57

In math, we define stuff like numbers and operators, then we go on to prove other stuff from those premises. When you ask: "Is 1 + 1 = 0?", a mathematician will just ask back: "With what definition of +?" If you assume natural numbers and the common definition of +, then this statement is false. If you assume numbers modulo 2 and + meaning XOR, then this ...


28

If you want to take a more constructive point of view, you need to reinterpret things accordingly. For example, "not P" should be interpreted as the assertion "P implies a contradiction". Accordingly, Fermat's last theorem says: Given any solution to xn + yn = zn, n > 2, you can deduce a contradiction To prove this statement, you don't need to examine ...


15

The math solution is, to find out properties of the things we work with, and prove those. Then we re-search those properties for more properties we can now prove. And on those more intricate properties, we build even more complex proofs. In the case of your parking lot, the mathematician might start by asking: What do I know about this parking lot? The ...


13

The answer is affirmative. The only hard fact is that the Hume's principle (bijective sets have equal sizes) and the part-whole axiom of Euclid (the whole is greater than its part) are incompatible for infinite sets. It is not that your intuition is "wrong", but rather that any extension of "size" to infinite sets will be paradoxical, it has to discard one ...


12

Russell's paradox arises within naïve set theory by considering the set of all sets that are not members of themselves. Such a set appears to be a member of itself if and only if it is not a member of itself. Hence the paradox. The "root" of the paradox is the so-called unrestriceted Comprehension Principle of naïve set theory: for every property φ(x) ...


12

It's actually misquoted. From: http://homepage.divms.uiowa.edu/~jorgen/hadamardquotesource.html A longer and more nuanced formulation appears (in English) in Hadamard's An Essay on the Psychology of Invention in the Mathematical Field (Princeton U. Press, 1945; Dover, 1954; Princeton U. Press, as The Mathematician's Mind, 1996), page 123: "It has been ...


12

What makes mathematical statements about infinite domains work is a belief in realism, that is, a belief that these statements represent on face-value something real. If they represent something real then according to Michael Dummett this implies a belief in the principle of bivalence regarding these statements. With realism each of these statements has a ...


11

The hypothesis 1+1=0 is false in the domain of natural numbers. If the domain is the finite field of the integers mod 2, then one is no longer in the domain of the natural numbers and the statement 1+1=0 would be true in that domain. The question is why do we not consider these to be falsifications of each other? These are not contradictions or ...


10

Timm Lampert, cited by the OP, quotes Wittgenstein (§8 of Remarks on the Foundations of Mathematics, Appendix 3): ‘True in Russell’s system’ means, as was said: proved in Russell’s system; and ‘false in Russell’s system’ means: the opposite has been proved in Russell’s system. Lampert claims Wittgenstein is assuming what needs to be proven: Whether P =...


10

This question sort of leads in two directions. The first direction is proof theory, which describes how mathematical proofs work. They formalize a process of manipulating statements according to a set of rules, much like a game. Reach the statement you wish to reach, and you win the game. There are many games out there, with different sets of rules. ...


9

1) Semantic Information Let us start with what information is. Suppose we have a set of sentences we know to be true, this allows us to answer (some) questions about the world. As we learn more sentences to be true the amount of questions we can answer grows. In the epistemic logic this is measured by defining an epistemic space, consisting of all possible ...


9

You're right to see some resonances between intuitionism and Kant. However, there's no uncontroversial sense in which you could blithely categorize Kant as an intuitionist. I'm not sure I've ever seen anyone do it, as a matter of sociological note. More instructive, I think, than trying to get Kant to fit the intuitionist picture is to explore the subtle ...


9

We can do the same thing for a parking lot problem as we do for Fermat's Last Theorem. Suppose we want to determine whether one of the cars is both orange and not orange (see note). I don't think anyone would need to go through the parking lot or even give so much as a cursory look to any of the cars. We can do the same thing for a mathematical problem as ...


9

Perhaps against my better judgment, I'll take a stab at answering this - this incarnation of the OP's issue does come closest to being a real question in my opinion. First, let's trim away all the excess and unclear verbiage ("stipulated relation"? "conceptual truth"? "ontological engineering"?). You're supposing that we have some formal system T (say, ...


8

Pigliucci gives an interesting review of the Mathematical Universe based on personal conversations with Tegmark. Apparently, Tegmark does admit plurality of mathematical structures, at least hypothetically, but his plurality is much reduced compared to what even "one-truth" mathematical platonists admit. First, "Tegmark replied that perhaps only Gödel-...


8

"Conventionalism" was the original position of positivists, which came to be seen as a failure after Quine's criticisms of truth by convention and the analytic/synthetic distinction. Wittgenstein abandoned it even earlier. The idea was that science uses what Carnap called "linguistic frameworks" based on conceptual schemes, axiomatizing the concepts used, ...


8

There is a lot of writing both in favor and against AC from a philosophical standpoint - e.g. in favor see Penelope Maddy's Believing the axioms. However, there are also more mundane issues. I think that, whether or not it's ideal, a key point here is usability. An answer like this may seem dubiously appropriate at philosophy.stackexchange, but I think it'...


8

Mathematics works because mathematics has a defined set of rules for manipulating mathematical symbols and entities. If we start with a specific mathematical phrase, we apply the rules in some sequence to achieve different mathematical phrases until we reach an outcome we want (a contradiction, a scope limitation, a relation...). If there were solid rules ...


7

For an introductory exposition, you can see : Richard Zach, Principia Mathematica and the Development of Logic (2010). A more detailed exposition is into : Ivor Grattan-Guinness, The Search for Mathematical Roots, 1870-1940, Princeton UP (2000), Ch.7. The issue is that the "standrad" proof of 1+1=2 from Peano's axioms is quite simple: it needs very few ...


7

If I understand your question correctly, you are asking in effect how do we distinguish logic from non-logic? Logical expressions give rise to valid arguments and logical truths, that is, arguments where if the premises are true it is impossible for the conclusion to be false, and truths such that there is no way for them to come out as false. But this ...


6

Logic, paraconsistent or not, does not exactly make something happen, it is applied to reshuffle information already contained in a system. Paraconsistent logic does not even have to be applied to inconsistent systems, and even when it is, derivable contradictions do not have to be interpreted as "true". What we need is not logic but semantics, although ...


6

Ambiguity does not rule out truth. An ambiguous statement is one with two meanings. You may not know which meaning to apply but whichever it is the statement, given either or both meanings, may be true. For example, 'She attends the small girls' school'. This is ambiguous between : She attends the school for small girls. & She attends the small ...


6

One person's foundations are another person's pointless pedantics. What follows is my own view of the foundational side of mathematics. Many of these results are not widely known among mathematicians, and so their foundational significance is definitely up for debate - is it really important if nobody needs it? - but to my mind these are results which tell ...


6

1 + 1 = 0 is false. Meanwhile, (1_2) +_2 (1_2) = 0_2 is true. Here +_2 is a different operation than +, and 1_2 and 0_2 are different things than 1 and 0. So it's not surprising that one equation is true while the other is false. The problem is that we do not like to write "_2" everywhere, so we often write 1 + 1 = 0 when we mean 1_2 +_2 1_2 = 0_2. This ...


6

I don't think there is much philosophical significance in what he said. Basically, he is saying that the complex field is a nice field to work with---and indeed it is. For example, every n degree polynomial in C[x] has exactly n roots in C, while R[x] does not enjoy this property. Of course, there any many reasons why C is nice. Another one is that the ...


5

A true random number is one that is unpredictable, even knowing the state of the Universe beforehand. In the special case of a random series of numbers, each number has to be generated with probability independent of all the previous numbers. It's not possible to do this with a mathematical formula or computer program, but it is possible to use principles ...


5

The infinitesimal instants are non-zero-it is infinitely small. Infinitesimals have a colorful history. In the later part of the 18th-century continuity of a function was taken to mean that infinitesimal changes in the value of the argument induced infinitesimal changes in the value of the function. With the abandonment of infinitesimals in the 19th ...


5

The most important development in the rejection of positivism, from the metaphysical perspective, is that it allowed for a revisit and reexamination of the philosophical systems thinkers, like Aristotle, Spinoza, Kant and Hegel. This resulted in a renewed appreciation for the fact that any study of knowledge or science or reality could not be reduced to any ...


5

There are two major issues here: the connection between existence and exact representability, and the notion of exact representability itself. There is no justification given for the implicit claim that only numbers which have exact representations can exist. The OP merely makes the claim that numbers which can't be "marked" don't exist. This is not ...


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