New answers tagged

1

Logical entailments of what, in particular? Clearly Newtonian physics -- the set of logical entailments of Netwon's principles, is not included. Newtonian definitions clearly involve guessing and observation and are definitely not apriori. Logic may be how synthetic apriori content has to be related internally, since the illogical is not really "known". ...


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You are right to say there is an intuitive use of logic. Even little children can understand basic deductions and use logical operators in their speech. Consider a child who says, "If I go to bed, then tomorrow I can have breakfast AND desert?". While not the only example, clear a mastery of the conditional and the conjunction. It is a trope that children ...


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I will respond to your embedded answers from the POV of modern math. 1) You can get the gist of model theory by looking at problems that were solved by just this sort of model: Model theory is instantiated in modern abstract algebra. Beyond syntax and embedding in a known domain, it provides a third way of looking at patterns, by giving them a range of ...


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Model theory, as it is today understood, is a formal way to study how bits of language manage to represent the world. The fundamental idea of model theory is that you have a structure that assigns interpretations to bits of language in such a way that the structure makes each sentence in the language either true or false. You know this. The metalanguage thus ...


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Formalism would need a first-order logic. Wikipedia describes this as follows: First-order logic uses quantified variables over non-logical objects and allows the use of sentences that contain variables, so that rather than propositions such as Socrates is a man one can have expressions in the form "there exists x such that x is Socrates and x is a man" ...


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Alan Weir distinguishes between game formalism and Hilbert's formalism. He describes Hilbert's formalism as follows: The Hilbertian position differs [from game formalism] because it depends on a distinction within mathematical language between a finitary sector, whose sentences express contentful propositions, and an ideal, or infinitary sector. Where ...


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A list of more than three items could be viewed as a relationship where each item in the list is connected by "and". For example, "Today I spent $20 on food, $40 on housing, $10 on entertainment and $30 dollars on transportation." I could symbolize my spending (S) as one part for food (f), one part for housing (h), one part for entertainment (e) and one ...


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If you're asking about predicate valency then there are a few examples of tritransitive/quadrivalent verbs with four arguments. For example: I1 bet you2 five dollars3 that it would rain4.


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I can't quickly think of ones very obviously corresponding to a word in a natural language (such as "between" for ternary). The best I can think of is "surrounded by", since presumably (living on a two-dimensional surface) one needs at least three things to be surrounded by -- one step up from between! In principle one could make up a word for all kinds of ...


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Physics is about what exists in reality and how it behaves. Maths is about abstractions. To the extent that a particular mathematical abstraction has the same properties as some aspect of reality, that mathematical abstraction can be used to help us understand physics. Now, you write: Math concerned on pure universal truths and physics concerned on ...


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The issue here is not numbers, but our perception of things as whole entities. Our perception determines there's one or two clouds, but a cloud-unit is not something that exists physically. Clouds are just optical effects caused by a certain density of water molecules in a region, which is not physically determined, but it is bounded by our mind. In other ...


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Well, the question of whether numbers "exist" is kind of awkward when placed beside your intuitive understanding of what we mean when we say they "exist". This isn't a criticism; it's a perfect valid question to ask here. It just makes it tricky to disambiguate. I'll briefly explain three concepts relevant to your question, and that should clarify things. ...


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Do whole numbers other than zero actually exist? Do positive integers other than 0 exist? Most mathematicians believe so, and for good reason, it is fair to say that an integer like 1 corresponds to a quantity in the real world, coheres with other mathematical truths, and makes number systems work. As for the process of determining real numbers, they do ...


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are there any published, comprehensive studies that thoroughly explore the empirical foundations for the idea that deductive logic is a reliable tool for obtaining new knowledge of the "natural" world outside the formalistic framework of logic? As I see it, deductive logic is fundamentally nothing but a capacity of the brain. As such, there is no reason ...


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If we go back to the roots of mathematics — operations on natural sets (e.g. counting) and basic geometry — we can see that mathematics is based in the measurement of physical experience. Of course, the focus of study for mathematics quickly shifted to the more formal question of how we can systematically compare, relate, and transform measurements: thus the ...


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I'll leave my two cents here :) So I had a question also, much like the OP. But mine was, "IS there such a thing as a whole number? For example, the number 1 is made up of infinite small pieces/parts/decimal places/points. So in reality everything is an infinite fraction of tiny pieces of itself, so 1, is really ALL. There would be no reason to ever ...


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First of all there can be science without math. But let's leave that for now and stay with the physics-math pair. Nature of numbers : exact Vs inexact In math: The billionth or trillionth digit of pi is no less exact than 1,2,3,4 being 1,4,1,5 In physics: Constants are commonly used with hardly 1 significant digit g = 9.8m/s² And I can't imagine ...


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This is a great philosophical question, as you are sorting out what truth really is! This is a question whose answer depends on the metaphysics involved. For instance, what do you consider truth? There are general theories of truth (correspondent, coherent, pragmatic), and then mathematical theories of truth. Model theory of truth is essentially a ...


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Math concerned on pure universal truths and physics concerned on inferential thruth so how can we could explain physics with math? There is no intrinsic, fundamental difference between the theories produced in mathematics and the theories produced in physics. There is in this respect no fundamental difference between mathematics, physics and what people ...


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I'm not sure if Moore's paradox should count. It does "talk about things like truth," but probably not in the way you intend. It can be constructed from any simple declarative statement. Here is the presentation which SEP quotes from Moore originally: I went to the pictures last Tuesday, but I don’t believe that I did. The "I did" refers to the first ...


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tl;dr- Having i apples is just gibberish, like having ZOOOOM! apples is gibberish. That the gibberish includes math-related language isn't really relevant. Having i apples is gibberish. You can't have i apples for the same reason you can't have AHHHH-IS-THAT-AN-EEL-ON-YOUR-FACE?!?! apples: because the statement doesn't mean anything. It's literally just ...


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What does "logic has no notion of order" mean? "No notion of chronological order" is a specific instance of this, and still doesn't make sense. Logic itself has no notion of anything. Order is simply one of many concepts that logic can be applied to, as in defining this rule: "A comes before B" and "B comes before C", therefore "A comes before C". Logic ...


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Since time is an intuition and time is continuous, the continuum is an object in intuitionism. There is no problem with geometry, Euclidean or non-Euclidean, for instance. But because of things like Zeno's paradox, we have to accept our intuition fails to make the points on the real line clear as individual objects that can be constructed. There is ...


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Mathematics is not truth. Neither is it green, ten or auspicious. Mathematics describes truth, or what is believed to be truth. Some aspects of the truth it describes are not applicable to other aspects. You can have 6 apples, and green apples, but not 6 greens*. This is not a "fault" of mathematics. *In that 'example' the failure is due to improper ...


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It depends on the theory. The best definition of mathematics I've ever heard was "The study of precisely defined ideas". In mathematics, you have a set of axioms, which are just precisely defined statements you assume to be true, and you use logical inference to draw conclusions that are necessarily true given the axioms. There's nothing in principle that ...


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Can I prove a philosophical theory mathematically? If yes? How? For example, can the theory of materialism be proved mathematically? The crucial issue would be the standard of proof. The standard of proof in physics is not the same as the one in mathematics. In mathematics, you don't prove anything true, you prove them valid or not valid. In physics, ...


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Is mathematics a mental idea? Mathematics is an abstraction, an idea, a thought, a meaning and is not physical. According to this answer, a mental idea cannot exist without a mind. It depends on who you ask. Descartes would disagree with his dualism. Daniel Dennett would disagree with Descartes. Theology still tends towards dualism since it preserves ...


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"Why does mathematics work in the physical sciences?" I don't know if this is what the OP was looking for, but I think the existing answers are missing an important interpretation of this question. Mathematics provides only an approximation; it makes no claims of having a perfect correspondence with the real word. In Fourier analysis for instance, any ...


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One approach to mathematics as a mental idea is intuitionism. Here is Rosalie Iemhoff's description of it: Intuitionism is based on the idea that mathematics is a creation of the mind. The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves ...


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The way to falsify an hypothesis in mathematics is to provide a counterexample. One way to characterize an hypothesis as "good" is if there exists an effective method for finding a counterexample that could be used to falsify the hypothesis. Here are the criteria for an effective method from Wikipedia: A method is formally called effective for a class ...


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Nice question. It seems to me Materialism can be disproved mathematically. It is a form of monism that by reduction places all that exists in one set, and this immediately gives rise to a well-known paradox that prevents the theory from being fundamental. In philosophy the consequence for Materialists is a miraculous Creation event or an endless pile of ...


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From a mathematician, not a philosopher: We can use mathematics to model parts of reality. Part of what makes mathematics such a powerful tool is that the same mathematical construct can model many different physical phenomena, often in truly unexpected ways. For instance, natural numbers can count discrete things (we have 3 apples). The positive real ...


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If you mean, can a philosophical theory be proven exclusively by mathematics, then the answer is no. This is because the two bodies of knowledge cover potentially overlapping, but distinct domains of discourse. Often times arguments in philosophy, mathematics, and science overlap to some degree, so it is possible that some aspect of a theory might be ...


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Long comment Idea is a complex philosophical term. According to Descartes’ Theory of Ideas the mind is an existing substance, and thought or thinking is its attribute. An idea is a mode of thinking. In being a mode of thinking, an idea is understood as a way of being (an instance of) thinking, or an idea is way in which an instance of thinking is ...


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Lovely question Michael. If that @RodolfoAP answer is literally exactly true, science is based on math math is based on perceiving mind my perception (mind) is all = solipsism Since all modern life is science-based solipsism is the ultimate philosophy. As a secondary proof of Anything is true if we will it strongly enough (The Secret!) consider: ...


0

Yes and no - the nature of truth is to some extent a matter of choice. Let me explain: Yes: If a statement is derived, by means of mathematical logic, from a statement known to be true, then it is true. No: At the very foundations of any mathematical theory, we make choices about what we regard as true without proof: these are the axioms. They are, in most ...


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Counting, the way you are used to it, isn't real. I mean, it makes lots of sense in our current low-entropy universe, where distinct things that are pretty similar are around, and we have lots of brains to notice it. But 1 cookie, 2 cookies, 3 cookies -- that isn't a fundamental thing about our universe. There is a bunch of stuff. When ridiculously ...


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From the point of view of Intuitionism in mathematics, it is because if it did not work, the mathematics would be different. It would not just be the same mathematics differently encoded, because that is not different. It would be essentially different. There are two different philosophies of mathematics that cover the vast majority of people's views of ...


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In his article, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences", Eugene Wigler offers an explanation: (page 230) A possible explanation of a physicist's use of mathematics to formulate his laws of nature is that he is somewhat an irresponsible person. As a result, when he finds a connection well known from mathematics, he will jump ...


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This is a perennial question in philosophy, which should be taken as an indication of its importance. Note that to find a satisfying answer to some extent on your metaphysical presumptions. Needless to say, whatever your blend of rational and empirical thinking is will determine the approach you take. In analytical philosophy, statements from math and ...


1

I looked at reddit, and they said that it's not surprising it does, just because that's what it's there for. Well, that is the correct answer... just stated very simply. How did math start? It started with some basic problem that needed solving, probably of the type "I need to describe the size of some collection, because I need to convey this size to ...


0

Mathematics works only to the extent that it is logical. There is in this respect nothing specific to mathematics as compared to our other modes of representation. They will all work as long as our modelling remains logical. Language works. Diagrams work. Pre-linguistic thought works. Any model works, as long as it is kept logical. Thus, the value of ...


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I'm just trying to make sense of the success of applied mathematics... what it means... I'm asking whether its success suggests anything According to James, success suggests something is true, in agreement with reality: What does it mean to call a proposition or belief “true” from the perspective of pragmatism? This is the subject of James’s famous ...


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Mathematics itself isn't truth, but all its results can be said to be true. Everything in mathematics begins with a set of assumptions and definitions. All proofs are pure deductive reasoning based on those assumptions and definitions. Every proof implicitly or explicitly begins with "Assuming A, B, and C are true, then … .". There is no claim whatsoever ...


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I think it is a mistake to assume that there exists something like a context-independent notion of truth. Let me explain what I mean with the context dependence of truth. Consider the following simple question: Did Han shoot first? Now you can observe that in the real world, as far as we can tell, Han didn't exist at all. Obviously a person that doesn't ...


1

in mathematics there are purely imaginary numbers which cannot be represented directly in reality. I think the latter part of this statement is invalid. Hence I question the validity of this question entirely. This part of the statement to be precise "which cannot be represented directly in reality". What defines reality? Do you mean that the physical world ...


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tl;dr: Yes to pragmatists; no to everybody else: For them, mathematics is about correctness, not about truth. While it is true that mathematics obviously was — and, perhaps less obviously, still is — inspired by our (perceived) reality, it is one of the essential traits of mathematics that it quickly and rigorously abstracts from that reality.3 ...


1

As an engineer, I would say that if something can be proven to be useful, then it is true in some sense. This philosophical stance that something being "true" is related to positive consequences or outcomes can be produced / derived from it has some specific fancy-pants Latin name which I knew when I was younger, but sadly seem to have forgotten now. ( ...


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Ill formed question. Mathematics (specifically, logics) define what truth is. You are trying to test the validity of the tool with the tool itself. The answer would be a plain "yes". Otherwise (if you discuss mathematics as an issue of perception) you fall into Rusi's answer. Yes, you can have i apples, if you define the domain of i (i is not just a ...


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“You can’t have i apples” As @Conifold points out you cannot even have √2 apples. I'd go further. Can you have -2 apples ⅓ apples? I'd say (from a certain pov) no. All physics is based on measurements All measurements come from instruments Which can only ever deliver integral non-negative bounded multiples of least count Note: I added the “bounded” ...


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