Infinity, Causation, and Paradox

Alexander R Pruss
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 OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

Infinity, Causation, and Paradox

OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi
 OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

Infinity, Causation,
and Paradox

Alexander R. Pruss

1
OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

3
Great Clarendon Street, Oxford, OX2 6DP,
United Kingdom
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© Alexander R. Pruss 2018
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First Edition published in 2018
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 OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

Contents

List of Figures xi
Acknowledgments xiii

1. Infinity, Paradox, and Mathematics 1

1. Paradox and Causal Finitism 1
2. Some Mathematical and Logical Notes 4
3. Modality 7
3.1 Metaphysical possibility and necessity 7
3.2 Rearrangement principles 7
3.2.1 Defeasibility 7
3.2.2 Causal powers 9
4. Finitism: An Alternate Hypothesis 10
4.1 Time and finitism 10
4.2 Non-causal paradoxes: An advantage? 11
4.3 Mathematics: A disadvantage 13
4.3.1 Infinitely many primes 13
4.3.2 Potential infinity 15
4.3.3 ∗If-thenism 15
4.4 Future infinities 17
5. ∗Defining the Finite and the Countable 18
5.1 The finite 18
5.2 Acceptable models for the axioms of arithmetic 20
6. Evaluation 23
Appendix: ∗ Counting Future Things 23
2. Infinite Regresses 25
1. How to Violate Causal Finitism 25
2. Infinite Causal Regresses 26
3. Type (i): Uncaused Regresses 27
3.1 Viciousness 27
3.2 Vicious regresses and the Hume–Edwards Principle 29
3.3 Regresses and explanatory loops 30
4. Type (ii): Causation Passing through Infinitely Many Steps 32
5. Type (iii): Outside Cause Directly Causing Each Item 33
5.1 Options 33
5.2 Regresses with outside overdetermination 35
6. ∗Analogy with Axiom of Regularity 36
7. Evaluation 37
Appendix: ∗ Two Kinds of Violations of Causal Finitism 37
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vi contents

3. Supertasks and Deterministic Paradoxes 40

1. Introduction 40
2. Thomson’s Lamp Revisited 40
2.1 Introduction 40
2.2 Causal finitism 40
2.3 Non-standard analysis 41
2.4 Special Relativity 42
2.5 Benacerraf ’s solution and the Principle of Sufficient Reason 43
2.6 Two counterfactuals 44
2.7 Evaluation 46
3. Grim Reapers 46
3.1 Introduction 46
3.2 Causal finitism 47
3.3 The absurd conclusion objection 48
3.4 A rearrangement objection 49
3.5 The mereological objection 50
3.5.1 Fusion 50
3.5.2 Necessary emergence of organic wholes 52
3.6 Uncaused lighting 52
3.6.1 Objection 52
3.6.2 The Causal Principle is true 53
3.6.3 Is the lamp lighting really uncaused? 54
3.6.4 A mysterious correlation 55
3.7 Discrete time 55
3.8 Evaluation 56
4. Infinite Newtonian Universes 56
4.1 An argument against causal finitism and a riposte 56
4.2 Smullyan’s rod 58
4.3 The conditional 60
5. Another Eternal Life 60
6. Time Travel and Causal Loops 61
6.1 Grandfathers and togglers 61
6.2 Time travel and backwards causation without causal loops 63
7. Evaluation 63
4. Paradoxical Lotteries 64
1. Introduction 64
2. Countably Infinite Fair Lotteries 64
2.1 Background 64
2.2 Expected surprise 65
2.3 A guessing game 66
2.4 Symmetry 66
2.4.1 Symmetry and lotteries 66
2.4.2 ∗ Symmetry and expected utility 68
2.5 Bayesian manipulation 71
2.5.1 The paradox 71
2.5.2 ∗ A switchover point? 74
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contents vii

2.5.3 ∗ Countable additivity and conglomerability 75

2.6 Improving everyone’s chances 77
3. Constructing Paradoxical Lotteries 79
3.1 Fairness and paradoxicality 79
3.2 Lucky coin-flip sequences 79
3.3 What it is to construct a countably infinite fair lottery 81
3.4 ∗ Coin-flips and the Axiom of Choice 83
3.5 Random walks 85
4. Objections 86
4.1 Infinite lotteries and uniform distributions 86
4.1.1 The problem 86
4.1.2 Response I: No continuous distributions 87
4.1.3 Response II: Measurement of infinite precision data 88
4.1.4 Response III: The use of the Axiom of Choice 88
4.2 ∗ A non-normalizable quantum state 89
4.3 Limitations on our reasoning 90
5. Evaluation 91
5. Probability and Decision Theory 93
1. Introduction 93
2. Guessing with Finitely Many Errors 93
2.1 Doing a little better than one can 93
2.2 A contradiction 95
2.3 Doing much better than one can 96
2.4 ∗ Construction of strategy guaranteeing at most finitely many errors 98
2.5 A multipersonal synchronic version 98
2.5.1 An angelic announcement 98
2.5.2 An objection and a tweak 100
2.5.3 ∗ Making the paradox robust 101
2.6 A parody? 102
2.6.1 The story 102
2.6.2 Evaluating the parody 103
3. Satan’s Apple 106
3.1 The story 106
3.2 Synchronic version 107
3.3 Diachronic version 108
3.4 Objection: Scores, desires, and promises 108
3.5 Evaluation 110
4. Beam’s Paradox 111
4.1 ∗ The mathematical formulation 111
4.2 ∗ Synchronic version 112
4.3 ∗ Diachronic infinite future version 113
4.4 ∗ Diachronic supertask version 114
4.5 Evaluation of Beam’s paradox 114
5. Evaluation of Decision-Theoretic Paradoxes 114
Appendix: ∗ Proof of the Theorem from Section 2.1 115
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viii contents

6. The Axiom of Choice Machine 117

1. Less Technical Introduction 117
2. ∗ The Axiom of Choice for Countable Collections of Reals 119
3. ∗ Paradoxes of ACCR 120
3.1 Die-guessing games 120
3.2 Non-measurable sets 120
3.3 Banach–Tarski paradox 121
4. ∗An Argument for ACCR 122
5. ∗A Choice Machine 125
5.1 Strange mathematics and paradox 125
5.2 Coin-flips and Dutch Books 129
5.3 How to construct a Choice Machine 130
5.3.1 Angels 130
5.3.2 A four-dimensional machine 131
5.3.2.1 Making the machine 131
5.3.2.2 Using the machine 133
5.3.2.3 Causal infinitism and verifying the machine’s match 134
5.3.3 A three-dimensional machine 136
5.3.4 ∗∗ Is AC needed? 136
5.3.5 Luck 137
6. Evaluation 137
Appendix: ∗∗ Details of Coin-Toss Rearrangement 138
7. Refinement, Alternatives, and Extensions 140
1. Introduction 140
2. Refinement 140
2.1 Event and trope individuation 140
2.2 Histories generated by partial causal relations 142
2.3 A closer look at Grim Reapers 143
2.4 Objections to causal finitism involving partial causation 147
2.5 Absences and omissions 148
3. Some Competitors to Causal Finitism 150
3.1 Finitism 150
3.2 No infinite regresses 151
3.3 No past infinities 151
3.4 No infinite intensive magnitudes 153
3.4.1 The basic theory 153
3.4.2 Some infinite intensive magnitudes 154
3.4.2.1 Center of mass and moments of inertia 154
3.4.2.2 Mental life 154
3.4.2.3 Black holes 155
3.4.2.4 Particles 155
3.4.3 Huemer’s intensive magnitudes 156
3.4.3.1 Speed, Thomson’s Lamp, and Hilbert’s Hotel 156
3.4.3.2 ∗ Smullyan’s rod 157
3.4.3.3 Immaterial minds 158
3.4.4 Evaluation 159
3.5 No room 159
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contents ix

4. Why is Causal Finitism True? 161

4.1 The question 161
4.2 Some explanatory suggestions 161
5. Further Extensions 162
5.1 Causal loops 162
5.2 Explanatory relations 164
6. Overall Evaluation 165
8. Discrete Time and Space 167
1. Introduction 167
2. Causal Finitism and Discreteness 167
2.1 The basic argument 167
2.2 From discrete time to discrete space? 167
3. Two Kinds of Discreteness 168
3.1 Subdivisibility and fixeity 168
3.2 Refining the Aristotelian picture 169
3.2.1 An objection to Aristotelian discreteness 169
3.2.2 Internal and external discreteness 170
4. Physics 172
4.1 An objection to causal finitism 172
4.2 Causation and physics 172
4.3 Quantum collapse 174
4.3.1 Some background 174
4.3.2 Causation 176
4.3.3 Back to discrete time 177
5. Fields and Discrete Space 178
6. Evaluation 180
9. A First Cause 181
1. Introduction 181
2. An Uncaused Cause 181
2.1 The quick argument 181
2.2 Towards a necessary being 182
2.3 Support for the Causal Principle 183
2.4 The Kalām argument 184
3. Compatibility with Theism? 184
3.1 Theism 184
3.2 Divine motivation 184
3.3 Divine knowledge 186
3.4 Divine action 188
3.5 Limits on metaphysical possibility 191
4. Evaluation 192
10. Conclusions 193

References 195
Index 201
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 OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

List of Figures
All illustrations in this volume have been compiled by the author.
1.1 Thomson’s Lamp 1
1.2 Correspondence between natural numbers and even natural numbers 5
1.3 Interval notation 6
2.1 The two ways of violating causal finitism: regress (left) and infinite
cooperation (right) 25
2.2 The testimony of unicorn experts 28
2.3 A theistic non-vicious regress? 33
2.4 Here, I1 = {a1 , a2 , a3 }, L1 = {e, a1 , a2 , a3 }, and M2 = {a4 , a5 , . . . } 38
3.1 Some representative Grim Reaper activations 47
3.2 Some representative reversed Grim Reaper activations 48
3.3 Smullyan’s rod with exponentially decreasing density and hence
exponentially decreasing quasi-gravitational pull 58
4.1 A lucky case where the lottery works, with the winner being the number 2 79
4.2 A traverse of a two-dimensional array 80
6.1 The (i)–(ii) betting portfolio that you should be happy to pay a dollar
for. The volume of each sphere is 1/100th of that of the cube 126
6.2 The (i )–(ii )
betting portfolio that you should be happy to pay a dollar
for if the argument works 127
6.3 The (i )–(ii ) betting portfolio you should be happy to accept for free 128
6.4 A slice of a Choice Machine 132
7.1 Benardete’s Boards 147
7.2 Four paradigmatic violations of (8) 163
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 OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi

Acknowledgments

I am especially grateful to Ian Slorach who gave me many very insightful and helpful
comments and criticisms both on my blogged arguments before I started writing this
book and while I was committing material to the book draft’s GitHub repository. I am
also particularly grateful to Miguel Berasategui, Blaise Blain, Trent Dougherty, Kenny
Easwaran, Richard Gale, Alan Hájek, James Hawthorne, Robert Koons, Jonathan
Kvanvig, Arthur Paul Pederson, Philip Swenson, and Josh Rasmussen. I am very
grateful to other readers of my blog as well as to my audiences at Baylor University,
Catholic University of America, University of Oklahoma, and the “New Theists”
workshop for their patience as I tried out versions of these arguments, and for their
critical commentary. Moreover, I am greatly in the debt of a number of anonymous
readers of this manuscript whose careful reading has resulted in much improvement
of the book. The remaining obscurities are my own accomplishment. Finally, I am very
grateful to Christopher Tomaszewski for his careful work on indexing this volume.
OUP CORRECTED PROOF – FINAL, 19/6/2018, SPi
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

1
Infinity, Paradox, and Mathematics

1. Paradox and Causal Finitism

A lamp is on at 10:00. Its switch is toggled infinitely often between 10:00 and 11:00,
say at 10:30, 10:45, 10:52.5, and so on. No other cause affects the lamp’s state besides
the switch. Thus, after an odd number of togglings the light is off and after an even
number it’s on. What state does the lamp have at 11:00? There seems to be no answer
to this question. Yet the lamp is either on or off then (Fig. 1.1).
This is known as the Thomson’s Lamp paradox (Thomson 1954). Potential answers
to a paradox like this fall into three general camps: logically revisionary, metaphysical,
and conservative. Logically revisionary answers resolve the paradox by invoking a
non-classical logic, say one in which the lamp can be both on and off at the same time,
and can use the paradox as support for such revision. Metaphysical answers resolve the
paradox by arguing for a substantive and general metaphysical thesis, such as that time
is discrete, that there are no actual infinities, or that it is metaphysically impossible
to move anything (say, a switch) at speeds whose limit is infinity (cf. Huemer 2016,
12.10.3), a thesis that explains why the story is impossible.
Conservative answers, on the other hand, refuse to revise logic or posit substantive
metaphysical theses, and come in two varieties. Particularist conservative answers
maintain that the particular story (and minor variants on it) is impossible, e.g., pre-
cisely because it is paradoxical. Defusing conservative answers maintain that the story
as given is possible and there is no paradox in it.
A particularist answer to Thomson’s Lamp paradox is simply that the story as given
is impossible, since if the story were possible a contradiction would result: the lamp
would be both on and off. A defusing answer, on the other hand, as given by Benacerraf
(1962), notes simply that there is no contradiction in saying that the lamp is on (or
off, for that matter) at 11 am: we just can’t predict the state the lamp will have from
the information given.

10:00 10:30 10:45 10:52.5 11:00

Fig. 1.1 Thomson’s Lamp.

OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

Other things being equal, conservative answers to a paradox are preferable to

metaphysical ones, while metaphysical answers are preferable to logically revisionary
ones. Nonetheless, other things need not be equal. For instance, while a given con-
servative answer may not invoke a metaphysical thesis, it may unexpectedly commit
one to such a thesis, and then the benefits of conservativeness are lost. For instance,
Benacerraf ’s solution is in tension with the Principle of Sufficient Reason. For even
if there is no contradiction in the lamp’s being on at 11 am, there seems to be no
explanation as to why it’s on then (and if it’s off, there is no explanation for that).
Furthermore, if a number of paradoxes are given and each can be resolved by
means of a different conservative response, nonetheless it could be preferable to
resolve them all in one fell swoop by a single elegant metaphysical hypothesis that
explains why none of the paradoxical stories are possible. For it is reasonable to prefer
unified explanations of phenomena.
In this volume, I will present a number of paradoxes of infinity, some old like
Thomson’s Lamp and some new, and offer a unified metaphysical response to all of
them by means of the hypothesis of causal finitism, which roughly says that nothing
can be affected by infinitely many causes. In particular, Thomson’s Lamp story is
ruled out since the final state of the lamp would be affected by infinitely many switch
togglings. And in addition to arguing for the hypothesis as the best unified resolution
to the paradoxes I shall offer some direct arguments against infinite regresses. It is
not the purpose of this book to consider all paradoxes of infinity—that would be an
infinite task—or even all the ones that have been discovered so far. Rather, I consider
a sufficient number to motivate causal finitism.1
The availability of an elegant metaphysical solution obviates the need for resorting
to logical revisionism. But we will need to be constantly on the lookout for conser-
vative solutions to the paradoxes. Nonetheless on balance causal finitism will provide
a superior resolution. Furthermore we will need to consider competing metaphysical
hypotheses that resolve some or all of the paradoxes. However, it will turn out that
each of the competing hypotheses suffers from one of the following shortcomings: it
is broader than it should be, it fails to resolve all the paradoxes that causal finitism
resolves, or it suffers from being ad hoc.
One can distinguish two ways of resolving a paradox: one can solve it by
showing how an apparently incompatible set of claims is actually compatible or
by showing how an apparently plausible assumption is no longer plausible after
examination, or one can kill it by arguing that the paradoxical situation cannot occur.2
In some cases, killing a paradox is not a tenable option. For instance, Zeno’s paradoxes
of motion can be solved, say by showing that they make assumptions about time or
motion that we can reject, or they can be killed by holding that motion is impossible.
Zeno, of course, wanted to kill the paradoxes, but since then most philosophers have
preferred to solve them.

1 For a more thorough survey, see Oppy (2006).

2
I am grateful to an anonymous reader for this distinction.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

paradox and causal finitism 

Whether killing or solving the members of a family of paradoxes is intellectually

preferable depends on the details of the situation. For instance, when the paradoxes
occur in situations that we have apparent empirical observations of—arrows flying
and faster runners catching up with slower ones, as in Zeno’s case—killing the paradox
by rejecting the actuality of the situations is apt to lead to an unacceptable skepticism,
pace Zeno. On the other hand, when the paradoxes occur in situations which we
merely intuitively think are metaphysically possible, killing the paradoxes by rejecting
the metaphysical possibility of the situations may be much more tenable, since our
intuitions about metaphysical possibility are unlikely to be as reliable as our empirical
observations.
We may have a certain intuitive preference for solving a paradox rather than
killing it. But unless the paradoxes are based on logically invalid reasoning, it will
be intellectually preferable to kill all the members of a family of paradoxes in a
unified way rather than solve them in a variety of different ways. One reason for
this is the simple fact that to solve a paradox based on logically valid reasoning we
have to reject a plausible premise, and hence to solve a number of such paradoxes
we have to reject a number of plausible premises. But it is typically preferable to
make a single assumption—especially if there is some independent reason to make
the assumption beyond the need to resolve paradoxes—than to reject a number of
plausible premises.
The main strategy of the book, then, will be like that of Zeno: rather than opt for
a number of different solutions to different paradoxes, they will be all killed through
the single assumption of causal finitism. But whereas the no-motion thesis that Zeno
defends is one we have very strong empirical reasons to reject, the thesis of causal
finitism is compatible with our observations (though arguing for this will take some
work in interpreting modern physics).
For most of the rest of the present chapter, after some important background
notes both technical and philosophical, I will consider one prominent alternate
hypothesis—full finitism—and argue that in order to get out of the paradoxes, it needs
to be married to a particular theory of time, the growing block theory, and that in
any case it causes serious difficulties for the philosophy of mathematics. While on
the subject of philosophy of mathematics, I will also offer an intriguing application
of causal finitism (and of finitism as well) to the problem of defining the finite and
the countable.
In Chapter 2, I will consider infinite regresses, which will give us some reason
to accept causal finitism independently of the paradoxes it can kill. Then, in the
succeeding chapters we will discuss several different kinds of causal paradoxes:
non-probabilistic paradoxes, paradoxical lotteries, other probabilistic and decision-
theoretic paradoxes, and paradoxes bound up with the Axiom of Choice from set
theory. At times we will also consider what will be seen to be an analogous question:
whether time travel and backwards causation are possible. I will then offer ways to
refine the rough thesis of causal finitism in light of the data adduced, and argue that
various alternatives to causal finitism are unsatisfactory.
OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

Finally, I will consider two potential consequences of causal finitism. That a theory
has consequences beyond what it was intended to explain gives some reason to think
the theory is not ad hoc. At the same time, such consequences make the theory more
vulnerable to refutation, since there might be arguments against the consequences.
The first apparent consequence is that time, and perhaps space as well, is discrete.
If this does indeed follow, that is intrinsically interesting, but also damaging to causal
finitism in that it appears to conflict with much of physics since Newton. We shall
consider whether the discreteness of time actually follows and whether the kind of
discreteness that is supported by causal finitism is in fact in conflict with physics, and
argue that causal finitism can cohere with modern physics.
The second consequence is clearer. If causal finitism is true, then there cannot be
backwards-infinite causal sequences, and hence there must be at least one uncaused
cause. There is also some reason to take this uncaused cause to be a necessary being.
Now the most prominent theory on which there is a causally efficacious necessary
being is theism. Thus, causal finitism lends some support to theism. Interestingly, this
will force us to consider whether theism doesn’t in turn undercut causal finitism.
I will occasionally use the convenient phrase “causal infinitism” for the negation of
causal finitism. Roughly, thus, causal infinitism holds that it is possible for something
to have an infinite causal history. (Note that causal infinitism does not say that there
actually is any infinite causal history.) Thus the point of the book is to argue for causal
finitism or, equivalently, to argue against causal infinitism.
Let me end this section by noting that I do not take Thomson’s Lamp to be a
particularly compelling version of a paradox motivating causal finitism. There will
be more discussion of it in Chapter 3, Section 2. But it is a helpful stand-in for many
of the more complicated paradoxes we will consider.

2. Some Mathematical and Logical Notes

We will need some technical terminology and symbolism as general background for
the book, and this will be introduced in this section. Additionally, the book contains
some technical sections marked with “∗ ” and very technical sections with “∗∗ ”. These
can be skipped without loss of continuity. Note that any subsections of something
marked with one of these markers can be presumed to have at least that level of
technicality. Note also that Chapter 6 is technical or very technical as a whole apart
from a less technical introduction and summary.
Start with the notion of sets as collections of abstract or concrete objects. The
statement x ∈ A means that x is a member of A. We say that a set A is a subset of
a set B provided that every member of A is a member of B, and that A is a proper
subset of B if it is a subset of B that does not include all the members of B. For any
set B and any predicate F(x) we write {x ∈ B : F(x)} for the subset of B consisting of
all and only the xs such that F(x) (sometimes when the context makes B clear, we just
write {x : F(x)}).
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

some mathematical and logical notes 

We can compare the cardinal sizes of sets as follows. If there is a way of assigning a
different member of B to every different member of a set A (i.e., if there is a one-to-one
function from A to a subset of B), then we say that A ≤ B, i.e., the cardinality of
A is less than or equal to that of B. For instance, if B is the set of real numbers between
0 and 1 inclusive, and A is the set of positive integers, then to every member n of A we
can assign the member 1/n of B (note that if n and m are different members of A, then
1/n and 1/m are different members of B).
We say that A has fewer members than B, and we write A < B, provided that
A ≤ B but not B ≤ A. We say that sets A and B have the same cardinality
when A ≤ B and B ≤ A. The famous Schröder–Bernstein Theorem (Lang
2002, p. 885) says that under those conditions there is a one-to-one pairing of all the
members of A with all the members of B.
Some sets are finite and some are infinite. A set is finite if it is empty or has the
same size as some set of the form {1, . . . , n} for a positive integer n. Otherwise, the set
is infinite.
Infinite sets are in some ways like finite sets and in others unlike them. Both their
likeness and their unlikeness to finite sets are counterintuitive to many people.
A way in which infinite sets differ from finite ones is that if A and B are finite sets
with A a proper subset of B, then A always has fewer members than B. But infinite
sets have proper subsets of the same size as themselves.3 For instance, if B is the
set {0, 1, 2, . . .} of natural numbers, then the proper subset A = {0, 2, 4, . . .} of even
naturals has the same size as B, as can be seen by joining them up one by one as in
Fig. 1.2.
On the other hand, just as the finite sets differ among each other in size, Georg
Cantor discovered that so do the infinite ones, if size is defined as above. The difference
in size between infinite sets is harder to generate, however. Simply adding a new
member to an infinite set doesn’t make for a larger infinite set. But given a set A,
we can also form the powerset P A of all the subsets of A. And it turns out that P A
always has strictly more members than A—this is now known as Cantor’s Theorem.4

0 1 2 3 4

0 2 4 6 8

Fig. 1.2 Correspondence between natural numbers and even natural numbers.

3 ∗∗ To be precise, this is only true for all Dedekind-infinite sets. If the Countable Axiom of Choice is

false, then there may be infinite sets that aren’t Dedekind-infinite (Jech 1973, p. 81). But standard examples
of infinite sets, such as the natural or real numbers, are still Dedekind-infinite. For simplicity, I will write
as if all infinite sets were Dedekind-infinite.
4 Here is a proof. It’s clear that P A has at least as many members as A does, since for each member

x of A, the singleton {x} is a member of P A. So all we need to show is that A does not have at least as many
members as P A. For a reductio, suppose there is a function f that assigns a different member f (B) of A
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

If A is finite and has n members, then P A will have 2n members (for we can generate
all the members of P A by considering the 2n possible combinations of yes/no answers
to the questions “Do I include a in the subset?” as a ranges over the members
of A), and n < 2n . But the Cantorian claim applies also to infinite sets: in general,
A < P A.
In particular, there is no largest set. For if A were the largest set, then P A would be
yet larger, which would be a contradiction.
Sets which are finite or the same size as the set of natural numbers {0, 1, 2, . . .},
which will be denoted N, are called countable. An example of an uncountable set
is P N. Another is the set R of real numbers, which in fact has the size as P N.
A useful notation for certain sets of real numbers is given by [a, b], (a, b), [a, b),
and (a, b] (see Fig. 1.3). Each of these denotes an interval from a to b, with the square
brackets indicating that the endpoint is included in the interval and the parenthesis
indicating that it’s not. Thus [a, b] is the set of all real numbers x such that a ≤ x ≤ b,
(a, b) is the set of all reals x such that a < x < b, [a, b) is the set of all reals x such that
a ≤ x < b and (a, b] is the set of all reals x such that a < x ≤ b. As long as a < b, all
the four intervals are uncountably infinite and of the same size as R.
Finally, it will sometimes be convenient to talk in terms of pluralities. When I say
(1) The members of my Department get along with each other
the grammatical subject of (1) is a plurality, the members of my Department. The verb
form that agrees with that plurality, “get”, is in a plural conjugation. The subject of the
sentence is not a singular object like the set of the members of my Department or
some sort of a mereological sum or fusion of the members, for that would call for a
singular verb, and it would make no sense to say that that singular object “get along
with each other”.
We can quantify over pluralities. We can, for instance, say that for any plurality of
members of my Department, the xs, there is a plurality, the ys, of people in another
Department such that each of the xs is friends with at least two of the ys. Plural
quantification is widely thought to avoid ontological commitment to sets. It also
avoids technical difficulties with objects that do not form a set. There is no set of all
sets, but it makes sense to plurally quantify and say that all the sets are abstract objects.

(−∞, −1) [0, 1] (2, 3) [4, 5) (6, 7] [8, ∞)

−1 0 1 2 3 4 5 6 7 8

Fig. 1.3 Interval notation.

to every different member B of P A. Let D be the set of all members x of A that are assigned by f to some
set B such that x is not a member of B, i.e., D = {x ∈ A : ∃B(x = f (B) & x ∈ / B)}. Let x = f (D). Note
that x is a member of D if and only if there is a subset B of A such that f (B) = x and x is not a member of
B. The only possible candidate for a subset B of A such that f (B) = x is D, since f assigns x to D and will
assign something different from x to a B different from D. Thus, x is a member of D if and only if x is not a
member of D, which is a contradiction.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

modality 

I will assume that pluralities have their elements rigidly. That is, if x0 is one of the xs,
then in any possible world where the xs exist, x0 exists and is one of the xs.

3. Modality
3.1 Metaphysical possibility and necessity
Fairies and water made of carbon atoms have something in common: they don’t exist.
But there the similarity ends. For although neither is actual, the fairy is possible, while
the water made of carbon atoms is not.
The kind of possibility at issue here is not merely logical. No contradiction can be
proved from the existence of a fairy, but also no contradiction can be proved from the
existence of carbon-based water. In each case, empirical work is needed to know the
item does not exist.
There are many theories of the nature of modality.5 The arguments of this book
are not tied to any particular such theory, but rather to intuitive judgments about
cases. These intuitive judgments about cases may themselves put constraints on which
theory of modality is plausible, though of course the reader will also find her theory of
modality affecting what to think about the cases. That is a part of why I give so many
paradoxical cases in this book: some cases may appeal to some readers while others
to others.
3.2 Rearrangement principles
.. defeasibility
If it is metaphysically possible to have one horse and two donkeys in a room,
intuitively it’s possible to have two horses and one donkey in a room as well. Lewis
(1986, Section 1.8) attempted to formulate a “rearrangement principle” that justifies
inferences such as this (see Koons 2014 for some more rigorous formulations). The
basic idea behind rearrangement principles is that:
(2) Given a possible world with a certain arrangement of non-overlapping spatio-
temporal items, any “rearrangement” of these items that changes the quan-
tities, positions, and orientations to some other combination of quantities,
positions, and orientation that is geometrically coherent and non-overlapping
is also metaphysically possible.
Unrestricted rearrangement principles sit poorly with causal finitism. Just multiply
the number of button flippings and you go from an ordinary bedside lamp being
turned off at night and on in the morning to Thomson’s paradoxical lamp. And even
if we restrict changes in quantity to be finite, an innocent forwards-infinite causal
sequence can be transformed into a backwards-infinite one.

5 In Pruss (2011) I defend a causal powers account of modality, but nothing in the present book depends

on that defense.
OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

Nonetheless, many of our arguments for causal finitism will depend on rearrange-
ment considerations. Isn’t that cheating?
To see our way to a negative answer, observe that unrestricted rearrangement prin-
ciples carry many heavy metaphysical commitments. They rule out classical theism,
since on classical theism God is a necessary being, and a situation that could coexist
with God could be rearranged, say by greatly multiplying evils and removing goods,
into a situation that couldn’t coexist with God (cf. Gulesarian 1983). They give an
argument for the possibility of a universe consisting of a single walnut that comes
into existence at some time, and hence for the possibility of something coming from
nothing. They rule out Aristotelian theories of laws and causation on which the
exercises of causal powers necessitate their effects in the absence of counteracting
causes. They sit poorly with the essentiality of evolutionary origins for biological
natural kinds and with the essentiality of origins for individuals. And they even rule
out certain colocationist theories of material objects. For colocationists will say that
wherever you have a clay statue you also have a lump of clay, but an unrestricted
rearrangement principle should let you have the clay statue without the lump!
Even philosophers of a Humean bent who are uncomfortable with theism and
essentiality of origins and have no problems with things coming into being ex nihilo
need to restrict rearrangement on metaphysical grounds. For instance, Lewis (1986,
p. 89) said that all rearrangements are possible “size and shape permitting”. The worry
is that it might turn out that material objects cannot interpenetrate, so one cannot
rearrange a world with a horse standing beside a cow into a world where they occupy
the same location. This seemingly purely geometrical constraint in fact needs to
depend on the metaphysics of the material objects in question. Perhaps indeed a horse
and a cow could not occupy the same location, but we have good reason to think that
multiple bosons like photons can occupy the same location, since two bosons can have
the same quantum state (Dirac 1987, p. 210). So what the “size and shape permitting”
constraint comes to depends on the metaphysics of the objects, namely whether they
can be colocated.
Thus, rearrangement principles like (2) should be curtailed in some way lest they
ride roughshod over too much metaphysics. One way to do this is to extend Lewis’s
strategy by giving a list of specific metaphysical constraints like his “size and shape
permitting”. But it is difficult to see how we could ever be justified in thinking that
our list of constraints is complete.
A better way is to stipulate that the rearrangement principles are defeasible, with the
understanding that it is best if defeaters for rearrangement principles are principled
rather than ad hoc. Theism, Aristotelian views of causation, essentiality of origins,
or causal finitism could each provide principled defeaters to particular cases of
rearrangement. But if one ruled out Thomson’s Lamp by saying that this particular
rearrangement of the ordinary bedside lamp situation is impossible, that would be
ad hoc. If, instead, we could rule out Thomson’s Lamp as well as a number of other
paradoxes by means of a single general principle, namely causal finitism, that would
be highly preferable. And it is this that is the strategy of the present book.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

modality 

At the same time, there is always a cost to introducing another metaphysical

principle like causal finitism that defeats particular instances of rearrangement.
But the cost is surmountable.
I do not want the arguments of this book to be hostage to a particular rearrange-
ment principle. Rather, I want to rely on the intuitive plausibility of the particular
rearrangements that I will make use of.

.. causal powers

One crucial question in formulating a rearrangement principle is what properties are
carried along with the objects as they are rearranged. I can rearrange a room with
a braying donkey into a room with two braying donkeys. But I cannot rearrange
a room with a solitary donkey into a room with two solitary donkeys. A standard
thing to say is that the properties that can be carried around by the items being
rearranged are the intrinsic properties: solitariness is not intrinsic, but braying might
be. But it is notoriously difficult to define an intrinsic property (see Weatherson and
Marshall 2014).
There is, however, one controversial choice that many of our arguments will require,
and this is a picture of objects and their activities as having a causal nature that is
carried along with their rearrangement. When one rearranges a lamp switch from
one location in spacetime to another, the rearranged switch continues to have the
same causal powers, and when put in the same relevant context (say, a lamp) these
causal powers will have the same effects. If intrinsic properties are what can be
carried along with rearrangements, then I am taking causal powers to be intrin-
sic properties.
This is a very intuitive picture of causal powers. It is, nonetheless, in conflict with
widely held Humean views on which causal facts supervene on the global arrangement
of matter in the universe. I take this conflict to provide an argument against the
Humean view. The possibility of rearranging things in the world while keeping fixed
the causal powers of things is intuitively more secure than the Humean theories
of causation.
One strength of making a case-by-case judgment about the possibilities of re-
arrangements of powerful objects rather than positing a single general principle is
that each such judgment can be separately evaluated by a Humean reader. The reader
may decide one of three things about a particular application of rearrangement:
(i) the application is incompatible with Humeanism and plausible enough to
provide significant evidence against Humeanism; or
(ii) the application is incompatible with Humeanism but not very plausible and so
instead Humeanism provides significant evidence against this application; or
(iii) the application can be made coherent with Humeanism, for instance, by
supposing many analogous background events sufficient to ground causal laws
that apply also to the rearranged case.
I leave such judgments to the reader.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

4. Finitism: An Alternate Hypothesis

4.1 Time and finitism
Finitism holds that there can only be finitely many things (including both substances
and events). Finitism, however, allows for potential infinities. Thus a collection of toy
soldiers to which a new toy soldier will be added every day would be potentially
infinite as for any number n, it would eventually have more than n elements. But
according to finitism there are no actual infinities. There are always only finitely
many things.
Finitism has an impressive philosophical history, going back at least to Aristotle’s
responses to Zeno’s paradoxes,6 and being the generally accepted philosophical
orthodoxy in the Middle Ages.
The exact upshot of finitism depends on which theory of time it is combined with.
The eternalist thinks of past, present, and future things as all ontologically on
par, and believes that (barring some catastrophe) our great-great-great-grandchildren
exist and Alexander’s great warhorse Bucephalus also exists. Of course, the great-
great-great-grandchildren and Bucephalus don’t presently exist. But they nonetheless
really do exist. The growing block theorist takes reality not to extend to the future, but
to include the past and present.7 Thus our great-great-great-grandchildren don’t exist
(though it might be true that they will exist), but Bucephalus does. The presentist, on
the other hand, only accepts presently existing entities as existing. I will, further, take
all of these theses about time to claim to be necessarily true.
Finitism plus eternalism straightforwardly entails causal finitism: if there can only
be finitely many things, and that includes past, present, and future, then of course
nothing can be affected by infinitely many causes. Thus any paradox ruled out by
causal finitism will be ruled out by finitism plus eternalism. But unfortunately finitism
plus eternalism also entails that the future must be finite—that there cannot be
infinitely many future events. But surely it is possible to have an infinite future full
of different events or substances, say with a new toy soldier being produced every day
forever. Thus, finitism is implausible given eternalism.
Given presentism, on the other hand, finitism is compatible with infinite sequences
of causes, as long as at no particular time are there infinitely many causes. Thus,
finitism plus presentism does nothing to rule out the infinitely many togglings of the
switch in Thomson’s Lamp. While the other paradoxes have not yet been discussed,
many of them will also have the diachronic character of Thomson’s Lamp and hence
will be untouched by presentist finitism.

6 Of course, in a degenerate way, Parmenides was a finitist, as he thought that there could be only
one thing.
7 There is also a variant by Diekemper (2014) that includes only the past. That variant will not be helpful

to the finitist, and I will stick to the canonical version that includes the present.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

finitism: an alternate hypothesis 

This leaves growing block plus finitism. If it’s necessarily true that a cause is earlier
than or simultaneous with its effect, then growing block finitism does entail causal
finitism, and hence can rule out all the paradoxes that causal finitism can. Given the
combination of (a) growing block theory, (b) finitism, and (c) the thesis that causes are
either temporally prior to or simultaneous with their effects, we do get causal finitism
again, and hence we can rule out all the paradoxes that causal finitism can rule out.
The finitist’s best bet at paradox removal is thus to adopt growing block together
with the thesis that causes are prior to or simultaneous with their effects.
Unfortunately, there is a powerful argument against growing block theory due
to Merricks (2006). Many people have thought thoughts about what date or time
it is, thoughts expressible in sentences like: “It is now 2012” or “It is now noon.”
If growing block theory is true, many of these thoughts are in the past, and most
have a content that is objectively false. On the growing block theory, the “now”
is the leading edge of reality, the boundary between the real and the unreal. The
thought expressed by “It is now 2012” is true if and only if 2012 is at the leading
edge of reality. But 2012 is not at the leading edge of reality. Moreover, my present
thought that it is now 2018 has no better evidence than the “It is now 2012” thought
that was in 2012. Since most thoughts of this sort, with the usual sort of evidence
for them, are false, I should be skeptical about whether it is now 2018. And that’s
absurd. Presentism escapes this argument by denying that the past thoughts exist.
Moving spotlight versions of eternalism, on which there is something like an objective
“moving spotlight” illuminating the “now” are also subject to this objection: most of
the “It is now t” thoughts are not illuminated by the spotlight, and yet “now” implies
such “illumination”. But B-theoretic eternalism (e.g., Mellor 1998), which claims that
the “now” is a mere indexical, rather than an expression of an objective changing
property (like being at the leading edge of reality or being “illuminated”), is not subject
to the objection. Hence, the finitist’s best bet at paradox removal requires adopting a
particularly vulnerable theory of time.
We now consider more advantages and disadvantages of finitism vis-à-vis causal
finitism as a way out of paradoxes.
4.2 Non-causal paradoxes: An advantage?
Imagine Hilbert’s Hotel—a hotel with infinitely many rooms numbered 1, 2, 3, . . . .
You can have lots of fun with that. Put a person in every room, and then hang up the
sign: “No vacancy. Always room for more.”8 When a new customer asks for a room,
just put them in room 1, and tell them to tell the person in the room to move to
the next room, and to pass on the same request. You can even have infinitely many
people vacate the hotel and still have it full. If all the people in the odd-numbered
rooms leave, you can tell each person in the even-numbered rooms to move to a room
whose number is half of their room number.

8
The sign suggestion comes from Richard Gale.
 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

While the stories I gave involved causation, that was only for vividness. To see
the paradoxicality, all we need to notice is that the guests can move around (even
causelessly, if that’s possible) to make space for a new guest, and that every second
guest can leave, while the hotel quickly regains its fullness.
The root of these paradoxes is that an infinite collection can be put in one-to-one
correspondence with a proper subset (cf. Fig. 1.2, above). If finitism is true, then of
course we can rule all such paradoxes out of court. This provides a simple argument
for finitism: If finitism is not true, then Hilbert’s Hotel is possible. But Hilbert’s Hotel
is absurd and hence not possible, so finitism is true.
But while Hilbert’s Hotel is indubitably strange, the strange and the absurd (or
impossible) are different, as is proved by the strangeness of the platypus. We could just
conclude from Hilbert’s Hotel that infinity is roomier than we previously thought.
One might think that an outright contradiction can be proved from Hilbert’s Hotel.
For instance:
(3) The collection of even-numbered rooms is smaller than the collection of all
rooms.
(4) The two collections can be put in one-to-one correspondence (matching room
n for even values of n with room n/2).
(5) Two collections that can be put in one-to-one correspondence are of the
same size.
(6) If A is smaller than B, then A is not the same size as B.
(7) So the two collections both are and are not the same size.
Assuming that “smaller” and “same size” are used univocally throughout, there are
two ways of rejecting the argument. First, one can reject (3). Granted, for finite sets a
proper subset is smaller than its proper superset. But we shouldn’t expect this to be
true in infinite cases. After all, infinite cases are different from finite ones. Alternately,
one can reject (5) (of course, here, “size” cannot be stipulated as was done in Section 2).
The question of how to gauge what is absurd and what is merely strange is a difficult
one. While I do not see much of a cost to rejecting (3) or (5), others will. Nonetheless,
in the case at hand there is a very strong reason to reject finitism, and thus accept the
possibility of something like Hilbert’s Hotel.
A related paradox is the following. Intuitively, there are more positive integers than
prime numbers. But now imagine an infinite collection of sheets of paper, with one
side red and one side green. It is as clear as anything that the number of red sides
equals the number of green sides. Now suppose that the green sides are numbered9
1, 2, 3, . . . , and suppose that the red side of a piece of paper that has n on its green
side contains an inscription of the nth prime number (we won’t run out of primes:

9 The “ink” will have to be a non-molecular sort, since in order to fit very long numbers on the page, the

numerals will have to get smaller and smaller.

 OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

finitism: an alternate hypothesis 

there are infinitely many of them10 ). Then the number of red sides equals the number
of primes, and the number of green sides equals the number of positive integers, and
since the number of red sides equals the number of green sides, we conclude that
the number of primes equals the number of positive integers, which contradicts the
assumption that there are more positive integers than primes.
Denying the possibility of an actual infinite kills the paradox. But one can also
solve the paradox by saying that the argument is a reductio ad absurdum of the initial
intuition that there are more positive integers than prime numbers. And denying the
possibility of an actual infinity only kills the paradox at the cost of undercutting this
initial intuition in a different way. For if actual infinites are impossible, then it seems
to make no sense to say that there are more positive integers than prime numbers,
since neither infinite plurality can actually exist. We will now consider an argument
against finitism along a similar line.
4.3 Mathematics: A disadvantage
.. infinitely many primes
A flatfooted mathematics-based argument against finitism is: “There are infinitely
many primes. So finitism is false.” (Note, too, how this argument doesn’t seem to affect
causal finitism, since numbers appear to be causally inert.)
But perhaps when the finitist told us that there couldn’t be an actual infinity of
things, she was thinking about concrete things like rooms and not abstract ones like
numbers?
This is not plausible, however. For while the finitist’s arguments are formulated
in terms of concrete things, the intuitions on size that underwrite arguments like
(3)–(7) apply just as much in the case of abstracta. The fact that there are as many
even numbers as natural numbers is in itself counterintuitive, and the Hotel merely
makes this more vivid. Thus the finitist cannot afford to restrict her view to concrete
entities, since doing so leaves unanswered paradoxes that intuitively are of exactly the
same sort as the ones she resolves.
Furthermore, the distinction between abstract and concrete objects is not a par-
ticularly clear one, and unless it is clarified it is hard to say why exactly one would
think that infinite collections of concrete things are a problem but infinite collections
of abstract things are not.
For instance, one clarification of the notion of concreteness in the literature is given
by Pruss and Rasmussen (2018), who say that an entity is concrete if and only if it
is possible for the entity to cause something. If this is right, then the finitist who

10 The classic proof is a reductio ad absurdum. If there are finitely many primes, let p be the product of
all of them. Then p + 1 is bigger than every prime. Moreover, p + 1 is not divisible by any prime, since
it yields remainder one when divided by any prime. But a number not divisible by any prime is a prime.
So p + 1 is a prime bigger than every prime, which is absurd.
OUP CORRECTED PROOF – FINAL, 14/6/2018, SPi

 infinity, paradox, and mathematics

thinks that there are infinitely many primes would need to answer why it is that
the possession of causal powers rules out infinities. After all, although the rooms in
Hilbert’s Hotel presumably have the possibility of causing things (for instance, a wall
can cause pain in a fist), nothing in the paradox depended on the causal possibilities
for the rooms. One might as well say that Hilbert’s Hotel is impossible because the
rooms possibly have color.
So objecting to the flatfooted argument from primes on the basis of a distinction
between abstracta and concreta is not promising. A better option, however, is to point
out that the argument depends on a Platonist interpretation of the sentence “There are
infinitely many primes.” But Platonism is not the only position in the philosophy of
mathematics. There are other options.
But not all of the other options are available to the finitist. The finitist labors under
the special disability that not only does she think that the infinitude of mathematical
objects does not exist, but she thinks that nothing with the relevant structure—
a central aspect of that structure being infinitude—could possibly exist. Thus the
mathematician is someone studying impossible situations. But while it has always
been a wonder that something as rarefied and abstract as mathematics should be
applicable to the actual world, it is a true miracle that the study of genuinely impossible
things should be of such relevance to us. How could the queen of the natural sciences
be the study of impossible structures?
Consider, too, that mathematics involves proofs from axioms. Certain axioms
are controversial, and hence they are not always assumed. Thus, in some contexts,
mathematicians go out of their way to flag that they did not assume the Axiom
of Choice in a proof. The reason for excluding axioms from the assumptions for a
proof is two-fold. The first reason comes from epistemic concerns about the truth
of the axiom. Since we’re not sure the Axiom of Choice is true, it is safer not to
assume it. The second is theoretical: even if the given axiom were true, we would be
interested in knowing what it would be like if we had a system that did not satisfy
that axiom.
But the finitist takes herself to have established on pain of absurdity that there
cannot be infinitely many things. If that’s really established, then we do not have
an epistemic reason to exclude the Axiom of Finitude—that there are only finitely
many things—from the axioms used in our mathematical work. And if the Axiom of
Finitude is necessarily true, then the question of what it would be like if a system failed
to satisfy the axiom is more of a logician’s or philosopher’s question than a question
for the typical mathematician to wonder about. While some mathematicians do in
fact study alternative collections of axioms for set theory, typical mathematicians are
happy to assume axioms of set theory they find intuitive. Likewise, if the Axiom of
Finitude were necessarily true, then most working mathematicians should assume it
and study its consequences, rather than engaging in the study of the per impossibile
counterfactual of what would happen if there were infinities. The resulting mathemat-
ical practice would be very different from ours, and we have good reason to doubt the
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Title: Lauluja

Author: Antti Rytkönen

Release date: October 31, 2023 [eBook #71994]

Language: Finnish

Original publication: Helsinki: Vihtori Alava, 1900

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LAULUJA

Kirj.

Antti Rytkönen

Helsingissä, Jakaa Vihtori Alava, 1900.

SISÄLLYS:

Syvänteet.
Kuohuissa.
Meri ja taivas.
Takatalvessa.
Kesää odottaissa.
Hän..
Sinä ja minä.
Lehdossa.
Sorsa se Saimaan aalloissa sousi.
Valkojoutseneni.
Valkamani.
Lauluni.
Se lempi.
Eloni.
Levoton.
Kevättuulessa.
Kaipuuni.
Kuusen alla.
Talvilehto.
Valtameri.
Laulu talvelle.
Oli metsä vihreä.
Merelle lähtijä.
Luojalle.
Rauhassa.
Rannan kuusi.
Myrsky-yönä.
Merenneitojen laulu.
Meriltä palatessa.
Kevät-iltana.
Tähtöselleni.
Samponi.
Syystunnelma.
Keväinen koski.
Muistojen mailta.
Tyhjä sija.
Sairas soittaja.
Niin syvästi särki se äidin mieltä.
Turhaan.
Manan morsian.
Mari pikkunen piika.
Paimeritytön kevätlaulu.
Tuliluulialei.
Ikävissä.
Paimenpoika ja paimentyttö.
Paimenpojan laulu.
Koti lahteen soutaessa.
Mistä kyynel.
Leppäkerttu ja tuomenterttu.
Musta lintu, merikotka..
Järven jäällä.
Köyhän koti.
Heilini.
Kevätkylmissä.
Jäiden lähtö.
Kalevaisten karkelo.
Kevätmietteissä.
Ainon kaiho.
Kerran lemmin.
12-13.IV.1597.
Korven keskellä.
Kylänkarkelo.
Tarina Pekasta ja vallesmannista.
Pimeän pesä.
Voiton saatte.
Maamiehen kevätlaulu.
Tahdon ikitulta.

Syvänteet.
 Syvänteihin katseleisin, mi lie siellä elo kumma?
Salaisuudetpa ne syvät kätkee multa meri tumma.

Syvänteistä myrskyt nostaa heleöitä helmilöitä. Selittäkää

helmet mulle rahtu syvänteiden öitä!

Sen vain kertoo helmet somat: syvänteet on pohjattomat.

Kuohuissa.

Raivoisana kuohuu meri,

tumma on sen pinta.
Kuohuu mullai sydänveri,
aaltoileepi rinta.

Lyöpi aalto louhikkohon,

kivirantaan aivan —
Sydän, kivihinkö kuohut? —
Tuoko maksaa vaivan?

Meri ja taivas.

Sua lemmin niinkuin voipi vaan

meri kuohuva lempiä taivastaan.

Sä oot niin korkeella, kaukana multa

kuin on meren kuohuista taivahan kulta.
 Ja niin syvällä sentään rinnassain
kuin meren pohjassa taivas vain.

Takatalvessa.

Tuo katse armas ja kaunoinen mun mieleni vallan hurmas,

vaan kevätkylmät ja kolkot säät pian toivoni kukkaset surmas.

Vaan sydämmessäkin vaihtuva lie kevään sekä talven

valta. Siks toivon, että nousta vois kevät taas takatalven alta.

Kesää odottaissa.

Kyllä se kuluu talvikin

ja kyllä se joutuu kesä,
tulevana kesänä tällä pojalla
on jo oma pesä.

Talvella hakkaan honkia

ja hirsipuita kannan,
tulevana kesänä mökkini nousee
luona lammin rannan.

Toisella puolen lampi päilyy

ja toisella vihanta vuori,
ja mökissä on herttaisen kaunis kukka,
ja se on oma kultani nuori.
Hän.

Hän on niin puhdas, kaunoinen kuin koitto aamuruskon, tuo

onnen, rauhan sydämmeen, tuo toivon, uuden uskon.

Myös on hän tulta, hehkua kuin tähti öisen taivaan, tuo

surun, tuskan sydämmeen, tuo epätoivon aivan.

Sinä ja minä.

Sä ruusu olet armahin, päivä olet kirkkahin, tähti olet

loistavin.

Mä lapsi olen kylmän jään, mä laine olen tuulispään, mä

lintu olen myrskysään.

Lehdossa.

Lehdossa tässä kerran mä istuin immen kanssa, hän silloin

mulle kertoi hartaimmat unelmansa.

Lehdossa tässä nytkin mä istun yksin vallan. Vaan unelmat

ja toiveet — ne saaliita on hallan.
Sorsa se Saimaan aalloissa sousi.

Sorsa se Saimaan aalloissa sousi kaipasi, kaipasi

kullaistaan. Mihin lie joutunut, kuka sen ties, joko lie ampunut
metsämies?

Sorsa se Saimaan aalloissa sousi, kanteli kaihoa

sydämmessään, syys oli synkkä ja kulta ei tullut, vilu oli uida
joukossa jään.

Sorsa se Saimaan aalloissa sousi, kuikutti kurja yksinään.

Valkojoutseneni.

Pois lensi valkojoutsen,

lens ulapalle päin.

Miks veit sä valkojoutsen

myös levon syömmestäin?

Mun valkojoutseneni,
sä miksi lensit pois?

Jos löytäisin sun kerran,

oi, silloin riemu ois!

Valkamani.
 Kun meri ärjyy, aaltoo ja tummana vaahtoaa, kun pilvinen
on taivas ja rinta ei rauhaa saa,

niin satamata tyyntä mä etsin harhaillen, ja luoksesi kun

pääsen, niin siellä löydän sen.

Kun nojaan rintahasi, meri kuohuva tyyntyy tää ja katsehes

päivän kirkas maan ääriä lämmittää.

Lauluni.

On lauluni kuin kukka tuo, ja lämmöstä se voimaa juo.

Vaan koleaks kun käypi sää, suruisna kukan painuu pää.

Se lempi.

Se lempi oli polttava kuin hehkuvainen tuli, ja kyllä kylmät,

kyllä jäät sen eessä kaikki suli.

Se lempi oli kuohuva kuin virta vaahtopäinen. Se lempi oli

unelma, unelma kevähäinen.

Eloni.
 Haaksi aaltojen ajama, vene vetten vierittämä on mun
vaivaisen vaellus.

Niinkuin lastu lainehilla,

vastavirrassa venonen
on koko eloni juoksu.

Kumpi ennen kääntynevi: vene virran viertehessä vai virta

venon mukana?

Levoton.

Miksi lie mun mieleni niinkuin meren laine? Rauhatonna,

levoton na kulkee oikeaan ja harhaan, onnen tahtoisi se
parhaan. Väliin sinne, väliin tänne häilyy, horjuu, kiertää,
kaartaa, väliin hetken levon löytää, jälleen järkkyy, väliin
kallioihin töytää — pirstoiks särkyy…

Kevättuulessa.

Jo kevättuuli hengittää, vaan syys on syömmessän. Se iäks

liekö riutunut vai heränneeköhän?

Jos eloni ei elpyne, niin iäks riutukoon! Jos vielä herää,

herätköön se kevään taisteloon!
Kaipuuni.

Valoa, päivän paistetta mun mieleni kalpaa aina, jos

kirkasta en näe taivasta, niin talvi mun mieltäni painaa.

Valo ja päivän lämpö ne ovat henkeni elinehto, vaan nää

jos puuttuu, kuihdun pois kuin hallan koskema lehto.

Kuusen alla.

Alla kuusen useasti istuin iltamyöhään asti, kuusi kuiski

kumeasti: "Lapsi, lähdet maailmaan, outohon ja avaraan, tiesi
ulapoille saavat, ulapat on aavat, aavat, siellä laineet
lakkapäiset iskee purteen monet haavat Ulapoilla usva on,
usva katoamaton; armas päiv' ei usviin koita, helposti ei valo
voita." Kuusen kuiskiessa nuokkui rannan kukat rauhaisasti,
kuusen kuiskiessa huokui meren aallot raskahasti.

Talvilehto.

Tuo lehto, jossa me leikittiin, nyt tuiman on talven vanki, ja

lehdon nuortean kukkaset jää kattaa ja harmaja hanki.

Vaan vielä se kerran sulaa jää

ja nuortuu lehdot ja vuoret,
ja vielä ne kerran heräjää
ne kevään kukkaset nuoret

Ja vielä me kerran lehdossa taas leikimme, impeni, illoin, ja

vielä me nautimme lemmestä kevätaika on armas silloin.

Valtameri.

Katso, katso valtamerta!

Milloin siell' on tyyntä, rauhaa?
Eikö siellä ainiansa
aallot paasihin vain pauhaa.

Rintani on valtameri,
miksi rauhaa etsit sieltä,
miksi tyyntä, sointuisuutta
ulapan ja aallon tieltä?

Laulu talvelle.

Talvi, lunta valkeinta, valkeinta, puhtoisinta sada immen

poskuelle, hipiälle hienoiselle. Talven, taruin valtakunta laula,
laula immelleni, kerro kaunokutrilleni utuisinta lemmen unta.
Kerro kuinka alta hallan, alta kylmän jäiden vallan nousee
toivon kukkasia, utuisia, armahia… Näytä unia kesästä, kerro
 omasta pesästä pienoisesta, sievoisesta, armaasta, ani
hyvästä.

Oli metsä vihreä…

Oli metsä vihreä, tuomessa kukka,

sitä ihaillen katselit impi rukka.

Sä ihailit kaunista kevätsäätä,

et muistanut kylmää, et muistanut jäätä.

Sä ihailit suuria Luojan töitä,

et muistanut harmaita hallaöitä.

Ja kuitenkin, ja kuitenkin
ne tuli ne keväiset kylmätkin.

Merelle lähtijä.

Läksi laiva uurtamaan meren tummaa pintaa, täytti kaiho,

kaipaus monta, monta rintaa — Eessä merta, merta vaan,
eessä pilvet harmaat, kotiranta kauvas jäi, sinne jäivät
armaat. Jäivät tuomet tuoksumaan, jäivät kuuset kukkimaan,
impi rantaan itkemään. Kotiranta kauvas jäi, sinne jäivät
armaat…
Luojalle.

Luoja, noita silmiä kyynelistä säästä, Luoja, tuohon rintahan

murheit' elä päästä.

Luoja, anna hänelle päivät paistehikkaat! Luoja, anna

hänelle elon riemut rikkaat!

Rauhassa.

Kun melske, myrsky raukenee, kaikk' uinuu rauhan unta,

niin silloin mulle aukenee viel' uusi valtakunta.

Nään silloin sinut luonani, nään silmäs ani armaat, ja on

kuin siirtäneet ne ois väliltä vuoret harmaat

Rannan kuusi.

Tääll' yksin seisot sorjana sä vihreä rannan kuusi; veden

alla toinen on maailma ja taivas siellä on uusi, sen
kauneuttako katsot sie, mihin miettehes silloin vie?

Kun laskeissaan kesäaurinko taas taivahan rantoja kultaa,

niin tunnetko silloin, tunnetko sä lämpöä, lemmentulta? Näät
kaikkialla sä rauhan maat, sopusointua, rauhaa kaiketi saat?
 Vai tunnetko kaihon tuskaisan
ja muistatko myrskytuulet?
Kun aalto se löi raju rantahan,
sen kuohuja vieläkö kuulet?
Sun toivehes, aattehes myrskykö kaas,
nyt kaihoin niitäkö muistat taas?

Sä lienetkin mun kaltaisen, sä vihreä rannan kuusi; kun

katson pohjihin syvyyden, ja kun taivas siellä on uusi, niin
oloni munkin oudoks saa ja kaiho mieleni valloittaa.

Ja muistuu mieleeni myrskysää ja kohina tuiman tuulen ja

murhe mieleni yllättää ja ma kuohuja kaukaa kuulen. Ne
rintaani kaikuja kummia saa, meri helmahan yön kun
uinahtaa.

Myrsky-yönä.

käy ulkona tuuli tuima, yö on niin myrskyinen, ja hurjana

aalto huima vain kuohuvi rannallen.

Ja haaksi aaltoja halkoo,

ja kohti kuohuja käy,
ja se keinuvi niinkuin palko,
ei muuta kuin vaahtoa näy.

Vaan haahdessa suojassa yksin

minä kanssasi olla saan,
me istumme vieretyksin,
mut tuuli se ulvoo vaan.

Te käykää te vihurit tuulen,

nyt teitä mä pelkäjä en!
Vain rintasi lyönnit ma kuulen,
ja sa painut mun rinnallen.

Mua katsehes kaunis huumaa

sysimusta kuin syksyn yö,
ja mun rintani tulta on kuumaa
ja se rintaasi vasten lyö.

Ja tuuli tuima se soittaa kuin mahtavat myrskyt on poviss'

suurien valtamerten ja povessa inehmon.

***

Oi myrskyjä meren pinnan, oi valkovaahtoja sen! Oi

kuohuja ihmisrinnan, sen kaihojen, poltteiden!

Merenneitojen laulu.

Tääll' on riemut, rikkaudet, riistat, riittävät tavarat, tääll' on

Ahdin kultalinna, täällä aartehet avarat Täällä lientyy maiset
huolet, täällä suistuu surman nuolet, täällä elo ihanaa. Täällä
tyyntyy tuskat, vaivat, alla aallon uudet taivaat täällä sulle
aukeaa. Täällä merten aaltoloissa, Ahdin kultakartanoissa
rintaraukka rauhan saa. Tänne riennä inehmo, tääll' on sija
sulle jo!
Meriltä palatessa.

Saapuu laiva mereltä kohti kotirantaa; tuhat tulta merelle

sieltä valon kantaa; tuhat tulta tuikahtaa sieltä mua vastaan,
yhtä tulta kaipajan, yhtä ainoastaan: immen ikkunasta vaan
tuloset ei tuikakkaan.

Kevät-iltana.

On kevät-ilta ja kuuhut niin kummasti kumottaa, ja kaiho

outo ja kumma mun mieleni valtoaa.

Kevät-iltana kuljin ennen

kanss' armahan ystäväin
ja unia ihanoita
minä kultani kanssa näin.

Ja mieleni oli niin raitis

kuin rannan aaltonen,
kun kullat kuuhuen taivaan
utukalvossa uiskeli sen.

Kuin ilmojen utuiset pilvet

myös ulapan pohjilla ui,
niin sielusi hellä ja herkkä
mun sieluuni kuvastui.

Ja tuota kun muistan, silloin mun mieleni summeutuu, ja

mun rintani kaihon tuntee kevät-ilta ja illan kuu.