You've already assumed 5 exists in order to assert that it's prime.
In any case existence of mathematical objects is a different meaning of existence to physical objects. We can say a mathematical object exists just by defining it, as long as it doesn't lead to contradiction.
I think your closing paragraph holds the key. 5 doesn't really exist, it's a constructor that parameterizes over something that does exist, eg. you never have "5", you have "5(something)". Saying 5 is prime is then saying that "for all x, 5(x) has the same structural properties as all other primes".
With numbers, I can give an explanation for the phenomenon I described above. If such reasoning cannot be done without reference to numbers, then, if such reasoning is correct, numbers must exist. If there is no other reasoning can be given that provides a good explanation, and as the explanation I gave for the phenomenon is compelling, then I think that a good reason to conclude that the reasoning is correct, and that therefore those particular numbers exist.
In particular, I would expect that if numbers don’t exist, the explanation I gave of the phenomenon I described, couldn’t be correct.
You could say they exist as concepts, that are necessary to use for some reasoning processes, without having any kind of independent existence.
It's similar for the case of programs or algorithms. We can say that a sorting algorithm exists, or a chess-playing program or whatever, which means we know how to implement the logical process in some physical system, but it doesn't mean that they have some kind of existence which is independent of the physical systems. It's just a way of talking about patterns that can be common to many physical systems
My view is that something exists iff there is any statement that is true of it.
I of course don’t mean that mathematical objects (such as the number 2, or some sorting algorithm) have the same kind of existence as my bed. To make the distinction, I would say that my bed “physically exists”.
That sounds like the same circularity, since you'll have to assume numbers exist before proving any statements about them.
Physical objects aren't like that because you can discover that they exist by empirical investigation.
In mathematics the discoveries are about the logical implications of sets of axioms. Some of those axioms contain assertions of existence, like a number 0 in Peano arithmetic or the empty set in set theory, and then you can prove statements about these objects based on the axioms. It's circular to infer from these conclusions that the axioms are true.
What's interesting is why certain axiom systems are so useful and fruitful. Personally I think it's because they evolved that way from our investigations of the physical world, but that's another matter
Haha, It reminds me some old movie called Andromeda I think... Space human probe crashlanded on earth and contained some greenish patches of stuff on it. It was space dwelling orgamism that direcly used energy to matter conversion for growth. It was pretty decent movie actually :)
I am particularly interested in coops as a model for a tech startup. How would one go about structuring a new corporation to best ensure it remains a coop?
Plenty of worker owned software consultancies in germany, and specially in Hamburg for some reason.
From the site of one that I that I used to work for(they are very friendly so you can hit them up if you want some advice on setting up one)
> How exactly are you structured?
> In our search for a structure that consistently implements the principles of responsible ownership, we came across the veto-share model. First, the principles are enshrined in the company's articles of association. Then, company shares are transferred to a controlling shareholder. This controlling shareholder is granted veto rights, which must be used to prevent any future deviation from the principles. We are delighted to have the Purpose Foundation on board as our controlling shareholder!
> We want to offer every (new) team member the long-term prospect of assuming entrepreneurial responsibility as a co-owner. The criteria for this are already defined in the articles of association. To simplify joining and leaving, we have established dyve Trust eGbR, a partnership that holds 99% of the voting rights in dyve.
Mondragon is often used by people who don't know much about the subject as a counter example, but if you look under the hood, you'll see it doesn't even compare: Less than half the workers own shares and the majority of shares is held by a small circle of executives in the richest branches. That's basically a standard capitalist pig with some socialist lipstick to give pseudo-Marxists an argument in these discussions, when there is none to be had in the first place. In fact Nvidia has a higher percentage of employees owning shares than Mondragon. Would you call that "worker-owned" too?
None of this should be surprising either, because anything that truly benefits workers will be at odds with things that benefit the business. In society there are always people who primarily get paid and people who primarily work. Even the real Marx realized that. If you really want to defeat capitalism, then you have to do more than just play pretend.
In which case the business will have a harder time competing. Not sure why we needed to circle back to that. This is not rocket science. If you believe you can somehow bypass the way the world works and create a socialist utopia using such a trivial approach, you should check if your opinions are based on facts or on dogmatic beliefs. Because the facts are quite clear once you really get into the material.
Yes, the L^p spaces are not vector spaces of functions, but essentially equivalent classes of functions that give the same result in an Lebesgue integral. For these reason, common operations on functions, like evaluating at a point or taking a derivative are undefined.
If you care about these you need something more restrictive, for example to study differential equations you can work in Sobolev spaces, where the continuity requirement allows you to identify an equivalent class with a well-defined function.
I agree with the ethos but "banning installing" wouldn't have been correct here.
There should be terminology for installing from the source of your choice which doesn't carry the marginal or sinister connotations of "sideloading" though.
"Freeloading" would have been a good one but... yeah
If we're being pedantic, the user still has to perform the final action before the install begins. I think it' more "Google has to allow you to install apps on your phone"
It's a misleading comparison. This is only the average amount of heat that diffuses through an ordinary patch of surface, and has more or less nothing to do with how a geothermal plant works, since they don't harvest heat by covering a large area of surface with conducting material.
The surface heat flow is low because rock acts as an insulator. If you drill down to where it's hot and draw the heat up you obviously get orders of magnitude larger flows of energy to the surface.
Are you suggesting to basically harvest the thermal energy in the rock in a non-renewable fashion? I don't think that is very promising, the heat capacity of rock is not that huge.
Back of the envelope calculation is drawing 1 GW from a cubic Kilometer of rock would lower the temperature by 1 degree C every 25 days. So I think you'd deplete a typical borehole quite quickly?
> Are you suggesting to basically harvest the thermal energy in the rock in a non-renewable fashion?
As I understand it, that's how many geothermal plants work, effectively mining the heat underneath them, but at a rate of extraction that means they would become uneconomical over a span of decades rather than months.
Exactly. The only exception to this are very rare sites like the one in Iceland where you can get close to a magma cell which has a much higher thermal gradient and possibly magma convection replenishing it.
If there really were baskets of corn there, it seems more likely it was used for storage, and situated along the ridge not too far from the valley floor to protect the crops from floods and perhaps raids. Picture (a) in the article has a defensive settlement marked nearby.
In any case existence of mathematical objects is a different meaning of existence to physical objects. We can say a mathematical object exists just by defining it, as long as it doesn't lead to contradiction.