Questions tagged [axiom]
The axiom tag has no usage guidance.
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Basic truths as self-justified or parajustified
Some foundationalists maintain that basic truths are self-justifying, which means they are allowing, in some exceptional cases at least, a form of circular reasoning; petitio principii or begging the ...
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Reference request : Defining concepts
Shafi Goldwasser's interview
In the video attached, Goldwasser talks about course(s) on abstract subject matter like definition of concepts, definienda, etc. I am looking for a link to course/book ...
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Could we use the foundation axiom to generate counterexamples to almost any substantial axiom?
Here's the argument scheme I have in mind ("F" refers to a substantial/positive property/description; negative qualifiers like "inaccessible" do not sustain this scheme correctly):
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Filling in the gaps in an erotetic argument for pluralism about the Continuum Hypothesis
Syntax assumption/stipulation. I have decided to work with an erotetic function that is parenthetical. So I will not start with some proposition A and then have A? as its associated question, but I ...
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Can there be a solution to these three problems?
I have read many times that some problems or logical propositions do not have solutions or are outright impossible. These are three examples of such problems:
[The Russell's paradox] which is deemed ...
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Do set theories have inconsistency strengths, on top of consistency strengths?
Caveat: this question is fairly technical in nature, and I have reason to believe it would be more fitting for the MathOverflow, especially in terms of potentially informative responses (there are ...
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Is the fact that ZFC implies that 1+1=2 an absolute truth?
This question is somehow of a follow up to to this other one, and it's something that has bugged me for a while.
I understand the notion that there's no "absolute truth" in math, in the ...
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If axioms are subjective, how could anything be objective?
Axioms are subjective (?), and, since propositions are based on axioms, isn't everything subjective? (of course, the answer should be from the perspective of someone who believes in objectivity)
Or am ...
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Does Tarski Indefinability theorem impose a computational lower bound on the axiomatization of the reality?
Based on the Tarski's Indefinability Theorem (TA in standard model is not arithmetic (no FOL formula can represent TA,
a formula represent a predicate relation definition: under Tarski's first order ...
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Is the axiomatic method an inherently well-founded method?
It occurred to me a little while ago, that there is a trichotomy in set theory that maps to the positive solutions to the problem of the regress of inferential reasons. Namely, well-founded sets map ...
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The Anthropic principle as a physical embodiment of the Axiom of Choice
This question pertains to physical reality, but it would be deemed inappropriate in Physics.SE because of its speculative, metaphysical nature. It also touches on the subject of singling out elements ...
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Is there a proof that we can't prove a physical theory?
I am thinking of physical theories (e.g. Newtonian Mechanics) as axiomatic systems. We have a list of axioms and from there we can derive theorems, make predictions etc. If the prediction don't agree ...
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What does it mean for one geometrical axiom to be considered _equivalent_ to another geometrical axiom?
What does it mean for one geometrical axiom to be considered equivalent to another geometrical axiom?
For example consider Playfair`s axiom:
In a plane, given a line and a point not on it, at most one ...
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Are these valid examples of axiomatic statements?
I'm trying to understand if a couple of statements would be considered axiomatic:
Example 1: "murder is the unjustified killing of a person; if there was a murder, then a person was killed ...
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Axiom of Choice: correspondence or derivability?
I'd like to ask about a specific impression that I have about issues concerning the Axiom of Choice.
It seems to me that either one claims that the axiom is an obvious fact about the modelled concept (...
user46880
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In what way, if any, can the axioms of set theory be themselves well-ordered?
In "Independence and Large Cardinals", Peter Koellner writes:
... it turns out that when one restricts [attention] to those theories that "arise in nature" the interpretability ...
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Are pursuing the well-being and reducing the suffering of sentient beings objectively good things?
I think most people intuitively agree that increasing their own well-being and minimizing their own suffering are the right things to do. Everyone wants to be happy, enjoy a good health, etc. The ...
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How to construct logic from nothing as a step by step procedure? [closed]
I am thinking of constructing logic from scratch. I tried the law of thought as a fresh start but not totally convinced this is a correct way for the following reason:
if law of identity is the first, ...
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Do contemporary logicians generally claim that classical logic can be simply reduced to these 5 logic principles?
Do contemporary logicians generally claim, as Wikipedia does, that classical logic can be simply reduced to the 5 logical principles below? Or is it more complex than that and are there principles not ...
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What is the current status of Foundation-of-Mathematics programmes?
I have been reading 'A Very Short Introduction to Mathematics' by Timothy Gowers and at one point he mentions that most of the mathematical proofs can be finally resolved to a set of logical ...
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Is faith required to believe any axiomatic assumption the scientific method is built upon?
It's my understanding that the scientific method builds upon certain axiomatic assumptions, such as uniformitarianism and the principle of induction. Is faith required to believe these axiomatic ...
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Is it possible to create an axiomatic system where 1+1 doesn't equal 2? What would be the consequences of such a system? [closed]
1+1=2 is a result (perhaps arguably more of a definition than a theorem?) of Peano Arithmetic, as well as other systems such as ZFC. I understand that 1+1 doesn't necessarily have to equal 2 if we ...
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Why do we rely upon scientific approach when its foundational axioms are assumed and agreed without proof?
Why do we rely upon scientific approach when its foundational axioms are assumed and agreed without proof?
Foundation of the scientific explorations are seem to be the mathematical axioms at its root....
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Do philosophers generally reject that philosophical reasoning relies on axioms?
The way I've always thought that philosophy worked is that philosophers have a certain set of tools (deduction, laws of thought, basic sources of knowledge) which they use to come to reasoned answers ...
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In a deductive reasoning system, what happens if we have unfounded axioms? [closed]
What if our axioms are false? What happens then?
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What are the most rational basic beliefs?
I understand that this question might be difficult or even unresolved. But within a foundationalist view of knowledge, has anyone proposed a set of basic beliefs that seem to be the most rational for ...
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Do complex quantities and irrational numbers exist in nature?
Completeness Theorems of Model Theory, a branch of Mathematical Logic. Together, these two Theorems show that: under the Field Axioms (the rules of the game for scalars) existence of rationals is ...
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Is there an infinity of axioms in mathematics?
As I was trying to find a list of mathematical axioms used in modern branches of mathematics, I wondered if there's any meaning to the question of "how many mathematical axioms are there ?", and then ...
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What are the axioms of personalism?
I've read some authors whose works are held as belonging to personalism.
However while their works elaborated based on principles of personalism, they never clearly explained what the axioms were ...
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How do philosophers formally characterise mathematical objects?
In the Stanford Encyclopedia of Philosophy article, 'Platonism in the Philosophy of Mathematics', the following formalisation is given for the existence of a mathematical object:
Existence can be ...
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Does reality have axioms?
Mathematics is considered the queen of sciences as it allows us to build simplified but functional models of the reality that surrounds us.
However, I do not understand if this isomorphism could be ...
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What kinds of proofs can be given for axioms, e.g. the modal axiom S5?
From John Bigelow and Robert Pargetter's book, titled, 'Science and Necessity', they assert the following:
. . . . The resulting system, S5, contains all the theorems of S4 and all the theorems of ...
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Is it possible to prove that a particular statement cannot be disproved without creating a contradiction?
In the following link (http://www.importanceofphilosophy.com/Metaphysics_ExistenceExists.html) the authors are basically arguing that there are statements that we cannot deny without contradicting ...
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What does it mean for a statement if we cannot disprove it?
In the following link (http://www.importanceofphilosophy.com/Metaphysics_ExistenceExists.html) the authors are basically arguing that there exists some truth that we cannot disprove by any other ...
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How do we call a statement that is unthinkable for any person to not be the case? [duplicate]
The example of such an unthinkable or unimaginable thing for a person could be non-existence, therefore argument against existence seems to be so absurd. Aren't we calling such things axioms or ...
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Do we have a name for the following axiom?: We can never know for sure whether we know everything that exists
Let's assume that our words and sentences are able to describe the truth. It is clear that whatever we know - even if we have knowledge about an entire universe and every position and momentum of its ...
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Axiomatization of philosophy?
In mathematics, many theories are built on assumptions that are taken to be true, and they are most often called axioms, and then, with the help of logical laws and definitions and with various ...
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Is constitution of a country simply a set of axioms?
Is it valid to think of a constitution or law in general as an axiomatic system?
Because what they do is actually stating some rules one-by-one which we just accept. This means we accept also all ...
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Term for the idea that regardless of our philosophy, only the observable/physical matters
I'm looking for terms that define the following presuppositions:
every action should be valued based on its outcomes, not choosing is a choice
impact of an action must be valued based on the ...
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Can social sciences have a non-subject -related axiomatic foundation? Why not?
Can social sciences have a non-subject -related axiomatic foundation? Why not?
Problems:
Interpretation is always relative to subjective interpretation. Which is not similar in natural sciences. In ...
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Are axioms in mathematics comparable to hypotheses in experimental sciences?
Remark: my question deals more particularly with the axioms of set theory, arithmetic, probability theory, etc. I think the status of the axioms in geometry is clearer.
The French fictitious ...
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What is the difference between depth and surface information?
I was looking for an answer to this question: Was Euclid's method of proof axiomatic?
While doing so I ran across an abstract of Jaakko Hintikka for an article "What is the axiomatic method?" ...
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Your irreducible ground truth [closed]
What is the ground you build your arguments on? By this I mean something that would be the ultimate base for your reasoning, something irreducible, a (self) evident and therefore unprovable fact, an ...
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Can a solid theory ever exist without any axioms?
In math, numbers and addition are logically defined by Zermelo Set Theory, a small group of axioms upon which everything else can be built. Could it be possible to have a working theory, (in any field ...
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What are the assumptions adopted by the scientific community?
What are the core assumptions of the modern scientific community with which they use to view the world and formulate theories etc?
By assumptions I mean premises taken as fact (about the universe/...
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Are axioms assumptions and should they be minimized?
Are axioms nothing but assumptions and, if yes and in accordance to Occam's Razor, should they be minimized?
When postulating scientific theorems which, unlike axioms, are subject to the scrutiny of ...
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