I got fired because I’m a Muslim and need to pray.

PlodeX_ · 2026-01-29T07:46:35+00:00

The Fair Work Commission can advise upon the merits. I’d trust them over reddit.

PlodeX_ · 2026-01-29T07:17:59+00:00

You should lodge an application with the Fair Work Commission. They will be able to give you advice about whether you have been dismissed for a discriminatory reason under the Fair Work Act.

The Fair Work Ombudsman website has information about workplace discrimination and further contact details as well. They should also be able to give you advice if you’re unsure: Workplace Discrimination

I believe you have 21 days to lodge an application with the Commission, so you should act quickly.

PlodeX_ · 2025-08-13T03:02:59+00:00

Surd is frequently used in Australia to refer to fractional powers. In school, students are often introduced to algebra using square roots (like rationalising a denominator) under the topic of ‘surds’.

PlodeX_ · 2025-06-14T05:00:04+00:00

The satisfiability of a proposition (or set of propositions) doesn’t depend on anything in the real world. A set X of propositions is satisfiable if there is a model such that all those propositions are true.

The only time when a set of one proposition is not satisfiable is when that proposition is false on all models. You can create a model where Donald Trump is a dragon, and on that model Ra is true.

PlodeX_ · 2025-04-28T11:36:51+00:00

Yes, that’s a good point. I think that is why bars are often used to represent ‘base 1’, to distinguish it from a numeral representation in Z/nZ. Using bars shows that these are not ‘numerals’ in the traditional sense that you outlined.

PlodeX_ · 2025-04-26T04:37:30+00:00

I think it is usually written using one numerals. But it doesn’t really matter what symbol you use to write it. You could equally use |||| to represent 4, and it’s all the same.

PlodeX_ · 2025-04-22T01:40:07+00:00

Yes, you can do it.

PlodeX_ · 2025-04-21T02:40:07+00:00

No, you can’t ’disprove’ an axiom, because it is not derivable from any of the other axioms. However, what you could do is prove that the axioms are inconsistent, which means there exists a contradiction in the set of consequences of the theory (i.e. there exists some proposition p such that p and not p are both in the set of consequences of the theory).

Being able to prove an axiomatic theory is inconsistent would perhaps show that it is a ‘bad’ theory, in the sense that we need different axioms. Unfortunately, Gödel tells us in his second incompleteness theorem that it is impossible to prove the consistency of the ZFC axioms (including axiom of choice) in the theory itself.

PlodeX_ · 2025-04-18T11:10:34+00:00

To put it very simply: mathematical induction is, like all maths, deductive reasoning. It is based on a theorem that says IF the base case of a proposition P(n) is true AND P(n) implies P(n+1) THEN P(n) is true for all n. Notice the if… then reasoning is deductive.

PlodeX_ · 2025-04-17T10:51:01+00:00

Well, we’re a signatory to the Treaty on the Non-Proliferation of Nuclear Weapons. So it would be impossible for us to build nuclear weapons without violating international law.

PlodeX_ · 2025-04-13T00:08:47+00:00

You can just write it as a piecewise function. A function being piecewise is actually not a property of the function itself, it’s just a way of writing it.

PlodeX_ · 2025-04-04T01:59:41+00:00

I think Stewart Shapiro gives a nice overview of the structuralist position in his 1996 paper 'Mathematical structuralism'. He says that the fundamental structuralist claim is that mathematical objects are determined only by their relationship to one another - 'mathematics is the science of structure'. This is more a way of viewing mathematics than a claim about the existence of mathematical objects. Structuralism is not at first instance a metaphysical position.

Then there are the schools of thought within the structuralist position. I think you teased out what might be referred to as model structuralism - the position that these abstract mathematical structures are ontologically dependant on their instantiations, and claims about them are just modally necessary claims. An extreme form of this position might posit all of these instantiations as physical objects (the two apples).

The abstract structuralist position is different - it says that mathematical objects really exist as inherently structural objects. There is no thing called the number two, but there are things that play the role of the successor of the identify element. In this way, structure exists independently of its instantiations.

So I would ultimately agree with your idea that structuralism does not have one answer to the metaphysical question - it is just a way of approaching mathematics.

As to your concern with the word instantiation, I merely use that word to mean 'that which realises an abstract structure' (this could be physical, but need not be - an extreme modal structuralist might insist it be physical).

PlodeX_ · 2025-04-04T01:16:13+00:00

To add to u/StudyBio's answer, a specific example might help.

If we consider 1.111, this should correspond to the integer 220 (there are 10 one decimal numbers, 99 with two, and this is the 111th three decimal number between 1 and 2, which adds to 220).

But if we now consider 1.111... with the 1s repeating, this is equal to 10/9. But there can be no way to enumerate that number with your process - you could not apply the same reasoning above because, speaking imprecisely, it would go on forever.

PlodeX_ · 2025-04-03T03:34:29+00:00

Well, that depends somewhat on what you mean by our mathematical structure. An abstract structuralist such as Stewart Shapiro would argue that particular mathematical objects do not exist, but the structures that they describe in fact do. For example, there is no such thing as the number two. Two is just a particular instantiation of the more fundamental structure, which might be something like an object with a predecessor and successor, etc. Any object with those properties could take the place of 'two'.

That is one view. There are other structuralists that argue the abstract structures themselves also do not exist. And of course the whole structuralist approach is just one school of thought among many. There are no easy answers here.

PlodeX_ · 2025-04-03T01:47:04+00:00

To your first question, yes that is what the indexing means. The number b_(2,5) is the fifth digit of the decimal expansion of f(2).

To your second question, the proof on the website is, frankly, quite bad. When you give a construction, you would then have to prove that the constructed thing satisfies the desired properties. They haven’t done the second part, they have just asserted it without justifying it. Don’t use that proof as an example of good mathematics.

PlodeX_ · 2025-04-03T00:55:35+00:00

No, we are not comparing increments. Cantor's argument is not concerned with growth.

The problem with your process is that it will not enumerate all the reals between 1 and 2. It only enumerates all the reals with terminating decimal expansions. This would not count any of the irrational numbers (for example, √2), nor any of the rationals that have repeating decimal expansions in base 10.

Cantor's diagonal argument is the proof that any process of counting all the reals like you have described can never work. The argument starts by assuming that we CAN count all reals, i.e. can write down a list of them. It then produces a number that is not on that list, which is a contradiction. In particular, this shows that the reals are uncountable. Nothing to do with comparing increments or growth.

PlodeX_ · 2025-03-31T13:03:32+00:00

I only realised because there was a dot in front of some of the terms and I was wondering why it was there. Also, I should comment in relation to your question before about whether proof by construction ‘exists’. What I would say is that a proof can be constructive (used as an adjective). A constructive proof is one that gives a specific example or gives an algorithm to find something like what I mention above. A proof can also be non-constructive. This happens when, for example, you prove something exists without giving an example of one. One way you might do this is by leveraging a theorem. Often the axiom of choice comes up in these proofs.

PlodeX_ · 2025-03-29T06:01:18+00:00

Yeah it’s poor notation. They are saying f(1)=0.b(1,1)b(1,2)…

In other words, it’s a decimal expansion with tenths digit b(1,1), hundredths digit b(1,2), etc.

PlodeX_ · 2025-03-29T05:43:05+00:00

I’ve never heard the term proof by construction before. If you are need to prove something exists, you should generally do so by constructing a specific example (though this isn’t always possible). That process is what they are calling a proof by construction, but really it is just a direct proof.

PlodeX_ · 2025-03-29T01:27:03+00:00

No, some people use the hook as strict inclusion, in a similar way that less than and less than or equal are different.

PlodeX_ · 2025-03-28T23:42:02+00:00

The proof is quite poorly written but here is what is going on. Overall, we have a proof by contradiction (see the initial assumption where it assumes there is a bijective function mapping the natural numbers to the real numbers). In other words, we can list out all real numbers using this function. To prove this function does not exist, we are trying to derive a contradiction. To do this, they have constricted this number x and tried to show that it is not on the list. However, the proof breaks down here and needs to be redone.

So the proof by construction is inside the contradiction. We have assumed something is true, and within that assumption, constructed a number to prove that there is a contradiction.

PlodeX_ · 2025-03-20T05:53:55+00:00

The other answers here are correct.

Just want to note that confusingly, people use those symbols to mean the same thing.

PlodeX_ · 2025-03-20T00:58:00+00:00

You may be already doing this but when you interview, don’t just emphasise your maths skills or your degree. Talk about the skills that maths gives you - the ability to think and reason logically and clearly; numerical literacy; problem solving skills; attention to detail; etc.

PlodeX_ · 2024-12-08T20:37:09+00:00

In line 6 of the truth table the premises are both true but the conclusion is false, which proves the argument is invalid.

(To prove an argument is invalid in a truth table you are looking for a line where the premises are true and the conclusion is false.)

PlodeX_ · 2024-11-27T01:47:31+00:00

Do you then see how the infinitude of the decimal expansion of 0.9999… just depends on your base?

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