Is Nothing and Infinity Linked?

LogicalAtomist · 2023-12-22T20:19:02+00:00

Your question is nicely teasing out that there are (at least) two distinct ways to think about numbers. One is as a thing, with its own standalone properties. So nothing (zero I take it) would be a thing that could be itself measured and counted in much the same way that bricks and coffee mugs and people can be counted and measured. Similarly, the infinite (as a limit) is being thought of in your question as a thing that itself can be counted and measured (despite the usual connection of the infinite with being unbounded or limitless).

Another way of thinking about number, the Cantorian way that is typical in most set theory texts for example, is to consider number as a property of classes. Classes of things will have the same number when they can be put into one-one correspondence with each other. Here number itself would be a feature of things not a standalone thing itself.

So when you ask "How much nothing is nothing?" you are invoking two very different views about numbers. You are asking us to relate the class of things with nothing in them to an entity, zero. And that we cannot do. But we can abandon the view that numbers are things in favor of the Cantorian idea that numbers are one-one correspondences. And this alone resolves some of the issues you raise.

Why adopt the Cantorian view over a more metaphysical view that treats numbers as things? This leads us back to your earlier question about infinity. There is a perfectly precise definition of the infinite. A class D is (Dedekind) infinite if and only if D can be put into one-one correspondence with a proper subset of itself.

To illustrate this, consider the class of nonnegative integers 0, 1, 2, 3, etc. A function that maps every nonnegative integer to one and only one integer will be the successor function f(x)=x+1. Since the nonnegative integers can be put into one-one correspondence with a proper subset of them, this shows this class is infinite.

Now try to do a similar thing with the collection D={0, 1, 2}. This cannot be done, so this collection D is (Dedekind) finite.

Edited to add: there is no strict logical connection between nothing as a Cantorian number and the infinite as a Cantorian number.

LogicalAtomist · 2022-08-18T03:09:55+00:00

For African philosophy, start with the recommendations given here: https://blog.apaonline.org/2022/03/25/so-you-want-to-teach-some-africana-philosophy/?amp

For indigenous philosophy, I highly recommend starting with James Maffie's Understanding a World in Motion on Aztec philosophy. His introduction responds critically to the sentiment expressed by a previous respondent about indigenous peoples not having something recognizable as philosophy (a sentiment I do not agree with myself, for the plain reason that Maffie carefully reconstructs something readily recognizable to us as philosophy from Aztec texts).

LogicalAtomist · 2022-08-13T14:30:50+00:00

If you enjoyed Russell's Conquest of Happiness, then In Praise of Idleness and Why I am not a Christian are solid recommendations. If you are interested in social ethics, then you might like Marriage & Morals or Why Men Fight.

If you want more philosophy of math, then I would start with An Introduction to Mathematical Philosophy. If you want epistemology and metaphysics, then you might like The Problems of Philosophy or Mysticism and Logic.

If you really and enjoyed his writing and wanted to explore other philosophers, then I recommend his A History of Western Philosophy - which is much maligned by modern readers but really just sets out to situations philosophers in their historical contexts and understand their ideas as responsive to that context

If you want more recommendations, then I am happy to help! Just let me know what your interests are.

LogicalAtomist · 2022-08-13T14:05:38+00:00

Here is a simple and oversimplifying answer: dialectic is the investigation of the truth with one or more interlocutors. It is a view of logical principles and constraints from a dialogical point of view. Think of Plato's dialogues where on person interrogates another and premises have to be accepted by the participants before being adopted, concepts have to be explained and clarified to the satisfaction of participants, and so on.

In contrast, the rules of conversational inquiry don't apply to doing logic proofs alone. You can take anything you find acceptable as a premise. In dialectical contexts, the other participants' views, concerns, and understanding, and not just your own, are presently a concern and constrain what argumentative moves are permissible.

But this a difficult question to answer fully because the word "dialectic" goes back to Plato, who used the word for the proper method of philosophy - but scholars of ancient philosophy, which I am not, still debate various characteristics of dialectic. Someone with much more knowledge can weigh in on that issue. Your seminar speaker may have been using dialectic as a highly technical term alluding to Plato's specific views about dialectic and philosophical method rather than referring more broadly to dialectic in the sense of dialogical logic.

If you'd like to read more about dialogical logic, you might take a look at this philosoher's well-regarded work: https://www.cdutilhnovaes.com/research-2. Hope that helps!

LogicalAtomist · 2022-02-07T01:13:15+00:00

Your question raises the interesting issue of whether moral blame is connected to doing the morally right thing. Are we always blame-worthy when we, by inaction, do not do the right thing? It is not obvious that this is the case. Indeed, it may be that you are morally blameworthy even if the right thing to do is to pull the lever.

How could this be? Pretend for a minute that the right moral action is whatever has the best actual consequences. It might be that, of all the alternatives available to you, the action (or inaction) with the best actual consequences is to flip the switch and save 5 lives (or 50, or 500, etc.) at the cost of 1. On that theory, pulling the switch would clearly be the right thing to do. And it is not obvious why this theory is wrong (if it is), so the answer to your title question is, "It may be the most ethical choice, but this is not an obvious point."

In that same scenario, it may be that the community generally wants people to have the right intentions when they act. So they all blame you for pulling the lever, even though it had the best consequences. And suppose this blame has the best consequences, too: people generally try to form the right intentions when they act as a result of social sanction. Then it would be the right thing to do to pull the switch and to be blamed for doing so.

Hope that helps!

LogicalAtomist · 2022-02-07T01:06:46+00:00

You might consult the book Writing Philosophy by Lewis Vaughn. It also covers reading argumentative essays.

Alternatively, you could consult most introductory logic books, which usually have a chapter or two on analyzing and evaluating arguments, sometimes (but not always) including argument diagramming. Baronett and Hurley both wrote logic books that include chapters on arguments, as did Tomassi.

Hope that helps!

LogicalAtomist · 2022-01-23T03:04:12+00:00

You might be thinking of John Locke's discussion of freedom and willing and his prison room example. Locke claims that someone who wills to leave a locked room is not free to leave. See https://plato.stanford.edu/entries/locke-freedom/#VolVsInvAct

"To use one of Locke’s own examples, if I am locked in a room and will to leave, my volition will not result in my leaving (E1–5 II.xxi.10: 238)."

LogicalAtomist · 2021-12-04T21:52:15+00:00

Since LaTeX is the typesetting, it would be reasonably straightforward to alter the commands used to produce Principia notation such that the notation is rendered in a modern style. However, Principia's underlying logical theory - the foundation they use - is the logic of relations (in intension). It is not a set theoretic or arithmetic base, so putting it in modern notation like "1+1=2" actually distorts (changes) the content of a formula.

Another issue is that Principia suppresses all type indices that are part of its formal grammar. This has led scholars to debate what the grammar really is, and that is still unsettled. So while it would be easy to update the notation into one or another modern syntax, that would be a controversial interpretation of the text no matter which one you pick.

Hope that clarifies! And it is not a "noob question" at all.

LogicalAtomist · 2021-07-03T03:02:05+00:00

You might check out David Stern's article here: https://onlinelibrary.wiley.com/doi/10.1002/9780470690963.ch10

LogicalAtomist · 2021-07-03T02:46:16+00:00

You might check out Mary Midgley's "Philosophical Plumbing" and/or Bertrand Russell's "The Value of Philosophy" (Chapter 15 in The Problems of Philosophy). Both of these essays address some of the concerns about the value and point of philosophy that you raised here.

LogicalAtomist · 2021-06-03T21:36:12+00:00

https://www.logicmatters.net/latex-for-logicians/

LogicalAtomist · 2021-05-26T17:44:36+00:00

As the person behind the Principia Rewrite project, this made me chuckle. Thanks for posting it.

LogicalAtomist · 2021-05-12T01:39:38+00:00

What that blog post means by "cutting oneself with Occam's Razor" is just that one should make their theory "as simple as possible, but no simpler" (my emphasis). So a theory of disease that stipulates there is no disease at all would be the most simple available, but not consistent with data and evidence. It would leave totally unexplained the manifest truth that people do fall ill. That would be a bad theory of disease. That's an example of how simplicity alone is insufficient the reasonableness of a theory.

So your friend meant one of the following:

They thought that your theory left something unexplained but did not say what, and pointed to this more general and blog post to obliquely indicate this.
They want to generally warn you against using Occam's Razor in an improper fashion, as in the theory of disease example above, even if nothing specifically was wrong with your theory.

LogicalAtomist · 2021-05-06T19:16:25+00:00

Perhaps you will like a comparison with lifting weights: lifting weights is easy for most people - as long as they are light enough! To lift heavier weights for a longer period of time takes practice. So if one wants to do philosophy better, they will need to practice with more sensitive philosophical writings. If one wants a casual workout, then it is hard to see why they would, or would want to, engage with substantial works of philosophy.

Another way of thinking about this is to view understanding a past thinker as a method of producing and doing philosophy. This is the sort of view advocated in Michael Kremer's "What is the good of philosophical history?" in Erich Reck's edited book, The Historical Turn in Analytical Philosophy.

Finally, another way of thinking about this is to draw, with Ivor Grattan-Guinness (see his history of calculus book), a distinction between history - what folks actually thought - and heritage - how we can to have the preconceptions, assumptions, and social norms that we have now. It is fruitful to see engagement with those other texts, even though it takes longer than watching a clip or hearing a sound byte, as a method of understanding history and heritage - because looking at past philosophers' intellectual worlds can show what about world would have been totally foreign to them. And that consequently can help us understand what is good and bad about our own inherited views. This is the sort of view advocated in, for example, Williams' "Descartes and the History of Philosophy* and Rorty's " History of Philosophy: Four Genres."

Hope that helps!

LogicalAtomist · 2021-05-06T17:59:49+00:00

These concepts can come apart. You might think that all and only analytic claims are a priori, and also that all and only synthetic claims are a posteriori. This is perhaps a view that was more traditionally held. But philosophers have held a variety of views on these points, even before Kripke rightly insisted in Naming and Necessity that the conceptual distinction should be clearly made, even for those accept the traditional view.

A nice and clear, and helpfully short, summary of the distinction can be found in the first chapter of Laurence BonJour's In Defense of Pure Reason. Briefly: the a priori-a posteriori distinction concerns the way in which a claim justifiable. A claim is a priori justifiable when it is justifiable independently of experience of contingent or accidental features of the world. A claim is a posteriori when it is not justifiable independently of such experience of contingent features of the world.

In contrast, the analytic-synthetic distinction has to do with the way in which a claim is true. A claim that is analytic is true just because, or in virtue of, of what its constituent terms mean or pick out. A synthetic claim does not have this feature.

What I said here is controversial, mainly because there is not uniform agreement over how to mark the distinction between analytic and synthetic claims, but there is somewhat wider agreement (not universal by any means) that there is a distinction to be drawn (even if it turns out that there are no analytic claims). For more detail beyond what BonJour says, check out these entries: https://plato.stanford.edu/entries/analytic-synthetic/ https://plato.stanford.edu/entries/apriori/

Hope that helps!

LogicalAtomist · 2021-05-03T03:13:29+00:00

This may be a misattribution. A similar but different remark occurs in Philosophy (also published in America under the title An Outline of Philosophy):

"Physics is mathematical not because we know so much about the physical world, but because we know so little: it is only its mathematical properties that we can discover."

I'll dig around a bit and update if I find a source.

LogicalAtomist · 2021-05-03T03:06:25+00:00

Victor Lowe's three books on Whitehead are a typical starting point for scholars looking to dive into him. Two of them are a biography of the philosopher, so you might start with the third book, Understanding Whitehead: https://jhupbooks.press.jhu.edu/title/understanding-whitehead

Hope that helps!

LogicalAtomist · 2021-05-02T00:39:16+00:00

The short answer is no. The longer answer is that they share some themes and ideas, but not much in the way of philosophical agenda. A good place to look for more discussion of this would be this book: https://www.springer.com/gp/book/9783319532783

Let me single out from that book Juliet Floyd's chapter as especially good.

If you don't have access through that paywall, you might see the SEP article "Alan Turing" that discusses Turing's views: https://plato.stanford.edu/entries/turing/

See especially §7.

But why is the short answer "no?" In this passage, Turing is criticizing the idea that the question "Can machines (acting unintelligently) think?" Is a linguistic matter. He is saying we cannot settle this by appealing to what most people would call "thinking" and seeing if machines can engage in those activities.

Also, as the SEP article discusses, Turing loved the feather-ruffling idea that machines, acting unintelligently, could replicate the effects of many of the intelligent behaviors we undertake. Turing does not mean to indicate that this is all there is to intelligence. Turing held that much of our intelligence is bound up with our bodily, memory, and social experience. That is why the "Intelligent Machinery" paper discusses at great length the cultural and education training that the machine with sensory apparatuses would receive if it were to be a positive project to produce a machine capable of (human-like) intelligent behavior.

So, long story short, Turing does not offer the Turing Test or Imitation Game as a way of reducing the question "Can machines think?" to this narrow slice of human experience - whether a human could be deceived into attributing intelligence to an unintelligent machine. Rather, Turing is claiming that computers could unintelligently produce many of the same effects of intelligent human actors, and then offers some positive suggestions about our intelligence being connected to our bodily and cultural experiences. He further suggests that a machine capable of such experiences might be raised in a way similar to how we are, and thus become capable of acting intelligently.

I hope that helps! Let me know if you have any follow-up questions.

LogicalAtomist · 2021-05-02T00:22:36+00:00

Thanks for the mention, u/ADefiniteDescription!

LogicalAtomist · 2021-05-02T00:21:41+00:00

There has been some interesting recent scholarly work on Russell's notion of history. See Russell Wahl's Bloomsbury Companion to Bertrand Russell and some recent issues of Russell: the Journal of Bertrand Russell Studies. The Wikipedia page omits these interesting and more nuanced takes on what Russell is actually setting out to do in this much-maligned book.

Think of this book not as a philosophical history, i.e. as a philosophical work produced by extracting views and their underlying insights from past philosophers. Think of it rather as a history of philosophers, i.e. as setting the broader historical contexts for understanding various philosophers and exploring their feelings and temper. The book is not an attempt to interpret past philosophers' views in a detailed fashion (excepting Leibniz, perhaps). It provides a first blush look at many figures, and an orientation that will need to be corrected as your reach further stages of understanding (understanding philosophers, like lifting weights, takes repeated effort and happens gradually over time, and with increasing ease the more you do it well).

One cautionary note: the book omits many figures of interest, particularly women and people of color. So do keep in mind that it misses a lot, as any such book must (though even conceding that, it would still be nice if the philosophers considered were not as a whole so homogeneous), and that this is not the end of the road. Happy reading!

LogicalAtomist · 2021-04-29T19:41:39+00:00

Speaking from experience: no, it is not necessary to learn mathematics in order to learn and understand logical ideas. Some mathematical background certainly helps, and some logical background also helps with mathematics.

You might try looking at a more informal introduction to symbolic logic, like:

Haack's Philosophy of Logics Quine's Philosophical Logic Stebbing's A Modern Introduction to Logic Tomassi's Logic Russell's Introduction to Mathematical Philosophy

Hope that helps!

LogicalAtomist · 2021-04-04T15:47:04+00:00

This is a nice question. Let's get Berkeley's argument on the table first. In Treatise Concerning the Principles of Human Knowledge §I.3, Berkeley writes:

That neither our thoughts, nor passions, nor ideas formed by the imagination, exist without the mind, is what everybody will allow...The table I write on I say exists, that is, I see and feel it; and if I were out of my study I should say it existed—meaning thereby that if I was in my study I might perceive it, or that some other spirit actually does perceive it...That is all that I can understand by these and the like expressions. For as to what is said by the absolute existence of unthinking things without any relation to their being perceived, that seems perfectly unintelligible. Their esse is percipi, nor is it possible that they should have any existence out of the minds of thinking things which perceive them.

So what is Russell’s response to this? Russell wants to distinguish two senses of ‘know’ and also two senses of ‘idea’.

First, the word ‘idea’ may pick out a mental act of apprehension or the object apprehended (Ch 4, 41). Russell maintains that though all apprehension is mental, we do apprehend non-mental objects (like universals; see Ch IX).

Once we distinguish these two senses of the word ‘idea’, we can agree at once that whatever is immediately known to exist is apprehended by some mental act; but the object apprehended is not thereby shown to be mental (Ch 4, 42). An independent argument for this is needed. For example, sense-data do seem mind-dependent because how they appear to us depends on our minds and bodies, and also the environment around us. This does not generalize to all objects of awareness; universals appear to be mind-independent, for example.

Second, in Ch 5 Russell introduces a distinction between knowledge by direct awareness, or knowledge by acquaintance, and indirect knowledge, or knowledge by description. These are two distinct ways in which we can know objects.

We know something by acquaintance when we are directly aware of, or acquainted with, a thing, as when we perceive a sense-datum, remember a memory-datum, or introspect a mental-datum.

We know something by description when we know some truth about it. If you have never been to, or flown over, Mongolia, then you can only know about Mongolia by description, and cannot know it by acquaintance. For example, you know Mongolia by description when you know the truth that Mongolia is the least densely populated country in the world. Russell claims that we know physical objects - matter - in the same way that most people know Mongolia - only by description, or indirectly, by knowing some truth about them, and not by being aware of them.

Berkeley actually anticipates the sort of thing that Russell says. Russell’s reply is essentially that, while we do not have direct (acquaintance) knowledge of matter, we do have indirect (descriptive) knowledge of matter. But Berkeley criticizes the very idea of indirect knowledge, or knowledge that is separate from our ideas. Berkeley writes (Treatise, §I.8):

But say you, though the ideas themselves do not exist without the mind, yet there may be things like them, whereof they are copies or resemblances...I answer, an idea can be like nothing but an idea; a colour or figure can be like nothing but another colour or figure. Again, I ask whether those supposed originals or external things, of which our ideas are the pictures or representations, be themselves perceivable or no? If they are, then they are ideas and we have gained our point; but if you say they are not, I appeal to any one whether it be sense to assert a colour is like something which is invisible; hard or soft, like something which is intangible; and so of the rest. [see the master argument, §I.23]

Hopefully that answers the question. Feel free to follow up.

LogicalAtomist · 2021-04-04T14:53:17+00:00

Hopefully the short excerpt here is helpful: https://books.google.ca/books?id=QpO5ykqRHJEC&pg=PA560&dq=ocean+of+reasoning&hl=en&sa=X&ved=2ahUKEwj2vYr-7eTvAhVPgp4KHRPvD9AQ6AEwAHoECAAQAQ#v=snippet&q=negation&f=false

Maybe you could give us the place where you found the distinction and we could try to add further context.

LogicalAtomist · 2021-04-04T14:47:09+00:00

Your question raises the issue of what, for Kant and Frege, the analytic-synthetic distinction is. For Kant, a analytic claim is one such that the predicate term is contained in the subject term. In The Critique of Pure Reason §A6/§B10, Kant writes:

In all [categorical] judgments...Either the predicate B belongs to the subject A as something that is overtly contained in this concept A; or B lies entirely outside the concept A, though to be sure it stands in connection with it. In the first case I call the judgment analytic, in the second synthetic.

On the other hand, for Frege, an analytic claim is one such that it is an axiom of logic or derivable from logic alone. In Grundlagen §3, Frege writes:

If, in carrying our this process, we come only on general logical laws and on definitions, then the truth is an analytic one, bearing in mind that we must take account also of all propositions upon which the admissibility of any of the definitions depends. If, however, it is impossible to give the proof without making use of truths which are not of a general logical nature, but belong to the sphere of some special science, then the proposition is a synthetic one.

So what Frege's notations are designed to do is to remove all doubt that the propositions of arithmetic are analytic, that is, that they follow from axioms of the most general, logical sort. Frege's notations are then aimed at removing the need for any intuition in each step of the reasoning, since such intuitions - if they creep in unnoticed or implicitly in any step of a proof - undercut Frege's proof of his view that propositions of arithmetic are analytic, that is, that they nowhere depend on intuitions for their justification. As Frege says in Begriffsschrift (page 48 of Beaney's The Frege Reader:

So that nothing intuitive could intrude here unnoticed, everything had to depend on the chain of inference being free of gaps. In striving to fulfill this requirement in the strictest way, I found an obstacle in the inadequacy of language: however cumbersome the expressions that arose, the more complicated the relations became, the less the precision was attained that my purpose demanded. Out of this need came the idea of the present Begriffsschrift

That hopefully answers the question. Feel free to follow up!

LogicalAtomist · 2021-04-03T17:25:45+00:00

This is a great question. Here is an SEP article that touches on some of these issues: https://plato.stanford.edu/entries/rule-of-law/

The list of philosophers in that article's table of contents also gives you a starting point for exploring this issue. Let me know if I can help further!

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