Index

A

A field 23–27, 164. See also vector potential.

a priori knowledge, 3, 28 38–39, 44 46, 61 67–68, 72, 98, 110; fallibility of, 46, 84–85; and naturalism, 32–33, 35, 40, 48, 61; see also, intuition.

abstraction, 45.

abstract entities, 3, 11–13, 31–32, 44–45, 55–57, 74, 79, 114; and perception, 94–97, 117, 140; knowledge of, 103–107, 111–112, 133; and encoding properties, 125–126.

Aharonov-Bohm effect, 24–27.

algebra, 9, 72–73, 139.

algebraic geometry, 52.

Ampere's law, 23.

analogy, 15, 20, 77–79.

analysis, 58, 62, 143.

analytic/synthetic distinction, 67–68, 116.

anthropocentrism, 33, 44.

anthropology, 40–41.

anti-realism 22–23, 50, 75–76, 79, 90, 127, 144, 158.

appearance/reality distinction, 124–126.

applied mathematics 6–9, 11, 13, 14–15, 18, 27–29, 63–67, 113, 119.

arbitrary set, 150.

Aristotle 12, 37, 83, 90, 108; taxonomy of causes, 12, 161.

arithmetic 10–11, 41–42, 48, 58–59, 64, 67–68, 81–82, 142.

“innate arithmetic”, 72–73; addition, 7–8, 65–66; subtraction, 72–73, 76–77; multiplication, 16, 150.

Armstrong, D. 12.

atheism 138, 160–161;

See also religion.

autonomy of mathematics, 32, 39, 63 133–140.

axioms 46, 61, 79–80, 84, 96–97, 109, 131, 149–158; axiom choice, 53, 97; justification of, 109; new axioms, 149–158, (see also axiom choice); of set theory, 92, 150, (see also Zermelo-Frankel axioms, ZFC).

Axiom of Constructability, 118, 135–136, 149, 152.

Axiom of Measurable Cardinals (MC), 150–151, 153–154.

Ayer, A.J. 33, 100, 146.

B

B field 24–7.

Baker, A. 1, 3–4, 6, 29.

Balaguer, M. 112, 122–132.

Ballentine, L. 24.

Barwise, J. 103,

Batterman, R. 3, 9.

Bangu, S. 3.

behaviour 10, 30, 34, 37, 49, 53–54, 57, 114.

behaviourism, 57.

Benacarraf, 55, 103, 106; argument against platonism, 103–104.

Biederman and Shiffrar, 100.

biology 1, 6, 13, 33–39.

Bloor, D. 33, 42, 44, 146–148, 160.

Bohm, D. 24–27. See also Aharonov-Bohm effect.

Bolzano, 138.

Born, M. 60.

Brouwer, L.E.J. 33, 58, 90, 138, 140.

Brown, J.R. 9.

Bueno, O. 3, 28–29.

Butterworth, B. 48.

C

Campbell, 7.

Cantor, G. 86–88, 124, 138, 165.

Carey, S. 67–70, 72–73.

Carnap, R. 116.

causal theory of knowledge 55–56, 104–107.

causal closure of the physical, 34. See also physicalism.

causation 4, 12, 148, 161; formal 12–13, 105–107, 161; teleological 37, 161; efficient 4, 12–13, 34, 161. See also, Aristotle, taxonomy of causation.

cardinality, 86, 88, 91–93, 108, 150, 155.

CH, see continuum hypothesis.

chicken sexing, 100.

Choice, axiom of 52–53, 98, 108, 118, 124, 133.

Chomsky, N. 57.

cicadas 1–2, 4–6, 9–11; prime reproductive cycle, 5, 10–11.

class, 73, 80.

classical electrodynamics, 21–24.

classical mechanics, 66.

cloud chambers, 95–96.

cognition, 68–70, 73–75, 159.

cognitive perspective, 71–72.

Colyvan, M. 1, 3, 9, 28–29, 160.

comprehension principle for abstract entities, 125–126.

composition of velocities, 65.

concepts, 3, 15–16, 27–29, 40, 43, 46, 72–74, 80–82, 88, 128; acquisition of, 67–70, 130; conceptual truths 67–68; cluster concepts, 30; understanding and, 18, 20; grounded concepts, 67–68; of sets, 80–82, 109, 124; primitive concepts, 129. See also, metaphor, conceptual metaphor.

conceptual analysis, 35, 82, 115.

conceptual change, 66.

continuum 93, 118, 142–144; continuum mechanics, 8.

continuum hypothesis (CH) 91, 97 107–110, 151, 155–158.

consequence(s): logical 14; observable, 120, 141, 151.

consistency, 123–124, 127–128, 149.

constructivism, 60, 140, 144, 148.

conventionalism 35, 52, 144.

countable sets, 93, 108, 155.

Corfield, D. 170.

culture, 40–41, 159; cultural creation of mathematics, 41. influence on mathematics 83.

Cumulative hierarchy, 125, 149, 167.

current science vs. future science, 37–8, 123.

D

Darwin, C. 4, 35, 42 159–160; and naturalism 35.

Darwinian evolution, 15, 159–160.

Darwinian processes, 53.

Darwinian theories of concepts, 68.

Dedekind, R. 55, 86–87.

Dehaene, S. 160.

Descartes, R. 33, 37, 90, 116.

descriptions, 9, 18, 30, 50, 117, 123.

Devlin, 152.

Dewey, J. 40.

Drake, F. 152.

Dretske, F. 12.

E

eigenvalue, 16, 18–20, 123.

eigenvector, 16, 18–19.

eigenstate, 16, 143.

Einstein, 13, 162.

Einstein-Podolsky-Rosen setup (EPR), 56, 104–105.

electron, 25, 34, 46, 50, 124–125.

electron spin, 16–20.

eliminability, 79.

electromagnetic field, 18, 21–26, 28.

elementary particles, 95.

empiricism 2–3, 9, 11 32–33, 38–39, 96, 112, 121, 128–130. 151.

empirical grounding of mathematics, 48–49.

empty set, 81, 126, 129, 149.

epistemic grounding 48–50, 58.

epistemic holism, 2, 112–113, 116.

Ernest, 42.

essentialism, 83.

explanation 2–3, 9–10, 15–16, 146–147; inference to the best explanation 2, 115; causal, 4, 161; mathematical, 1, 4–6, 29; scientific, 1, 2–3, 9; and tracking 10; and understanding, 15–16.

explication, 80–81.

F

factoring, 10.

faith, 40, 71, 82–83, 117, 160.

fallibilism 60–62, 84–86, 96.

Faraday, 21–22.

Faraday's law, 23.

Feynman, R. 22, 24, 26

fiction, 8–9, 59, 127–128.

fictionalism, 127, 153.

Field, H. 2–3, 13, 106–107, 160.

formalism, 44, 46, 52, 59–60, 80, 128, 140, 153.

foundationalism, 33, 61.

Frege, G. 43, 79–80, 90, 127–128, 144.

Freiling, C. 91, 94, 97, 107–110, 155–158.

Freiling's Symmetry Axiom (FSA), 108–109.

Friedman, M. 146–148.

full-blooded platonism (FBP) 122–123, 128–130, 132. See also, plenitudinous platonism.

functions, 29, 46, 51, 69, 108–109, 120, 150, 152.

G

Gauss's law, 23.

Gauss, 55.

geometry, 9, 42, 64, 125–127; non-Euclidean, 13, 63.

General Relativity, 1, 13, 63, 118.

generalized continuum hypothesis (GCH) 150.

gestalt, 121.

Giere, R. 39–40.

God, 35, 43–44, 82–83, 123, 126–127, 161.

Gödel, K. 53, 60–61, 84–85, 94, 96–97, 124, 136–137, 152.

Goldman, A.I. 54, 106.

Gould, S.J. 5–6.

group, 45.

H

Hardy, G.H. 64, 89, 95.

Helmholtz, 7.

Hersh, R. 42, 88–90.

Hilbert, D. 52, 59, 127–128, 138.

Hilbert space, 16–17, 120, 143.

history of mathematics 60, 63, 66, 83, 135.

history of science, 117–118, 149.

homomorphism, 7–9, 11, 29, 65, 119.

Hondrich, 31.

I

idealizing theories, 59.

idealization, 59, 143.

ideal agents 58–60

if-thenism, 123

impredicative definition, 133, 135.

inaccessible cardinals, 150.

incommensurability 125.

inconsistent theories, 36, 125, 142–143.

independence proofs, 127.

indispensability argument, 2–3, 8, 9, 13, 15, 113–115, 142–143; Quine-Putnam, 2, 9, 113–115, 138–139; enhanced, 2.

induction, 42, 160; mathematical

induction, 68, 102.

infinite, 58–59, 86–88, 91–92, 150.

intellectual grasp, 33, 114, 140, 144. See also intuition).

“intelligent design” (movement), 38, 43–44.

intuition, 3, 33, 60–61, 84, 96–98, 106–107, 110–112, 114, 124–125, 139; intuitive knowledge, 102–103.

intuitionism, 128.

isomorphism, 29, 85, 103, 119.

iterative conception of set, 124.

J

Jenkins, C. 67–69.

Johnson, M. 71.

justification 2, 15, 32, 48, 56–58, 142.

K

Kant, I. 33, 59, 90.

Kelvin, 18.

Kitcher, P. 10, 33, 44, 48–65; account of practices, 49–50; epistemology of practice 55–57.

“knowing how”/“knowing that” 100.

knowledge by description, 123.

Kornblith, H. 31.

Kripke, S. 160.

Krantz, D.H, 6, 8.

Kuhn, T.S. 51–52, 126, 159.

L

large cardinals, 57, 122, 142, 157.

Langlands programme, 52.

Lakatos, I, 90, 117, 159.

Lakoff, G. and R. Núñez 71–82.

laws of nature, 12, 38.

Lebesgue measure, 107, 138, 150.

Leng, M. 3, 160.

Lewis, D. 122–123, 152–153.

linguistics 68.

linguistic plurals, 70.

linguistic resources 150, 152–153.

Linksy, B. and E. Zalta, 122–126, 130–131.

Lorentz, H.A. 24.

Lyon, A. 3.

M

Maddy, P. 49, 109, 118–123, 133–149, 151–158.

magnetic field 25–26.

Mancosu, P. 3.

mapping view, 8, 28.

Marx, K. 162.

mass, 4, 10, 14; imaginary, 14–15.

mathematical change, 62–63.

mathematical experience 126–127.

mathematical methodology 133–135, 153–154.

materialism, see physicalism.

material objects, 33, 36, 51.

material substance, 21.

matter 13, 15.

MAXIMIZE, 135–137, 154–155.

Maxwell, J.C. 18, 20–23, 147.

Maxwell's equations 22–23

McGinn, C. 107.

meaning, 36, 40, 76–79, 87–88, 126–127.

measure (of a set), 93, 150 measurable cardinal, 150–151.

measure theory, 94, 109, 150.

measurement 14, 16, 56, 104–105; ordinal, 8.

measurement theory, 6–10, 28.

Melia, J. 3.

membership relation, 51, 80, 121, 129–130.

metaphor, 75–76; Conceptual metaphor, 71–72, 75–76, 79; linking metaphors, 73, 77–78; grounding metaphor, 73, 76–77, 74; and contrastiveness, 78–79.

Metaphor Principle, 75–76.

Mill, J.S. 9–10, 35, 40, 42–43, 48–49, 90.

minds 34, 36, 44, 72, 75, 107, 159.

mind-body problem, 36, 107, 159.

modality, 36–37 See also possibility.

model, 3, 8–9, 11, 13, 17–18, 28–29; models, non-isomorphic 131; standard models, 131–132.

morality, 36–37.

Moschovakis, Y. 152.

Mumford, D. 93–94, 107, 109, 155–157.

N

Nagel, E. 7, 40.

natural selection, 15, 68. See also Darwinian processes.

naturalism (as a program) 3, 11, 30–43, 159–161; and epistemology, 30–31, 120. See also empiricism; and traditional epistemology, 35–36; strong naturalism 34–35; weak naturalism 34–35; and normativity, 34–36, 53–54; motivations for, 35–36; challenges to, 36–37.

natural number, see number.

Neurath's boat, 69.

neutrinos, 99.

Newton, I. 4, 42, 55, 147–149; Newtonian physics, 13, 119; theory of gravitation, 2, 9, 20 22; Newton's first law, 34; laws of mechanics, 113.

nominalism 2, 8, 11, 13, 96, 114, 160.

non-creativity, 79–80.

non-material entities, see abstract entities.

normativity, 34, 136–137, 146–147.

numbers, 1–8, 10–12, 20, 34–25, 45, 71–78, 91–94; complex numbers, 82, 143; imaginary numbers, 43, 128; natural numbers, 11, 41, 59, 72–73, 87, 91–92, 143; number concept 67–70; prime numbers 1–2, 5–6, 10–11, 41, 114 144; and sets, 61, 77; rational numbers, 142–143; real numbers 65–66, 85, 108–109, 118–119; transfinite numbers, 64, 128.

O

observation, 9, 50–52 97–100, 146, 149.

observation sentence, 115–117.

obviousness (of mathematics), 117, 125.

“one-knowers”, 69.

ontological commitment, 2, 114.

operators, 16, 18, 120.

objectivity 41, 68, 90.

P

Papineau, D 32.

Parsons, C. 117.

Pauli, W. 17–18.

Pauli spin matrices, 18, 20.

Peano Arithmetic 132.

pedagogy, 80–82, 85–90.

perception 49–53 94–96, 100–110, 120–121; mathematical perception 60–61, (see also intuition; abstract objects, perception of.) of practices 56–57.

Pettit, 31.

physicalism, 31–32, 34, 37, 39, 96.

Pickering, 42.

Pincock, C. 3.

picture proofs, 101–103.

Plato 33, 95; Plato's heaven/realm 3, 9, 12, 22, 43, 95, 122, 124, 128–129; Plato's epistemology, 95; Republic 79; Meno, 88–89.

Platonism 3, 44–47, 124, 94–103, 145–146; plenitudinous platonism, 122–124; ontology of, 45; epistemology of, 45–46, 91, 94–103; and metaphor, 79–82; and God's existence, 82–83, 160–161.

“Platonised naturalism” 122–123.

Poincare, H. 127–128, 138.

point (geometry), 85, 126–127, 129.

politics, 89–90.

Polkinhorne, J. 161.

possible worlds, 122–123, 152–153.

possibility, 36–37.

primitives, 51, 79.

probabilistic proofs, 155.

progress (mathematical), 64, 138.

proof, 46–47, 53, 65, 86, 92, 94, 97, 101–103, 110, 127.

proof techniques, 155–157.

propositions, 12, 36, 116, 128.

propositional knowledge, 48, 100–101.

pseudoscience, 140–141.

pure mathematics, 57, 62, 66, 112, 118, 139.

Q

quantum mechanics, 13 16–20, 24, 56, 63, 104–105, 113, 143.

Quine, W.V. 30, 57, 90, 112–120, 122–125, 134, 141–142, 151–153; and bootstrapping, 69; criterion of ontological commitment 113–114; and naturalism, 112–114; holism, 112; and bootstrapping, 69–70; and Putnam, 9, 29, 134, 138.

R

rationality, 62–63, 90, 137, 146–148.

rational intuition, see intution.

random variables, 93, 109, 155–156;

“real random variables” 93, 109. See also Mumford, D.

realism 2–3, 7, 9, 12–13 30, 52, 64, 67–68, 111–112, 115, 119–120, 144–145; ontology of, 86.

realism/antirealism debate, 90.

reasons 146–147.

reasoning practices, 49, 52–3.

“recreational mathematics”, 57, 152.

reduction, 12, 77–78, 158.

reference 74, 111; theories of, 103.

refutation, 13, 58, 92–93; logical, 63.

relativistic physics, 8, 22, 26, 66, 119.

relativism 90.

religion, 30, 43–44, 139, 147, 160–161.

reliabilism 54, 106–107.

representation, 6–8, 10, 14, 142–144; and electron spin, 18–19.

reproductive cycles, 2, 10–11.

Rosenberg, A. 40.

“Romance of Mathematics”, 74, 83–84.

Russell, B. 7, 89–90, 116, 161–162.

Russell's paradox, 124.

S

scale, 6, 8.

Schmitt, 31.

Schrödinger equation, 60.

Schilpp and Hahn, 40.

science chauvinism, 133–134.

scientific theories, 1, 13, 34, 36, 66, 99, 106, 115, 118, 140–141, 151.

scientific practice, 146–147. See also history of science.

second philosophy, 39, 119, 134–140.

Scott, D. 152.

semi-naturalism, 111–112, 119–120.

sets, 51, 80–82, 120–121, 124–125, 129; concept of set, 131–132. See also, set theory.

set theory, 51, 61, 72–63, 80–81, 82, 91–94, 96–97, 107–109 118–119, 121–125, 149–158; set theoretic imperialism, 51; arbitrary set 150–152. See also: Zermelo-Frankel axioms.

set theoretic paradoxes, 61, 84, 97.

singleton, 120.

Slawinsky, 9.

sociology of science 146–147.

Sokal, A. 90.

solenoid, 24–26.

special relativity, 56, 104–106.

Steiner, M. 3, 44,

Stern-Gerlach apparatus, 19.

supernatural vii, 40, 43–44, 114, 126.

T

Tachyons, 13–15.

Taniyama-Shimura conjecture, 52.

teleology, 37–39, 161. See also cause, teleological.

temporal durations, 11.

theism, 138.

theology, 137–140.

theories, 1–2, 9, 36, 58, 83, 96–99, 112–115.

theories of metaphor, 76.

Tooley, M. 12.

transcendence, 161.

U

Uhlenbeck and Goudsmit, 17.

unification of science 39, 141.

universals, 12, 45, 98, 123.

V

van Fraassen, B. 50, 144.

vectors, 16, 18.

vector field theory 23, 27.

vector potential, 23–27.

vector space, 45, 51, 113.

verificationism, 14, 127.

visualization 72, 102.

von Neumann, J. 77–78.

W

Wagner, S. and R.W. Warner, 31.

war, 161–162.

White, L. 40–42.

Wittgenstein, L. 33, 130–132, 146, 160.

Woodin, H. 157.

Z

Zalta, E. 112, 122–126 128–132.

Zermelo, 53, 149.

Zermelo-Frankel set theory, 92 149–150; with choice, 92, 126–127.

ZFC, see Zermelo-Frankel set theory, with choice.

Zorn's lemma, 53