I shall try again and again to show that what is called a mathematical discovery had better be called a mathematical invention. — Wittgenstien - Richard B — Wayfarer

Questions for Richard B

If addition was invented, who invented it?

Who invented the distributive law, i.e., that, for example 5*(3+7) always equals 5*3 + 5*7 ?

The birthday paradox of probability says that given 23 people, there is a 50% probability that two will have the same birth day (but not necessarily the same birth year). With 75 people, the probability is 99.9%. https://www.scientificamerican.com/article/bring-science-home-probability-birthday-paradox/

Who invented the birthday paradox? AND how did the inventor of the birthday paradox arrange it so that when I had about 23 people in one of my probability classes and decided to demonstrate the birthday paradox, that about half the time, two people had the same birth day?

I want to meet these inventors. Who are they?

When I ask what the number 7 is, you will point to the number, 7, and say that is what it is. But '7' is a symbol. That is an invention and can be represented in many different symbols: VII, SEVEN. What is not invented, is the meaning of the symbol. And that is what we all agree on. — Wayfarer

If you asked me "what the number 7 is?", I may want a little more clarity on what you mean by this question. In different contexts, it could mean different things. If I did point to the symbol "7", maybe I was teaching a child how to count with mathematical symbols. Or maybe I had used that symbol to show how to add, subtract, multiple, or divide. Or, maybe I showed how numbers select out an individual in a soccer match. Or, maybe how it can be used to title a movie. All created by humans to give a dead sign "life." These symbols are created by humans, and humans give it a use which gives it a meaning.

When we talk about math, we say things like "I figured out the solution to the problem", "I constructed a proof to demonstrate such and such", or "I determine the equation for such a figure." It would be odd to say after every addition problem, "Wow I discovered '1+45=46', '25+75=100', '7+1000=1007', etc"

When we talk about "discovering meanings, ideas, eternal objects", we belittle the creative aspect of human intelligence. It gives this picture that human go into the room called "Platonic realm", find aisles of bins labeled "meanings", "idea", eternal objects" and select the one we like, call it a discovery, and share it with the world. I don't know about you, but that is not how I learn. I read books, listen to lectures, debate other people, ask questions, draw diagrams, make errors, test a hypothesis, conduct experiments, etc...in order to learn ideas or come up with new ideas.

If you asked me "what the number 7 is?", I may want a little more clarity on what you mean by this question. — Richard B

It's very clear - 7=7 (or the sum of its factors).

Actually during this exchange, I've come to realise that what I believe is that whilst mathematical systems may be invented, numbers are discovered. I think that's the thrust of a saying by mathematical philosopher Leopold Kronecker, 'God invented the integers, all else is the work of man'. But even if mathematics is invented, it is dependent on that foundation, without which none of it could be invented. And having been invented, it has great predictive power, but only because it is grounded in reality, not simply in convention.

When we talk about "discovering meanings, ideas, eternal objects", we belittle the creative aspect of human intelligence. It gives this picture that human go into the room called "Platonic realm", find aisles of bins labeled "meanings", "idea", eternal objects" and select the one we like, call it a discovery, and share it it with the world — Richard B

I think that's an innaccurate depiction of what 'the ideas' actually represent, and we make it because we're accustomed to thinking of whatever as real as being 'out there somewhere'. But as already noted, universals (of which I am saying numbers are a subset) don't exist in that sense. They can only be apprehended by reason, which is a faculty unique to h. sapiens (although present in rudimentary form in some of the higher animals). My idea of the forms is that they're closer in meaning to 'principles' than to ghostly ethereal objects - and they're principles that, whilst independent of any particular mind, can only be grasped by the mind. Our thinking is thoroughly suffused with such principles.

and they're principles that, whilst independent of any particular mind, can only be grasped by the mind. — Wayfarer

In what sense is “principles independent of any particular mind”?

An example from an article referenced earlier:

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he says "that a pencil exists independently of grasping it. Thought contents are true and bear their relations to one another (and presumably to what they are about) independently of anyone's thinking these thought contents* - "just as a planet, even before anyone saw it, was in interaction with other planets." ' — Frege on Knowing the Third Realm, Tyler Burge

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* 'In his work, Frege used the term "thought contents" to refer to the meanings of sentences or propositions. He argued that the meaning of a sentence or proposition was not simply a matter of the words used, but rather the thoughts that the sentence or proposition expresses. He believed that language is a system of symbols that can be used to express these thought contents, which are themselves independent of any particular language.'

Frege believed that number is real in the sense that it is quite independent of thought: 'thought content exists independently of thinking "in the same way", he says "that a pencil exists independently of grasping it. — Frege on Knowing the Third Realm, Tyler Burge

“In the same way” is the mystery. I can picture a hand separated in space from the hand. And I can picture the hand moving to grasp the pencil. But what am I picturing when thought content is separate from thinking? This is not a spacial relationship Maybe it is more like the relationship between a triangle and three sides, you can’t imagine one without the other. So, it is unlike a hand and pencil. Thus, they are not independent of each other.

Maybe it is more like the relationship between a triangle and three sides, you can’t imagine one without the other. So, it is unlike a hand and pencil. Thus, they are not independent of each other. — Richard B

'Independent' in the sense that the concept triangle is not dependent on your thinking about it.

the Aristotelian-Thomistic tradition regards the intellect as a distinct faculty from the senses and the imagination. The objects of the intellect are concepts, which are abstract and universal, while the senses and imagination can only ever grasp what is (at least relatively) concrete and particular. Hence your sensation or mental image of a triangle is always of a particular kind of triangle – small, isosceles, and red, for example – while the concept of a triangle grasped by your intellect applies to all triangles, whether they are small or large, isosceles, scalene, or equilateral, red, green, or black. Sensations and mental images are also subjective or private, directly knowable only to the person having them, while concepts are public and objective, equally accessible in principle to anyone. Your mental image of a triangle might be very different from mine, but when we grasp the concept of a triangle, it is one and the very same thing each of us grasps, which is why we can communicate about triangles in the first place. — Edward Feser

The same principle is broadly applicable to all manner of geometric and arithmetic concepts, as well as to logical principles and scientific laws.

The elements of a set are logically prior to the set — Art48

I'm not so sure. Perhaps a platonist would consider all of the entities of set theory to be timeless. It'd merely be our presentation of them which would require first the source set and then the equivalence classes.



Consider a novel that written that's been translated into 20 languages. Somehow the original text and all who know the original language are removed (taken by aliens to Jupiter, for instance.) So now we have 20 'translations' which contain the 'same' idea. Do we think perfect translation is possible ? Is a perfect paraphrase in the same language even possible ? Or are we not dealing with a sameenoughness ?

So now we have 20 'translations' which contain the 'same' idea. Do we think perfect translation is possible ? Is a perfect paraphrase in the same language even possible ? — green flag

If you were dealing with a recipe, or a formula, or design blueprints, you'd better be damned sure they're accurate. (Remember that European Mars Lander that failed because an engineer confused imperial and metric?)

This is a long-standing interest of mine. Consider this question: if you have a string of text of the type mentioned above, it can be translated, not only into other languages, but completely different symbolic systems, like binary. In such cases, what changes, and what stays the same? I think the answer is, the symbolic form changes, but the meaning is constant. Same with number: we can invent all kinds of symbolic systems and relationships, but the meaning of '7' must remain invariant. That is what *I* think 'platonism' is intuiting, although I accept it's very much a minority view.

He believed that language is a system of symbols that can be used to express these thought contents, which are themselves independent of any particular language.' — Wayfarer

I think Frege was right about that (relative) independence. The sameenough idea can be put in lots of sentences in the same language or in some other language.

The sentences are containers metaphor has its advantages, but perhaps it can lead us astray if we forget other possibilities. Equivalence classes look to be a more neutral approach.

I think the answer is, the symbolic form changes, but the meaning is constant. Same with number: we can invent all kinds of symbolic systems and relationships, but the meaning of '7' must remain invariant. That is what *I* think 'platonism' is intuiting, although I accept it's very much a minority view. — Wayfarer

We pretty much agree here. I imagine that's the point of forms. Even Saussure talked of form. But equivalence classes do the same job with less commitment. The container metaphor is too spatial, in my view. Or maybe it's fine but we need more approaches. We tend to think when translating that we are unwrapping and rewrapping a Content instead of searching for a tool that does the same job.

(And there must be some other good metaphors out there besides just these.)

But what am I picturing when thought content is separate from thinking? — Richard B

I suggest a structuralist approach. Imagine a game that is basically Chess but every piece is carved differently and has a different name. Translating the bishop token (its 'content') would just be pointing out the piece that does the 'same thing' (plays the same role) in the other game.

I've learned that platonism in mathematics is regarded as highly non-PC - presumably because of its challenge to philosophical naturalism. Have a look at this article, which is in my current bookmarks list What is Math? Smithsonian Magazine

I tracked down and bought the (expensive!) textbook of the platonist Professor mentioned in that article, James Robert Brown (although you'd probably be able to make more sense of it than me). But note this passage from the essay:

Other scholars—especially those working in other branches of science—view Platonism with skepticism. Scientists tend to be empiricists; they imagine the universe to be made up of things we can touch and taste and so on; things we can learn about through observation and experiment. The idea of something existing “outside of space and time” makes empiricists nervous: It sounds embarrassingly like the way religious believers talk about God, and God was banished from respectable scientific discourse a long time ago.

Platonism, as mathematician Brian Davies has put it, “has more in common with mystical religions than it does with modern science.” The fear is that if mathematicians give Plato an inch, he’ll take a mile. If the truth of mathematical statements can be confirmed just by thinking about them, then why not ethical problems, or even religious questions? Why bother with empiricism at all?

Something which I would describe as 'inadvertantly revealing'.

This reminds me of how Pythagoreans viewed numbers to such an extent that it could be viewed as a religion. Apparently, they had a prayer to something called a Tetractys (sometimes called the "Mystic Tetrad"):

“Bless us, divine number, thou who generated gods and men! O holy, holy Tetractys, thou that containest the root and source of the eternally flowing creation! For the divine number begins with the profound, pure unity until it comes to the holy four; then it begets the mother of all, the all-comprising, all-bounding, the first-born, the never-swerving, the never-tiring holy ten, the keyholder of all.”

There's definitely a Pythagorean flavour to it. But then, Russell says in HWP that the mathematical mysticism of Pythagoreanism is one of the key differentiators of the Western cultural tradition from the Asiatic. So I don't think it is something to be belittled.

Also recall that in Platonism, knowledge of arithmetic and geometry was 'dianoia', which is higher than opinion concerning appearances, but not the highest level, which is 'noesis'.

There's a book I've noticed, Jerrold Katz, The Metaphysics of Meaning. (Reviews here and here). This book, and indeed most of Katz' career, was dedicated to critiquing Wittgenstein, Quine, and 'naturalised epistemology' generally. He also studied under Chomsky, but I think the basic drift is Platonist, i.e. meaning has to be anchored in recognition of universals as constitutive elements of reason - not simply conventions or habits of speech. — Wayfarer

I finally have had a chance to read this book. Thanks for mentioning it. I have not seen many sophisticated attempts that try to argue against later Wittgenstein, but this is one of them. On the positive side, Katz does a great job of elucidating both Wittgenstein’s and Quine’s philosophy. In fact, most of the time he agrees with Wittgenstein’s investigation into meaning and language. However, he believes, Wittgenstein’s criticism does not touch certain linguistic attempts at discovering the underlying structure of language. Additionally, he believes Platonic versions of philosophical theories can explain those linguistic proto-theories better than the naturalistic positions that Wittgenstein and Quine offer.

In the end, he believes he has shown that his vision is a justifiable alternative to the naturalism positions he attempt to critique.

I finally have had a chance to read this book — Richard B

Hey you're ahead of me. I have to renew my alumni membership at the uni library and make the trip to get it out. Glad you found it useful.

The online reviews summarizing Jerrold Katz's platonist critiques of "naturalism", "empiricism", "pragmatism", "conceptualism", "nominalism" & "antirealism" remind me of the platonic cosmologist Max Tegmark's mathematical universe hypothesis, presented in his (speculative memoir) Our Mathematical Universe, which I'd found rigorously compelling in spite of my own (anti-platonist) philosophical naturalism.

I can't quite wrap my head around Tegmark. I think Katz' book is more my cup of tea but I've yet to get hold of a copy.