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Approaching Infinity
Approaching Infinity addresses seventeen paradoxes of the infinite, most of which have no generally accepted solutions. The book addresses these paradoxes using a new theory of infinity, which entails that an infinite series is uncompletable when it requires something to possess an infinite intensive magnitude. Along the way, the author addresses the nature of numbers, sets, geometric points, and related matters.
The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes.
The book addresses the need for a theory of infinity, and reviews both old and new theories of infinity. It discussing the purposes of studying infinity and the troubles with traditional approaches to the problem, and concludes by offering a solution to some existing paradoxes.
294 pages, Kindle Edition
First published March 23, 2016
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Displaying 1 - 10 of 10 reviews
February 6, 2021
Huemer seeks to clarify the concept of infinity - what it is, what it isn't, and how it applies to reality. While infinity is used in mathematics, the philosophical implications are less clear. This is evident by the myriad paradoxes created throughout history (Zeno's paradox, Hilbert's Hotel, and Thomson's Lamp, to name a few). Huemer's hope is that by coming up with a better approach to infinity, he can more appropriately distinguish between viscious infinite regresses and acceptable infinite regresses - which in turn could help resolve at least most of the paradoxes proposed about infinity.
The book is an equal blend of profound and frustrating. Huemer's overall epistemology (Phenomenal Conservatism) presumes that it is reasonable for one to affirm that reality is as it seems. This means that one is justified in holding a position if that position seems right to them and they don't have defeaters for it. Huemer's chapter discussing this view is brief but insightful. It also helps make sense of his reasoning process throughout the book.
Another piece of significant insight comes from Huemer's overall philosophy of mathematics. Huemer argues that mathematical sets don't exist - one of the most controversial claims of this book - and in doing so undermines several basic axioms of mathematics as they apply to set theory. He also argues against the idea that space space is made up of physical points. His ideas may be novel, but they are at least thought provoking.
When it comes to infinity, Huemer ultimately proposes that infinity only results in metaphysical problems when whatever is infinite is metrical in nature. This means anything requiring an infinite measure of something (mass, motion, etc.) is metaphysically impossible because the physical implications of the scenario are nonsensical. This is a strange line of reasoning, to say the least. The way Huemer applies this to the paradoxes seems like he's bypassing the metaphysical concerns by only considering the strictly physical scenarios. Huemer denies this, but his argument is less than convincing.
Part of the problem with Huemer's arguments is he often dismisses objections or positions with little discussion because they seem to him to be obviously wrong. For example, he dismisses any argument against the actual infinity of space or time because space and time seem infinite to him. But he doesn't argue for this position at length. This may be a side effect of his Phenomenal Conservatism that - ironically - also makes it easy to dismiss some of his arguments because he takes for granted assumptions that may seem obviously false to many people.
Another weakness of this book is Huemer's side comments about arguments for the existence of God (and even the concept of "God" more generally). Sporadically he makes comments about the idea of God being nonsense, but his reasoning appears to be based on superficial misunderstandings of what philosophers of religion typically mean when they talk about God. Personally, it made me question how seriously he engages views that he disagrees with before critiquing them.
Overall, the book is worth reading and enjoyable. With that said, most discussions are highly technical, so if you've never taken a course on set theory or transfinite mathematics it may be difficult to follow large portions of the book. Unfortunately these portions are integral to the overall argument. But if you can handle the mathematics, I recommend you give this book a read. Agree or disagree, you'll come out with a better understanding of what infinity is (or isn't) and why that matters.
The book is an equal blend of profound and frustrating. Huemer's overall epistemology (Phenomenal Conservatism) presumes that it is reasonable for one to affirm that reality is as it seems. This means that one is justified in holding a position if that position seems right to them and they don't have defeaters for it. Huemer's chapter discussing this view is brief but insightful. It also helps make sense of his reasoning process throughout the book.
Another piece of significant insight comes from Huemer's overall philosophy of mathematics. Huemer argues that mathematical sets don't exist - one of the most controversial claims of this book - and in doing so undermines several basic axioms of mathematics as they apply to set theory. He also argues against the idea that space space is made up of physical points. His ideas may be novel, but they are at least thought provoking.
When it comes to infinity, Huemer ultimately proposes that infinity only results in metaphysical problems when whatever is infinite is metrical in nature. This means anything requiring an infinite measure of something (mass, motion, etc.) is metaphysically impossible because the physical implications of the scenario are nonsensical. This is a strange line of reasoning, to say the least. The way Huemer applies this to the paradoxes seems like he's bypassing the metaphysical concerns by only considering the strictly physical scenarios. Huemer denies this, but his argument is less than convincing.
Part of the problem with Huemer's arguments is he often dismisses objections or positions with little discussion because they seem to him to be obviously wrong. For example, he dismisses any argument against the actual infinity of space or time because space and time seem infinite to him. But he doesn't argue for this position at length. This may be a side effect of his Phenomenal Conservatism that - ironically - also makes it easy to dismiss some of his arguments because he takes for granted assumptions that may seem obviously false to many people.
Another weakness of this book is Huemer's side comments about arguments for the existence of God (and even the concept of "God" more generally). Sporadically he makes comments about the idea of God being nonsense, but his reasoning appears to be based on superficial misunderstandings of what philosophers of religion typically mean when they talk about God. Personally, it made me question how seriously he engages views that he disagrees with before critiquing them.
Overall, the book is worth reading and enjoyable. With that said, most discussions are highly technical, so if you've never taken a course on set theory or transfinite mathematics it may be difficult to follow large portions of the book. Unfortunately these portions are integral to the overall argument. But if you can handle the mathematics, I recommend you give this book a read. Agree or disagree, you'll come out with a better understanding of what infinity is (or isn't) and why that matters.
November 10, 2016
An altogether fascinating (albeit difficult) read by Professor of Philosophy Michael Huemer which sums up philosophers' and mathematicians' inquiries into infinity and tries to come to its own theory of infinity. He reviews many of the different paradoxes of the infinite (including but not limited to Hilbert's Hotel, Galileo's Paradox, etc.) and attempts to solve them all through his new approach, which tries to cut a middle ground between the traditional Aristotelian denial of the actuality of the infinite (which I side more with) and Georg Cantor's revisionist theories. As I said, I am more in agreement with Aristotle on this but I think Huemer brings up some excellent points in this most mind-boggling of debates, plus alongside learning about infinity, it provides some great introductions to philosophy of mathematics, epistemology, and metaphysics, all subjects of which I take a great interest. If you are interested in philosophy, I'd definitely recommend!
September 10, 2022
I have been a student of mathematics and the subject interests me, but I mainly started reading this because I was interested in this particular authors approach. I was not disappointed, this is a fun and interesting examination of the concept of infinity.
July 27, 2018
“the present theory of the impossible infinite – that an infinite series of preconditions cannot be satisfied – is false. Consider the beginningless Zeno series, in which the ball makes the series of motions: ... , ⅛, ¼, ½, 1. This is an infinite series of preconditions: to reach the endpoint, the ball must first reach the halfway mark; to reach the halfway mark, it must first reach the one quarter mark; and so on. Nevertheless, all of these conditions are in fact satisfied, every time an object moves. Or consider a line segment one meter long. The segment is made up of a left half and a right half. Typically, when a thing is made up of parts, we think that the thing depends for its existence on the existence of those parts, and that the thing exists at least partly in virtue of those parts existing. So the one-meter line depends for its existence upon, and exists at least partly in virtue of, the existence of the left half and the right half. Similarly, the left half depends upon the left quarter and the second quarter; the leftmost quarter depends upon the leftmost eighth and the second eighth; and so on. So there is an infinite series of dependencies; nevertheless, all these objects exist. The same point can be made using a time interval of some specified length. Now, what was wrong with the argument in favor of this theory of the impossible infinite? Earlier, we said that as one goes through the infinite series of preconditions, one is at every stage still infinitely far from reaching the end, and this is supposed to support the conclusion that none of the conditions can obtain. Perhaps the thought is that the first condition, C 1 , ‘starts out’ not obtaining, and will only get to obtain if we can get to the end of the entire infinite series of conditions. And perhaps the argument borrows from the notion, discussed in Section 4.1, that it is impossible to complete such a series, since the series is endless. This thinking would be understandable if the members of the series were actions occurring in a temporal sequence, and if each action required at least some minimum duration. Then it would be correct to say such things as ‘we are always infinitely far from completing the series.’ But (except in the case of the regress of causes; see below) we are not generally dealing with a series of temporal events. What is needed for C 1 to obtain is simply for C 2 to obtain, which in turn requires C 3 to obtain, and so on; so all told, what C 1 requires is that every member of the series obtains. But the members do not need to obtain at different times, in a certain temporal order; they may all simply obtain simultaneously (or they may all obtain timelessly). Nor is there any need for any human observer, or anyone else, to recognize or otherwise go through the members of the series. For example, in the infinite regress of Forms, there is no reason why anyone would need to identify each of the Forms in the series, or do anything else involving them that might count as ‘going through’ the series. The Forms themselves, if they exist, would all simply exist timelessly and independently of observers. Of course, this is not to say that the notion of an infinite series of Forms is plausible . The point is simply that it is not at all clear why there could not be an infinite series of objects standing in dependence relations, each one to the next. In the special case of the infinite regress of causes, there is a sequence of events that occur in a certain temporal order. Here, at least, do we have a problem? Well, it is really not clear what the problem would be. In most cases, an infinite series of events occurring one after another in time would be problematic because an infinite duration would be required to complete the series. For example, we should not postulate a belief arrived at by an infinitely long chain of argument, because it would take a person forever to complete such an argument – and we don’t have forever! But in the case of the regress of causes in the universe, we in fact have an infinite time to work with – time stretches back infinitely into the past, and the infinite series of causes occupies exactly that span of time. Why may not an infinite series of events transpire in an infinite amount of time?”
“the modifications made to the theory in this case are particularly suspicious: we are supposed to believe in all the various individual sets, but deny the existence of the set of all sets. Now, one of the things you are supposed to be able to do with sets is to explain semantics: according to many champions of sets, the meaning of a general term is to be explained, in part, in terms of the set of things to which the term applies. For instance, the meaning of ‘red’ is explained in terms of the set of all red things. 6 If this is correct, then if a general term has no associated set (N.B., it isn’t merely that the set of things it applies to is the empty set; it’s that there is no set of the things it applies to), the term must be meaningless. But the term ‘set’, according to modern set theory, has no associated set – there is no set of all the things it applies to, since that would have to be the set of all sets. Therefore, it seems, the term ‘set’ is meaningless. Of course, one might simply say that the meaning of ‘set’ is not given by the set of things to which it applies, but is to be understood in some other way. Suppose there is some such other way. Maybe the meaning of ‘set’ is given by the Platonic Form of Sethood. But if there is such a Form, then why wouldn’t there also be a Form of Redness, and why wouldn’t that explain the meaning of ‘red’? A similar question would seem to apply no matter what we say about the meaning of ‘set’ – which shows that sets really don’t help us explain semantics after all. None of this proves that there are no sets. It is not obvious that there isn’t anything that ‘set’ refers to. But it is also not obvious that there is , and there is no strong reason to believe that there is.”
“Aristotle is correct in thinking that infinity is no determinate quantity. He is also correct to ban the indeterminate from actuality. But the right conclusion is not that infinity can never be actual or completed . The right conclusion is, in a sense, that infinity can never be localized ; there cannot be any single individual or limited region that exemplifies infinitude. Strictly speaking, there are no infinite quantities: there is no cardinal number greater than all the natural numbers, nor is there any number larger than all the real numbers. Any ascription of infinitude must therefore be able to be paraphrased in such a way that the paraphrase ascribes only finite quantities to any individual. In the case of infinite cardinality, such a paraphrase can be given: we can say that the F’s are infinitely numerous, provided that for every natural number n , there exist more than n F’s. 1 This paraphrase only makes reference to natural numbers. It assumes that for every natural number n , there is a number larger than n – but it expressly avoids assuming that there is a number larger than every natural number. When asked what is the number that applies to all the F’s taken together, we should reply that there is no such number; the F’s are literally numberless. So too, an adequate paraphrase can be given for ascriptions of infinite extensive magnitudes. For example, space is infinitely extended, provided that for every (finite) volume v , there exist regions larger than v . This paraphrase only makes reference to finite volumes. It assumes that for every size, there is a larger size; but it does not assume that there is a size larger than every other size. If asked what is the size of all of space, we should deny that there is any such size. Similarly, time has infinite extent, in the sense that for any chosen temporal interval, there are intervals longer than it. But no such paraphrase can be given for an ascription of infinite intensive magnitude. If an object is ascribed an infinite intensive magnitude – for instance, an infinite mass density, or an infinite temperature – this can only mean that the object possesses some one magnitude that exceeds all other possible magnitudes of its type. This is metaphysically impossible.”
“The above views help to resolve the paradoxes of the infinite. For example, since there are no infinite numbers, it does not make sense to apply arithmetical operations to infinity. We can thus avoid various spurious calculations in which one derives that 1=0 and the like. We can similarly avoid Galileo’s Paradox. In answer to the question ‘which are more numerous: the natural numbers or the perfect squares?’, we should say, with Galileo, that neither are more numerous, nor are they equally numerous; both are simply infinite, and hence numberless. By denying the existence of points, we avoid the paradox in which it is said that an object composed of pointlike parts could be converted into two objects, each qualitatively identical to the original, merely by moving around the object’s parts. We can resolve Zeno’s Paradox by recognizing, pace Zeno, that an infinite series can be completed. Here we note that the Zeno series involves an infinite cardinal number of stages, but that its completion requires no infinite intensive magnitudes ; there is thus no objection to the completion of the Zeno series from within our theory of the infinite. On the other hand, a variety of other paradoxes do require infinite intensive magnitudes, including Thomson’s Lamp, Smullyan’s Rod, Benardete’s Paradox (including both the infinite series of walls and the infinite pile of slabs), the Littlewood-Ross Banker, the Spaceship, and Laraudogoitia’s Marbles. These scenarios require such things as infinite material strengths, infinite energy density, infinite electrical resistivity, and infinite attenuation coefficient (opacity). Each of these paradoxes thus posits a metaphysically impossible scenario, on my account. There is therefore no need to answer what would happen if such scenarios occurred. The Saint Petersburg Paradox can be resolved by noting that the paradoxical reasoning requires ascribing a nonzero probability to a certain infinite conjunction. But the conjuncts in that conjunction should in fact be assigned diminishing probabilities – or at least should not be assigned ever increasing probabilities, approaching probability one. As a result, the probability of the infinite conjunction is zero. The Delayed Heaven Paradox similarly depends on ascribing a nonzero probability to an infinite conjunction that should really be assigned probability zero. The puzzle about the Martingale betting system is resolved when we note that at any given time, the expected amount of money the gambler will have won by that time will be negative, and the expected winnings only decrease (that is, become more negative) as one considers longer finite time periods. Since it is not possible to have played infinitely many times, the strategy never yields an expected profit.”
“the modifications made to the theory in this case are particularly suspicious: we are supposed to believe in all the various individual sets, but deny the existence of the set of all sets. Now, one of the things you are supposed to be able to do with sets is to explain semantics: according to many champions of sets, the meaning of a general term is to be explained, in part, in terms of the set of things to which the term applies. For instance, the meaning of ‘red’ is explained in terms of the set of all red things. 6 If this is correct, then if a general term has no associated set (N.B., it isn’t merely that the set of things it applies to is the empty set; it’s that there is no set of the things it applies to), the term must be meaningless. But the term ‘set’, according to modern set theory, has no associated set – there is no set of all the things it applies to, since that would have to be the set of all sets. Therefore, it seems, the term ‘set’ is meaningless. Of course, one might simply say that the meaning of ‘set’ is not given by the set of things to which it applies, but is to be understood in some other way. Suppose there is some such other way. Maybe the meaning of ‘set’ is given by the Platonic Form of Sethood. But if there is such a Form, then why wouldn’t there also be a Form of Redness, and why wouldn’t that explain the meaning of ‘red’? A similar question would seem to apply no matter what we say about the meaning of ‘set’ – which shows that sets really don’t help us explain semantics after all. None of this proves that there are no sets. It is not obvious that there isn’t anything that ‘set’ refers to. But it is also not obvious that there is , and there is no strong reason to believe that there is.”
“Aristotle is correct in thinking that infinity is no determinate quantity. He is also correct to ban the indeterminate from actuality. But the right conclusion is not that infinity can never be actual or completed . The right conclusion is, in a sense, that infinity can never be localized ; there cannot be any single individual or limited region that exemplifies infinitude. Strictly speaking, there are no infinite quantities: there is no cardinal number greater than all the natural numbers, nor is there any number larger than all the real numbers. Any ascription of infinitude must therefore be able to be paraphrased in such a way that the paraphrase ascribes only finite quantities to any individual. In the case of infinite cardinality, such a paraphrase can be given: we can say that the F’s are infinitely numerous, provided that for every natural number n , there exist more than n F’s. 1 This paraphrase only makes reference to natural numbers. It assumes that for every natural number n , there is a number larger than n – but it expressly avoids assuming that there is a number larger than every natural number. When asked what is the number that applies to all the F’s taken together, we should reply that there is no such number; the F’s are literally numberless. So too, an adequate paraphrase can be given for ascriptions of infinite extensive magnitudes. For example, space is infinitely extended, provided that for every (finite) volume v , there exist regions larger than v . This paraphrase only makes reference to finite volumes. It assumes that for every size, there is a larger size; but it does not assume that there is a size larger than every other size. If asked what is the size of all of space, we should deny that there is any such size. Similarly, time has infinite extent, in the sense that for any chosen temporal interval, there are intervals longer than it. But no such paraphrase can be given for an ascription of infinite intensive magnitude. If an object is ascribed an infinite intensive magnitude – for instance, an infinite mass density, or an infinite temperature – this can only mean that the object possesses some one magnitude that exceeds all other possible magnitudes of its type. This is metaphysically impossible.”
“The above views help to resolve the paradoxes of the infinite. For example, since there are no infinite numbers, it does not make sense to apply arithmetical operations to infinity. We can thus avoid various spurious calculations in which one derives that 1=0 and the like. We can similarly avoid Galileo’s Paradox. In answer to the question ‘which are more numerous: the natural numbers or the perfect squares?’, we should say, with Galileo, that neither are more numerous, nor are they equally numerous; both are simply infinite, and hence numberless. By denying the existence of points, we avoid the paradox in which it is said that an object composed of pointlike parts could be converted into two objects, each qualitatively identical to the original, merely by moving around the object’s parts. We can resolve Zeno’s Paradox by recognizing, pace Zeno, that an infinite series can be completed. Here we note that the Zeno series involves an infinite cardinal number of stages, but that its completion requires no infinite intensive magnitudes ; there is thus no objection to the completion of the Zeno series from within our theory of the infinite. On the other hand, a variety of other paradoxes do require infinite intensive magnitudes, including Thomson’s Lamp, Smullyan’s Rod, Benardete’s Paradox (including both the infinite series of walls and the infinite pile of slabs), the Littlewood-Ross Banker, the Spaceship, and Laraudogoitia’s Marbles. These scenarios require such things as infinite material strengths, infinite energy density, infinite electrical resistivity, and infinite attenuation coefficient (opacity). Each of these paradoxes thus posits a metaphysically impossible scenario, on my account. There is therefore no need to answer what would happen if such scenarios occurred. The Saint Petersburg Paradox can be resolved by noting that the paradoxical reasoning requires ascribing a nonzero probability to a certain infinite conjunction. But the conjuncts in that conjunction should in fact be assigned diminishing probabilities – or at least should not be assigned ever increasing probabilities, approaching probability one. As a result, the probability of the infinite conjunction is zero. The Delayed Heaven Paradox similarly depends on ascribing a nonzero probability to an infinite conjunction that should really be assigned probability zero. The puzzle about the Martingale betting system is resolved when we note that at any given time, the expected amount of money the gambler will have won by that time will be negative, and the expected winnings only decrease (that is, become more negative) as one considers longer finite time periods. Since it is not possible to have played infinitely many times, the strategy never yields an expected profit.”
July 4, 2023
Huemer's characteristic wit and clarity are expectedly present. I really enjoyed his dissection of paradoxes involving infinity and his critique of Cantorian notions of infinity and numbers.
However, I find his resolutions to several paradoxes somewhat unsatisfying. One of his fundamental conclusions is the rejection of infinite intensive natural magnitudes. By rejecting these, he affirms that paradoxes such as Thompson's paradox is impossible. This also results in the rejection of the idea that a singularity is a point of infinite density and instead points to the ineptitude of modern scientific theories in adequately explaining them.
My primary gripe is with his treatment of Zenos paradox as showing that infinite series can be completed, and extending this as support for an infinite past. It seems that the completion of a convergent series (which might be more aptly described as the division of a finite extension into an infinite series) does not confirm the possibility to complete a divergent series such as what we'd expect for the passage of time.
Overall, this was a very enjoyable read, and I look forward to spending more time puzzling over questions raised.
However, I find his resolutions to several paradoxes somewhat unsatisfying. One of his fundamental conclusions is the rejection of infinite intensive natural magnitudes. By rejecting these, he affirms that paradoxes such as Thompson's paradox is impossible. This also results in the rejection of the idea that a singularity is a point of infinite density and instead points to the ineptitude of modern scientific theories in adequately explaining them.
My primary gripe is with his treatment of Zenos paradox as showing that infinite series can be completed, and extending this as support for an infinite past. It seems that the completion of a convergent series (which might be more aptly described as the division of a finite extension into an infinite series) does not confirm the possibility to complete a divergent series such as what we'd expect for the passage of time.
Overall, this was a very enjoyable read, and I look forward to spending more time puzzling over questions raised.
This entire review has been hidden because of spoilers.
June 3, 2023
Gripping read: I started reading it at maybe 1:00 AM and was up until about 4:00 AM, and finished it the next morning. Huemer's thinks like a lazer and writes like a dream. Some of his solutions to the paradoxes were unconvincing, e.g. the Saint Petersburg paradox solution, but nonetheless, very worth reading to solve many paradoxes.
October 2, 2018
A book I read slowly and with breaks between chapters. It covers a range of paradoxes in great detail and definitely stretches the mind! Much of the maths was too advanced for me to get my head around, but I greatly enjoyed delving into infinity.
August 3, 2018
Intellectual impostures all over again. A guru defining infinity on the same rigid format that Bergson used to use more than a century ago.
October 12, 2023
This Is a useful and inventive book on the paradoxes of infinity and their resolution.
June 23, 2023
You enjoy the book if you like logical and mathematical puzzles and reasoning.
I bet only a few handful of people in the world can appreciate the brilliance of it.
I'm not one of them.
I only suspect it's brilliant.
And I enjoyed reading it.
I'm impressed by the clarity of thought of Huemer. The subject is massively complex, yet I still felt I understood 60-70% (though it's likely less).
I bet only a few handful of people in the world can appreciate the brilliance of it.
I'm not one of them.
I only suspect it's brilliant.
And I enjoyed reading it.
I'm impressed by the clarity of thought of Huemer. The subject is massively complex, yet I still felt I understood 60-70% (though it's likely less).
Displaying 1 - 10 of 10 reviews