That doesn't make sense automatically because formalism is a program for foundations, platonism is an ontological claim. And idk what post of MU it is. — Lionino
I don't see what this all has to do with your claim that a concept like a number, 5, could have a physical instantiation . Fingers are fingers, and are therefore physical instantiations of fingers, not of numbers, not matter how many of them you have. — Metaphysician Undercover
Wittgenstein took up this issue in the Philosophical Investigations, showing why there is a lot more involved with learning a language than simple ostensive definition. Abstraction is very complex, and with complex concepts like number, an explanation of what it is about the thing which is being shown, which is being referred to with the word, is a requirement. — Metaphysician Undercover
A person cannot simply look at the fingers on a hand and apprehend the concept 5. — Metaphysician Undercover
An explanation about quantity, or counting is required. The concept 5 is learned from the explanation, not from the ostensive hand, therefore the hand is not a physical instantiation of the number. — Metaphysician Undercover
It can be judged by anyone. The issue though, is that many, like yourself refuse to make such a judgement. — Metaphysician Undercover
You say that there is no truth or falsity to mathematical axioms, they are simply tools which cannot be judged for truth. Since mathematicians tend to think this way, they are not well suited for judging truth or falsity of their axioms. But I've shown how axioms can be judged for truth. If an axiom defines a word or symbol in a way which is inconsistent with the way that the symbol is used, then it is a false axiom. — Metaphysician Undercover
So for example, if a mathematical axiom defines "=" as meaning "the same as", — Metaphysician Undercover
yet in applied mathematics the mathematicians use "=" to mean "has the same value as", then the axiom makes a false definition. — Metaphysician Undercover
This axiom will be misleading to any "pure mathematician" who uses it to produce a further conceptual structure with that axiom at the base, just like if anyone else working in speculative theories in other fields of science starts from a false premise. False propositions are fascinating, sometimes leading to theories which are extremely useful, because they are designed for the purpose at hand. — Metaphysician Undercover
Sorry, I don't understand what you mean by "meta-false". I am talking about "literally false". — Metaphysician Undercover
False to me, means not corresponding with reality. — Metaphysician Undercover
For example, if someone says that in the use of mathematics, "=" indicates "the same as", but in reality, when mathematicians use equations, "=" means "has the same value as", then the person who said that "=" indicates "the same as" has spoken a falsity. Do you agree that this would be an instance of "literally false"? — Metaphysician Undercover
That doesn't help. Numbers form discrete units, and discrete units cannot model an idealized continuum. — Metaphysician Undercover
There is an inconsistency between these two, demonstrated by those philosophers who argue that no matter how many non-dimensional points you put together, you'll never get a line. The real numbers mark non-dimensional points, the continuum is a line. The two are incompatible. — Metaphysician Undercover
5 is an attribute of the fingers on your hand, would you grant me at least that? — fishfry
I think of fingers as a physical instantiation of the concept of 5. But if you disagree, then we must be using the word differently. I'm ok with that. How about representation, in the same sense that the first cave man to kill five mastodons and make five marks in the ground to keep track. — fishfry
This is manifestly false. Not a matter of opinion or interpretation or language. Flat out false. — fishfry
On the contrary, it is exactly through the experience of looking at one's hand that one at first does apprehend the number 5; and only later, by analogy and induction, all the other natural numbers. — fishfry
Oh no. 5 is learned by bijection with the fingers, not with counting. Counting is a higher function. Bijection is more primitive or intuitive. If you've seen a mother cat missing a kitten from her litter, she is not going "One, two, three ..." She's comprehending the total number instinctively and knowing when she's one short. — fishfry
There is a modern trend of misspelling judgment, and I can't let it go by. No middle 'e' in judgment. — fishfry
If 2 + 2 is 5, then I am the Pope.
That is a true statement that does not correspond with reality. — fishfry
Statements assumed true in a fictional context so as to work out the consequences. — fishfry
An instance of literally false? — fishfry
Sorry, the devil made me do it. For some reason, out of all the words that have multiple spellings British/American mainly, people on this forum complain about judgement/judgment. Why is this worthy of a correction? You didn't correct me when I spelled color colour. — Metaphysician Undercover
This discussion has degenerated. Let's evacuate. — Metaphysician Undercover
Is judgement with the extra 'e' a Britishism? — fishfry
I don't know, but there are lots of US/Brit differences, the common one being the "o/ou", which most are familiar with. I'm Canadian so I'm stuck in between, getting it from both sides. For us, the 'proper' way is the Brit way, which my spellcheck hates. I have the keyboard option for Canadian English, but it seems to default to US. There are some interesting nuances, such as the practice/practise difference. We would use "practise" as a verb, an activity, but if a professional like a doctor, or lawyer, sets up a practice, we have the other form as a noun. It's not a very useful distinction, and difficult to figure out when you're writing, so screw it! What's the point in such formalities? — Metaphysician Undercover
How does a formalist typically account for the ontology of rules? What kind of existence do rules have? Consider the rule of how to spell "judgement" for example, how does that rule exist? — Metaphysician Undercover
It's call the arithmetization of analysis. It's a thing in late 19th century math. Basically founding math, including calculus and continuous processes, on set theory.
https://en.wikipedia.org/wiki/Arithmetization_of_analysis — fishfry
I explored this question somewhat in my Grundlagenkrise thread, specially in my chat with Banno, but there was no interest in the topic died after 3 days — folks prefer to go around circles about ethics instead and keep it shallow. The ontology of rules are ultimately derived from logic, be it first-order or second-order — and logical terms can be taken as primitives defined from their truth tables — and the usage of undefined terms, such as "line", "+", or, in the case of ZF, membership ∈. — Lionino
5 views per day. The title doesn't resonate with many apparently (including me). Nevertheless, an important movement. — jgill
If logic is following rules, as formalists seem to think, then to say that rules are derived from logic is circular. — Metaphysician Undercover
In the philosophy of mathematics, formalism is the view that holds that statements of mathematics and logic can be considered to be statements about the consequences of the manipulation of strings (alphanumeric sequences of symbols, usually as equations) using established manipulation rules. — Wikipedia: Formulism (Philosophy of Mathematics)
Virtually every professional mathematician lives in the world created by this movement. Nobody notices because it's like fish not noticing water. — fishfry
Thus, were set theory removed mathematicians would perish. — jgill
I think not. But mathematics would not be nearly as robust as it is today. My humble opinion. — jgill
Back in the late 1960s my advisor remarked on the separation of the nitty gritty at ground level and the efforts to fly high and look down on mathematics, an abstract perspective to see how the various parts fitted together and document how parts from one branch were like parts form another. He gave me a choice and I felt far more comfortable working in the lowlands, (particularly after learning a bit about algebraic topology). I came into the profession exploring convergence and divergence of analytic continued fractions and related material. Pretty much an extension of the efforts during the 1700s and 1800s to solidify those properties of series. Grubby stuff, but I still enjoy grovelling in it. :cool: — jgill
What "ontology of rules"? — Lionino
Consider it is the same sort of issue as the ontological status of numbers — Metaphysician Undercover
If we answer yes, there is such a thing as rules, then we may proceed to ask questions like are they objective, and if the answer to this is yes, then we are in Platonism — Metaphysician Undercover
? Those are set up by convention.and logical terms can be taken as primitives defined from their truth tables — Lionino
Rules themselves may be mathematical objects. Languages, axioms, rules, systems, theories, and proofs can be defined and named in set theory. Even informally, when, for example, we say "by the rule of modus ponens", the rule of modus ponens is a thing named by 'the rule of modus ponens'. — TonesInDeepFreeze
and logical terms can be taken as primitives defined from their truth tables
— Lionino
Those are set up by convention. — Lionino
It's not clear to me what you're claiming. Example? — TonesInDeepFreeze
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