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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 ...
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 ...
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 ...
6
Biological evolution is an undirected process driven by chance mutation. Many mutations have severe consequences, and even those that could be considered beneficial in certain contexts may have nasty side effects in different contexts. And of course even many of those changes which could be considered beneficial don't get passed on to the next generation.
...
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 ...
6
Considering
An important paper concerning the distribution of prime numbers was Riemann's 1859 memoir "On the Number of Primes Less Than a Given Magnitude", the only paper he ever wrote on the subject. Riemann introduced new ideas into the subject, the chief of them being that the distribution of prime numbers is intimately connected with the zeros of the ...
4
Your example from mathematics shows: To assess a mathematical statement one should first fix the context, the domain of validity of the symbols. Because in the context of natural numbers the statement 1+1=0 is false. While in the context of Z/2Z the statement is correct. Except the rare case of undecidable questions, in mathematics one can prove or disprove ...
3
The so-called Demarcation Problem is a little bit later, emerging with Vienna Circle and Popper.
But its origins are in the Tractatus :
4.03 A proposition communicates a situation to us, and so it must be essentially connected with the situation.
Thus, a true proposition is the picture of a fact.
4.05 Reality is compared with propositions.
4.06 ...
3
Hmmm. What about 1 + 1 = 10 ?
Is that equation, expressed in binary arithmetic, "false in the domain of natural numbers"?
My grounding in math and logic isn't very strong, but I understand the Wikipedia entry...I just don't think that the notions of truth and falsity can coherently apply to inductive inferences (abstract descriptions of unobservable ...
3
How should “knowledgeability” be defined? Is it merely w.r.t. to best scientific knowledge? [closed]
Well, mavavilj, definitions are easy - dictionaries are teeming with them. None of them is right or wrong, they're only more or less agreeable to whomever interprets them!
However, 'knowledge' isn't a scientific term, it's a philosophical issue. Science is more concerned with evidence, theory and statistical analysis.
Are you keeping in mind the ...
3
Indirect realism is the view that our contact with reality is mediated ("filtered") through our sense organs and/or cognitive faculties. We do not experience reality "as it is, out there". This isn't assumed by science per se, but it is probably supported by science. Colors for example, do not inhere in mind-independent objects, but in our perceptions of ...
3
Today, this can have a significantly different answer than a few years ago.
Let me explain why. Logically, an event would happen and you the observer would be irrelevant. Meaning, that tree would generate its sound and shockwave when falling no matter if you are there or not. This is physically valid and was practically demonstrated on numerous occasions.
...
2
Heisenberg wrote in his Physics and Philosophy p. 42 that the probability wave concept in quantum mechanics
was a quantitative version of the concept of "potentia" in Aristotelian philosophy.
See also:
"Act, Potency, and Energy" by Thomas McLaughlin, an expert in the Aristotelian definition of motion & principle of inertia
Rankine's "Outlines of the ...
2
I am not familiar with formulations of the non-existence of objective reality, but here's my stab at the question anyway:
Suppose you have the hypothesis "All apples are red", and you see a green apple. If objective reality exists, then the hypothesis has been falsified by the observation. If objective reality does not exist, then the hypothesis was false ...
2
The main skill of a philosopher, in my view, should be the ability to simplify. The complexity of most issues is an illusion caused by incomprehension. Comprehension requires simplification. KISS, or Keep it Simple Stupid, is the method.
The philosopher has only a secondary interest in groupology or even particular instances. The first thing to do is ...
2
It is hard to correlate Kant and Popper, at least I find it so, since their enterprises were so different. Kant's major epistemological concern was with what might be termed psychological epistemology. The forms of intuition (our sense of space and time) and the categories of the understanding (causality, quantity, quality, plurality, limitation, ...
2
Two understand why this is so, you have to understand the distinction between "false" and "incorrect".nYour question is caused by conflation of the two. I must admit that my usage of the two terms is not universal, but the distinction is common even when people use other terms for it (confusingly enough, textbooks on formal logic tend to use "false" for what ...
2
Before trying to ask 'How to measure', first you should confirm whether consciousness is a thing like light, waves, radiation etc. If you are quite sure about it, you may try to search a tool for measuring it.
We don't even know whether consciousness is zero dimensional, infinite dimensional or something more than these. Those who could realize ...
1
The discovery of gravitational waves in 2016 shows that spacetime is a physical object. The gravitational masses of two merging black holes act on spacetime.
Since 100 years we know the bending of light by curved spacetime. Hence spacetime acts on other physical objects like light rays.
Therefore, spacetime is accepted for membership in the ontology of ...
1
I believe this is one of the most important question in philosophy. It questions whether philosophy is worth doing.
The solution to the problem lies in seeing that Kant and Popper deny the possibility of certain knowledge, as does their tradition of thought. Thus the question of truth for them is confusing. This denial of the possibility of knowledge ...
1
To answer this, we have to know what "knowledge" means.
One answer is to take the radical skeptic's approach to say that we can know nothing, but that doesn't quite fit with the line of questioning you're after, so let's narrow the definition to assume that knowledge is something that can indeed be obtained.
The most common criteria for knowledge in ...
1
That change in species occurs is evident from both paleontology and Darwin's theory. It also seems, if one looks at it from a high enough level, that a progress-like pattern from simpler to more complex life forms appears to characterize it.
If there is real progress going on then an explanation for that would likely be some holistic cause effectively ...
1
Your link gives two definitions. The first is "Forward or onward movement towards a destination." Evolution is "movement", in the metaphorical sense, from earlier forms to later forms. If we take "forward" to mean "going from earlier to later", then evolution does indeed constitute progress.
But the second definition is "Development towards an improved or ...
1
Depends on what you would define as progress. If you say that being fit for nature, the environment, and getting more and more adapted to it, then yes, I would argue evolution is a progress of process. However, part of evolution is selection, and that includes the death and extinction of those living organisms, which are unfortunate enough to not be well-...
1
I can't really answer your question, but I would like to comment on one of the social sciences - political science.
Whether or not it really qualifies as a science is uncertain, because political science is in reality a combination of science (or "social science") and philosophy.
I like to break it down into more manageable pieces.
First, there's the ...
1
Apart from any (a)theological or biblical references, non-interventionist evolution and young-earth creationism (YEC) are comparable. Both make claims about the physical world, present and past, that in principle can be tested. The challenges of testing claims about the distant past applies to both.
Their key claims are comparable: (1) the evolutionary tree ...
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