Animal	cognition,	species	invariantism	and	mathematical	realism Helen	De	Cruz This	is	a	draft	of	a	paper	published	in	Aberdein,	A.	and	Inglis,	M.	(2019).	Advances	in Experimental	Philosophy	of	Logic	and	Mathematics.	London:	Bloomsbury	Academic (pp.	39-61). 1.	Introduction What can we infer from numerical cognition about	mathematical realism? In this paper, I will consider one aspect of numerical cognition that has received little attention in the literature: the remarkable similarities of numerical cognitive capacities across many animal species. This Invariantism in Numerical Cognition (INC) indicates that mathematics and morality are disanalogous in an important respect: proto-moral beliefs differ substantially between animal species, whereas proto-mathematical beliefs (at least in the animals studied) seem to show more similarities. This	makes	moral beliefs	more susceptible to a contingency challenge from	evolution	compared	to	mathematical	beliefs,	and	indicates	that	mathematical beliefs	might be less vulnerable to evolutionary debunking arguments. I will then examine to what extent INC can be used to flesh out a positive case for mathematical	realism.	Finally, I	will	review	two	forms	of	mathematical	realism	that are	promising in the light	of the	evolutionary evidence	about	numerical cognition, ante	rem	structuralism	and	Millean	empiricism. 2.	The	contingency	challenge Moral	realism	is	the	view	that	moral	claims,	such	as	"slavery	is	wrong",	or	"Jane	is	a good	person",	are	about facts	and	that	we	know	some	of these facts. Moral facts are normative: they not only describe what is the case (e.g., slavery is wrong, or giving	people	their	freedom	is	right),	but	also	what	ought	to	be	the	case	(e.g.,	people should never be enslaved). Such facts are different from natural facts (e.g., that water	is	composed	of	H2O),	but	that	does	not	make	them	any	less	true	in	the	eyes	of the	moral realist. By contrast,	moral antirealists contend that there are no	moral facts.	Traditionally,	moral	antirealists	have	argued	that	moral	claims	do	not	describe beliefs,	but	emotions	(e.g.,	violence	makes	me	feel	bad,	slavery	makes	me	feel	sorry for	enslaved	people).	More	recently,	authors	such	as	Sharon	Street	(2006,	2008)	and Richard Joyce (2006) have argued against	moral realism on evolutionary grounds. They worry that human moral intuitions, and their resulting judgments are influenced	by	the	peculiar	evolutionary	history	of	our	species.	Arriving	at	the	correct moral	beliefs,	given	the	contingency	of	human	evolution,	would	be	a	formidable	and inexplicable	instance	of	luck.	It	would	be	as	if	one	set	sail	in	the	hope	that	the	winds and	tides	will	get	one	to	Bermuda	(Street	2006,	121).	As	Street	writes, There is a striking coincidence between the normative judgments we human beings think are true, and the normative judgments that 2 evolutionary	forces	pushed	us	in	the	direction	of	making.	I	claim	that	the realist	about	normativity	owes	us	an	explanation	of	this	striking	fact,	but has	none	(Street	2008,	207). Street	(2006)	lists	some	moral	concerns	that	are	similar	across	many	species,	such	as that	survival	is	good,	or	that	an	obligation	to	care	for	one's	offspring	is	greater	than the	obligation	to	help	complete	strangers.	However,	many	other	moral	concerns	are the result	of the	peculiar	quirks	of	human	evolution.	For	example,	humans	believe that helping unrelated strangers is a good thing, or that one should punish group members	who	do	not follow	social	norms (Henrich	et	al.	2006). Such (proto)moral sentiments	are	not	present	in	other	primates	(see	e.g.,	Silk	and	House	2011). In	the	Descent	of	Man	(1871)	Darwin investigated,	among	many	other	topics, the	evolution	of	the	moral	sense	in	humans.	The	overall	project	of	that	book	was	to show	that	although	the	difference	between	human	cognitive	capacities	and	those	of others	was	substantial,	including	the	human	sense	of	morality,	beauty,	and	religion, it was only a difference in degree and not in kind. Darwin sought to establish precursors	of	the	moral	sense	in	other	animals.	He	conjectured	that	the	evolution	of moral capacities became unavoidable in cognitively complex social animals. As he wrote "any animal whatever, endowed with well-marked social instincts, the parental	and filial	affections	being	here included,	would inevitably	acquire	a	moral sense	or	conscience,	as	soon	as	its	intellectual	powers	had	become	as	well,	or	nearly as	well	developed,	as	in	man."	(Darwin	1871,	71-72).	But	he	also	thought	that	their moral	beliefs	would	vary	depending	on	the	social	context	in	which	members	of	the species	would	evolve.	To	illustrate	this	point	vividly,	he	offered	the	following	thought experiment: if humans had evolved from animals with a eusocial structure, our moral beliefs would be very different from the ones we currently hold: "our unmarried females	would, like the	worker-bees, think it a sacred	duty to kill their brothers,	and	mothers	would	strive	to	kill	their	fertile	daughters;	and	no	one	would think	of	interfering"	(Darwin	1871,	73).	This	is	because	in	eusocial	societies,	the	longterm survival of the group trumps concerns of individual workers. Eusociality has evolved three to	eleven times independently in	nature, in various clades including insects, shrimps, and mammals (West and Gardner 2010). If there are rational creatures	on	other	planets the	kinds	of	actions	we	think	are	morally reprehensible could	be	obligatory	for	them	and	vice	versa. A moral realist could respond to this challenge by arguing that in eusocial structures, nests rather than individual workers are the right-bearers, so the difference may not be so vast after all. One could also argue that morality is a uniquely	human	domain,	which	does	not	even	arise	in	other	animals.	Other	animals have	proto-morality	at	best:	dispositions	that	lead	them	to	helping	behavior,	or	that lead	them	to	prefer	prosocial	over	anti-social	individuals,	but	no	explicit	moral	norms governing social interactions. Still, Darwin's bee thought experiment stresses the contingency of moral beliefs upon our specific evolutionary history. Our moral beliefs are not just the product of evolution; they are the peculiar outcome of human	evolution,	a	haphazard	process	that	has	favored	unique	social	structures	and behaviors. Lillehammer (2010, 365) terms challenges of this kind the Contingency Challenge:	"we	would	have	had	very	different	beliefs if	certain	things	about	us	had 3 been different, even supposing the relevant ethical facts to remain the same". A related,	but	distinct,	challenge	is	the	Inflexibility	Challenge,	which	says	that	we	would have had the same beliefs, even if the relevant facts had been different. Street's (2006) Darwinian dilemma for	moral realists stresses the inflexibility of our	moral beliefs: even if pain were morally good in some realist sense, we would still be inclined to disvalue it because evolution through natural selection leads us to disvalue pain, as it decreases survival and reproductive success. Both kinds of challenge	are	part	of	a	broader	kind	of	purported	failure,	which	spells	bad	news	for moral realism, the	Tracking	Failure (Lillehammer	2010).	The	evolutionary	challenge to	ethics	does	not	amount	to	the	claim	that	moral	beliefs	are	likely	not	truth-tracking because	they	are	the	outcome	of	a long,	evolutionary	process (this is	probably	the case	for	all	our	beliefs,	and	thus	would	render	the	challenge	trivial).	Rather,	it	is	the more	specific	claim	that	moral	beliefs	are	not	truth-tracking	because	they	depend	on contingent	facts	about	our	evolutionary	history. In this paper, I will not be concerned with the evolutionary debunking literature	on	moral realism,	but rather,	with the	question	of	whether	evolutionary challenges	to	moral	realism	could	be	extended	to	the	mathematical	domain	(see	also, e.g.,	Clarke-Doane	2012,	2014,	De	Cruz	2016).	This	fits	in	a	broader	literature	of	the so-called	"companions	in	guilt"	arguments	where	relevant	features	of	moral	realism are	argued	to	occur	in	other	domains	(e.g.,	logic,	perception,	mathematics)	(see	e.g., Rowland	2016).	The	basic	outline	of	such	an	argument	holds	that	these	two	domains fall	or	stand	together:	if	a	challenge	to	moral	realism	proves	fatal,	it	will	also	be	fatal for that other domain. The question here is whether evolutionary debunking arguments, if successful against	moral realism,	also	damage	mathematical realism. Mathematical	realism-analogous	to	moral	realism-is	the	claim	that	mathematical statements such as "2 + 2 = 4" are about facts. In order to assert this, the mathematical realist posits that	mathematical entities (e.g., the natural numbers), relations or structures exist. A dominant position in mathematical realism is platonism,	which holds that	mathematical objects exist. They are abstract entities that	exist	independently	of	human	minds,	cultural	constructs,	language	and	symbols. Clarke-Doane	(2014,	manuscript)	has	connected	the	contingency	challenge,	normally applied to the question of moral realism, to the Benacerraf-Field challenge to mathematical realism. Benacerraf (1973) originally formulated the following objection	to	mathematical	platonists: if	mathematical	objects	are	outside	of	spacetime, how can	we establish a causal link between these objects and the	minds of mathematicians? How can the physical brain of the mathematician get access to these	remote	mathematical facts?	Field	(1989)	reformulated	the	challenge in	more general terms. Field's formulation does not require a causal link (which is a controversial requirement in any case), but hinges on the fact that mathematical realists	cannot	explain	the	reliability	of	mathematical	beliefs: We	start	out	by	assuming	the	existence	of	mathematical	entities	that obey the standard	mathematical theories;	we grant also that there may	be	positive	reasons	for	believing	in	those	entities	...	Benacerraf	's challenge ... is to ... explain how our beliefs about these remote entities	can	so	well reflect the facts	about them ... [I]f it	appears in principle	impossible	to	explain	this,	then	that	tends	to	undermine	the 4 belief in mathematical entities, despite whatever reason we might have	for	believing	in	them	(Field	1989,	26). How	we	should	cash	out	"explain	the	reliability"	is	not	easily	resolved.	Clarke-Doane (manuscript) argues that it should be spelled out in terms of safety: "In order to "explain	the	reliability"	of	our	mathematical	beliefs it is	necessary	to	show	that	we could	not	have	easily	had	false	ones	(using	the	method	that	we	actually	used	to	form them),	even	if,	had	we,	they	would	have	been	false."	This	formulation	responds	both to the contingency challenge: our mathematical beliefs are not easily false (for instance, because they are not contingent upon our evolutionary history in a	way that is pernicious), and it also responds to the inflexibility challenge: our mathematical beliefs are not inflexible. They track something that is independent from	them,	and	that	could	be	false. Clarke-Doane (2016) argued that mathematical realists might not face an access	worry	because	mathematical	beliefs	are	arguably	safe.	This	might	be	because our "core" mathematical beliefs could be evolutionary inevitable. However, in a more	recent	paper	(manuscript)	Clarke-Doane	argues	that	it	would	be	very	hard	for	a mathematical realist to show that	mathematical	beliefs, say, about set theory, are safe,	because it	would	be	difficult to	show	that	"we	could	not	have	easily	believed different	axioms	of	set	theory".	Indeed,	the	fact	that	mathematicians	disagree	about many core claims in every	mathematical area shows that	mathematical beliefs do not	meet this criterion. For instance, Edward	Nelson rejected the successor axiom (every natural number has a successor). Because mathematical beliefs are presumably less colored by irrelevant influences such as religion and cultural background, Clarke-Doane (2014) thinks that this puts mathematical realists in a worse position than moral realists, as the latter can at least explain away moral disagreements	as	a	result	of	distorting	irrelevant	factors,	but	the	former	cannot. This	interpretation	of	the	Benacerraf-Field	challenge	places	the	bar	for	realists quite high, some might argue, impossibly high. Moreover, some epistemologists have	argued	that	safety	is	not	a	useful	criterion	for	knowledge	(e.g.,	Bogardus	2014). Nevertheless, spelling	out the	Benacerraf-Field challenge in terms	of safety can	be useful if	we	consider the	evolutionary	origins	of	mathematical	beliefs, in	particular numerical beliefs. In this paper, I will not be concerned with set theory or other mathematical propositions, but with the evolutionary basis of our ability to form beliefs	about	numbers	at	all. There is a large literature that supports the view that formal mathematics depends	on	evolved	capacities	to	deal	with	number,	which	is	collectively	sometimes referred to as "the number sense" (e.g., Dehaene 2011). Two capacities are hypothesized	to	underlie	animals'	ability	to	deal	with	numbers:	the	object	file	system (OFS)	which	might	underlie	our	ability to	enumerate	and	keep in	working	memory small collections of items (up to three or four) precisely, a capacity that is called subitizing, and the approximate number system (ANS), which may underpin our capacity	to	estimate	and	compare	larger	collections	(Feigenson	et	al.	2004). Many cognitive scientists hold that the OFS and ANS lie at the basis of our ability to engage in	more formal arithmetic abilities. Elizabeth Spelke (e.g., Spelke and	Kinzler	2007)	has	argued	that	ANS	supplemented	with language	allows for the ability	to	engage	in	formal	arithmetic.	She	finds	support	for	the	role	of language	in 5 studies indicating that people who speak languages without exact number words cannot perform basic calculations exactly (e.g., 6 2 = 4), but their approximate numerical	cognition is	on	par	with	numerate	adults (Pica	et	al	2004).	Although it is limited	in	that	it	only	allows	for	approximate	numerical	calculations,	the	ANS	already allows for	abstract	numerical representations	across	modalities:	preschool	children can	add	and	compare	arrays	of	dots	and	sound	sequences	(Barth	et	al.	2005).	Carey (2009) sees the OFS at the root of more formal arithmetical capacities. Her bootstrapping	account	emphasizes	the	role	of	subitizing	in	children's	ability	to	learn the	successor	function	in	arithmetic.	Children	learn	to	associate	the	meanings	of	the first	words in	a	count list (in	English,	"one",	"two",	and	"three")	with	collections	of one,	two	and	three	items,	which	they	can	subitize.	This	explains	why	children	tend	to learn the meanings of number words in the same order: they first become oneknowers, then two-knowers, next three-knowers, and very occasionally, fourknowers. But because subitizing stops at 3 or 4, they need to	make an inductive generalization	to	learn	the	next	words	in	the	counting	sequence.	According	to	Carey, children	then	make	the	following	induction:	if	"x"	is	followed	by	"y"	in	the	counting sequence, adding an individual to a set	with cardinal value x results in a set	with cardinal	value	y. The	idea	that	these	two	capacities	play	a	critical	role	in	our	ability	to	engage	in formal arithmetic is not universally accepted (see e.g., Rips et al. 2006, Rips et al. 2008). Some authors have argued that non-numerical sensory properties, such as visual	density	and	circumference	can	explain the	animal	data	and	have	questioned the	existence	of	the	ANS	(e.g.,	Gebuis	et	al.	2016).	That	being	said,	the	ANS	and	the OFS are still the predominant theories to explain animal numerical cognition. Authors	such	as	Lourenco	et	al.	(2012)	have	argued	that	people's	ability	to	engage	in approximate arithmetic (both symbolic and non-symbolic) correlates with their mathematical	abilities.	This	has	been	confirmed	in	a	recent	meta-analysis,	although the correlation between mathematical skills and symbolic numerical abilities is stronger	(Schneider	et	al.	2017).	If	formal	arithmetic	is	dependent	(in	some	causal	or psychological sense) on the evolved number sense, it becomes relevant for mathematical realists to explore it in more detail. In particular, explaining the reliability	of	our	mathematical abilities	will involve reference to the	number sense and	the	way	it	forms	beliefs	about	magnitudes. 3.	Invariantism	in	numerical	cognition Numerical cognition is a well-researched domain of higher cognition. While obviously it is not identical across species (for one thing, humans use Arabic numerals whereas mosquito fish do not), I will here examine striking similarities between the numerical capacities of animals from a wide variety of species and clades,	which	I	will	call invariantism	in	numerical	cognition (INC). I	will	here look	at four features of numerical cognition across species to argue the case for INC: numerical	cognition	is	present	in	many	different	animal	species,	including	in	animals with	small,	simple	nervous	systems	such	as	insects	and	spiders	(3.1),	it	plays	a	crucial role	in	animal	adaptive	decision	making	(3.2), it	shows	similarities	in	computational characteristics	and	limitations	across	species	(3.3)	and,	to	the	extent	that	it	has	been investigated, there is	evidence that	numerical	cognition is the result	of	convergent 6 cognitive	evolution	rather	than	common	descent	(3.4). 3.1.	Numerical	competence	is	present	in	a	wide	variety	of	clades Most research on numerical competence has been conducted with primates, including	rhesus	monkeys,	capuchin	monkeys	and	chimpanzees.	For	example,	rhesus monkeys are able to order collections of items from	1 to 9 (Brannon and Terrace 1998). Other mammals, including brown bears and dogs, are also capable of discriminating	numerosities.	For	example,	brown	bears	were	trained	to	select	among two	screens	the	display	that	had	the	largest	number	of	dots	(even	if	sometimes	that meant the	overall lowest surface	area,	because the	dots	were	smaller),	using food reinforcements	(Vonk	and	Beran	2012).	Domestic	dogs	were	tested	using	a	violationof-expectation	paradigm,	where	they	saw	simple	calculations including	"1	+	1	=	2", "1	+	1	=	1"	and	"1	+	1	=	3".	Dogs	looked	longer	at	the	incorrect	outcomes,	which	is interpreted	as	showing	they	did	not	expect the incorrect	outcomes	and	thus	know that 1 + 1 = 2 (West and Young 2002). Birds, including pigeons, chickens (even newborn	chicks)	and	crows,	are	capable	of	calculating	and	estimating	collections	of items	(e.g.,	Scarf	et	al.	2011,	Ditz	and	Nieder	2016,	Rugani	et	al.	2008).	Although	not all	experiments	control	for	non-numerical	cues,	such	as	the	total	surface	area	or	the density	of	displays,	many	experiments	have	done	so.	For	example,	mosquito	fish	can discriminate between smaller (e.g., 3 vs. 2) and larger groups (e.g., 8 vs. 5), even when	controlling for the	density	of the fish	and the	overall space	occupied	by the group	(Dadda	et	al.	2009).	For	larger	groups	(e.g.,	8	vs.	4),	total	area	and	the	amount of movement of the fish in both groups matter (Agrillo et al. 2008). Such experiments strongly suggest that it is numerical cues-rather than	non-numerical continuous	variables-that	animals	are	responsive	to. A	recent	domain	of	inquiry	is	numerical	competence	in	insects	and	spiders	(see Pahl	et	al.	2013	for	a	review).	Although	insects	have	small	nervous	systems,	they	are very	adept	at	integrating	complex	information,	such	as	the	relative	returns	of	nectar by particular types of flowers, even depending on times of the day and the probability of yields (Real 1991). Numerical information is one such source of information	that	insects	and	spiders	use	in	their	everyday	ecological	decisions.	Portia africana	spiders,	for	example,	practice	communal	predation,	sharing	their	prey	with another	resident	conspecific.	Juvenile	Portia	africana	prefer	to	settle	when	there	is one	conspecific	present,	preferring	this	outcome	to	zero, two	or	three	conspecifics (Nelson and Jackson 2012). Dacke and Srinivasan (2008) designed a carefully controlled experiment	where bees	were trained to fly in a long tunnel	where five landmarks	consisting	of	identical	yellow	strips	were	placed	at	irregular	intervals,	and the feeder was hidden at one of those landmarks. In the test condition, the researchers	examined	whether	bees	would	look	for	the	feeder	close	to	the	number of landmark	they	were	trained	they	could	find	it.	The	bees'	accuracy	was	very	high up	until	3,	but	became	more	erratic	at	4	and	5.	Bees	not	only	are	able	to	discriminate numbers sequentially, but also to visually discriminate different numerosities of displays.	They	can	match	displays	of	two	blue	dots	and	two	yellow	stars,	and	can	do so	up	to	3,	and	their	performance	drops	at	chance	level	at	4	(Gross	et	al.	2009).	This is	a	striking	similarity	to	other	animals,	suggesting	bees	may	be	subject	to	the	same limitations	of	the	OFS	as	human	infants	(Starkey	and	Cooper	1980). 7 3.2.	Numerical	cognition	plays	a	crucial	role	in	animal	adaptive	decision-making While	early	authors	writing	on	numerical	competence	in	animals	tended	to	dismiss	it as	a	last	resort,	to	which	animals	only	turn	if	there	is	no	other	information	available (Davis and	Pérusse 1988), the current consensus is that animals use their number sense in adaptive decision-making. The best-studied ecological situation in which animals rely on numerosities is food choice: given that they need to travel to a source	of food	and	use	up time	and	energy	doing so, it	makes sense to go to the source	that	has	the	most	food.	Research	indicates	that	animals	tend	to	"go	for	more", selecting	maze-arms, feeders and other experimental setups that have the largest number	of food items. For	example, free-ranging	adult salamanders	placed in	a	Tshaped	enclosure	that	could	choose	between	the	ends	containing	either	1	or	2	live flies,	or	2	or	3	live	flies,	chose	the	arm	of	the	enclosure	with	the	most	flies	(like	other amphibians,	salamanders	can	only	visually	see	small	stimuli	if	they	move).	However, they showed no preference if the choice	was between 3 and 4 or 5 and 6, again revealing	limits	to	the	OFS	(Uller	et	al.	2003). Petroica australis, a food-caching songbird, shows sophisticated reliance on numerical information when storing, retrieving and pilfering caches of food (mealworms).	The	birds	could	watch	food	being	put	in	a	pair	of	artificial	cache	sites, and	could	choose	one	of them.	They	were	successful in finding the	cache	with the most	mealworms (experimenters controlled for duration and other non-numerical confounds) in caches up to 12 items. The experimenters also did a violation of expectation	experiment,	where	birds	watched	a	number	of	mealworms	being	stored, but	only	a	subset	was	findable,	and	they	examined	whether	these	birds	would	take	a longer time searching for the remaining worm(s). This study revealed that birds looked	longer	in	2	vs.	1	in	3	vs.	2,	but	not	in	8	vs.	4	conditions,	perhaps	because	they were subject to the limitations of OFS which is especially operative for keeping numerosities in	working	memory	(Hunt	et	al.	2008). Animals	also	use	numerical	information	for	selecting	their	territory	(Nelson	and Jackson	2012),	and	for	choosing	whether	or	not	to	attack	a	rivaling	group,	based	on	a comparison	of	that	group's	size	and	the	own	group's	size	(e.g.,	McComb	et	al	1994 for	a	study	with	wild	lionesses).	Shoaling	fish	choose	to	aggregate	with	shoals	based on	their	perceived	size,	for	example	guppies	prefer	a	shoal	of	8	over	a	shoal	of	4	to aggregate	with	(Bisazza	et	al.	2010). 3.3	Similarities	in	computational	characteristics	and	limitations As	we	have	seen,	the	Object	File	System	(OFS)	and	the	Approximate	Number	System (ANS) are the dominant ways to explain human numerical cognition. Both have specific	limitations	and	characteristics.	The	OFS	is	accurate	for	collections	of	items	up to	3	or	4.	It	allows	animals	to	make	comparisons	and	calculations	across	modalities. For example, rhesus	monkeys (Jordan et al. 2005) and human infants (Jordan and Brannon	2006)	can	match	the	number	of	voices	they	hear	to	the	correct	number	of speaking heads they see on a monitor. It also supports addition and subtraction. Infants, as well as domestic dogs, show surprise at unexpected additions and subtractions, such as 1 + 1 = 1 or 2 – 1 = 2 (Wynn 1992,	West and Young 2002). Above	3	or	4,	the	OFS	is	not	able	to	make	calculations	or	comparisons	anymore.	For example,	chicks	can	discriminate	between	displays	of	1	and	2,	and	between	2	and	3 items,	but	not	between	3	and	4,	or	4 versus	5,	or	4 versus	6 (Rugani et al. 2008), 8 although chicks can tell the difference between larger numbers when the ratio difference between them is large enough, e.g., 2 vs. 8 and 8 vs. 32 (Rugani et al. 2015). We saw above that this limitation was also observed in bees and in salamanders, and in human infants. Human numerate adults can, of course, distinguish	between	collections	of	3	and	4,	or	4	and	6.	Yet	even	adults	are	subject	to the limitations of the OFS: they are much more accurate in enumerating small collections	of	items	(up	to	3)	than	larger	collections,	with	a	steep	decline	in	precision after 3 (Revkin et al. 2008). The explanation for this limitation of the OFS is that there	are	inherent	limitations	to	working	memory.	The	OFS	works	by	putting	mental representations	of	discrete	objects	(e.g.,	two	bananas,	one	sound	and	one	dot)	in	a placeholder format as slots that are kept in	working	memory (Feigenson	& Carey 2005). The ANS, unlike the OFS, does not have a strict limit on how much it can represent, although experimental setups typically stay under 100. This system handles	the	approximate	representation	of	numbers,	and, like	the	OFS, it	can	work across	modalities,	and	it	supports	addition	and	subtraction	(Barth	et	al.	2005).	Next to these features, its outputs show the Weber-Fechner signature: the discriminability	of	two	magnitudes	(numerosities) is	determined	by	their	ratio.	As	a result, numerical judgment improves with increasing distance (e.g., it is easier to discriminate	2 from	8 than	7 from	8,	not	only if this is	presented	as collections	of dots	but	even	in	symbolic	format	(see	Moyer	and	Landauer	1967	for	the	first	classic study	to	show	the	distance	effect	in	symbolic	format). Comparative research indicates that rhesus monkeys' performance on approximate arithmetical tasks is similar to that of college students. Students and rhesus	monkeys	were	required	to	mentally	add	a	number	of	dots	and	select	a	display that	showed	the	sum	(e.g.,	for	displays	of	1	and	7	dots,	the	display	containing	8	dots had to be selected).	Next to the	display showing the correct sum (e.g., 1 + 7 = 8) there	was	a	distractor	display	that	contained	an	incorrect	number	of	dots	(e.g.,	1	+	7 = 5), which had a cumulative surface area close to the correct solution. Although adults were more correct (94% correct answers, compared to only 74% for the monkeys), their response	patterns	were very similar, showing similar sensitivity to the ratio	between	the	numerical	values	of the	sum	and	choice	stimuli, in line	with the	Weber-Fechner	law	(Cantlon	and	Brannon	2007).	In	a	direct	comparative	study, pigeons	performed	on	a	par	with	primates	in	numerical	tasks	such	as	ordering	cards with different numbers of items in ascending order, showing very similar distance effects,	i.e.,	better	performance	if	numerosities	lie	further	apart	(Scarf	et	al.	2011). There have been a few systematic studies that have examined	whether the ANS in non-human animals other than primates obey the Weber-Fechner law. Gómez-Laplaza	and	Gerlai (2011) showed	that	angelfish (Pterophyllum	scalare) can choose	the	larger	of	two	shoals,	and	that	their	number	discrimination	is	sensitive	to the	ratio	difference	between	the	two	groups,	e.g.,	they	prefer	to	aggregate	with	the larger	shoal	if	the	differences	are	4:1	(e.g.,	12	vs	3),	3:1	(9	vs	3),	and	2:1	(8	vs	4),	but not	at	smaller	ratio	differences,	e.g.,	1.5:1	(9	vs	6	and	6	vs	4).	Carrion	crows	(Corvus corone) can discriminate numbers up to 30 (in displays that controlled for total surface	area)	in	line	with	the	Weber-Fechner	law	(Ditz	and	Nieder	2016).	While	more research	would	need	to	be	carried	out	to	see	how	far	this	generalizes,	the	research so far supports similar cognitive mechanisms of OFS and ANS underlying animal 9 numerical	competence	in	a	wide	range	of	species. 3.4 Neural correlates of numerical cognition shows evidence of convergent evolution In human brains, several areas of the neocortex are associated with numerical cognition,	in	particular	the	bilateral	intraparietal	sulci.	This	area	is	active	when	adults engage	in	calculation	with	Arabic	digits	and	dots	(e.g.,	Dehaene	et	al.	1999)	or	even participants are	merely	passively looking	at	or listening to	Arabic	digits	or	number words	(Eger	et	al.	2003).	The	intraparietal	sulci	are	also	active	in	four-year-olds	and in adults	when presented	with visual displays of collections of items that differ in number	(Cantlon	et	al.	2006).	Homologous	areas	in	the	primate	parietal	cortex	and prefrontal cortex are responsive to numerosity. Recordings of single neurons responses	in	the	brains	of	monkeys	show	that	there	are	number-sensitive	neurons	in the lateral prefrontal cortex and the intraparietal sulcus of the posterior parietal cortex.	These	number-sensitive	neurons	selectively	respond	to	a	specific	number	of items	in	visual	dot	displays,	including	zero.	Their	response	does	not	vary	with	other spatial features, such as the size of dots, but seems to be number-specific.	While they preferentially fire at a given number of dots (say, 3), they	will also respond, albeit less frequently, to other numerosities (say, 2 or 4), with response patterns following a	Gaussian curve around the preferred numerosity (Tudusciuc	&	Nieder, 2007). Bird numerical cognition is situated in the endbrain,	more specifically in the nidopallium caudolaterale. The neurons of crows in this part of the brain fired selectively for different numerosities, just like they did in rhesus	monkeys, e.g., a neuron	selectively	tuned	to	4	items	also	responded,	but	to	a	lesser	extent,	to	3	and	5 items (Ditz and Nieder 2015). Primates and birds have markedly different brain structures.	Their	last	common	ancestor	lived	about	300	million	years	ago,	at	a	time when the six-layered neocortex (which hosts, among others, the neurons responsible for numerical cognition) had not evolved yet in mammals. Thus, the similarities	between	crows	and	rhesus	monkeys	in	neural	representation	of	number show	a	striking	convergent	evolution. The similarities between insect and	mammalian (including	human)	numerical cognition cannot be due to homologous neural structures either. The European honeybee	only	weighs	0.1g,	and	its	brain	only	weighs	0.001	g,	with	a	total	size	of	1 mm3,	and	about	1	million	neurons.	Compared	to	the	human	brain	with	its	100	billion neurons, it has only 1/100,000th of the number of human neurons. The main functions relating to memory and adaptive decision-making are situated in the mushroom bodies and the central complex, so this is likely also where numerical cognition	takes	place.	Unfortunately, it is	not	possible	at	present	to	find	the	neural correlates for	numerical	cognition in	such	a	small	brain (see,	however,	Greco	et	al. 2012	for	recent	advances	in	scanning	brains	of	live	bees).	Given	these	similarities	in processing, in spite of very different neural implementation, insect numerical cognition	presents	another	case	of	convergent	evolution. 4.	The	metaphysical	significance	of	INC The	behavioral	and	neural	similarities	in	the	numerical	cognition	of	a	wide	diversity of	species	and	clades is	a	remarkable	phenomenon,	which I termed	invariantism	in 10 numerical	cognition	(INC).	It	cannot	be	explained	by	homology	(similarities	due	to	a shared	ancestral	trait)	given	how	divergent	insect,	avian	and	mammalian	brains	are. If homology cannot explain INC,	what alternative do	we have?	Homology is often contrasted	to	homoplasy	(similarity	due	to	independent	evolution),	but	homoplasy	is a portmanteau term for several distinct evolutionary patterns (Hall 2013). One of these is convergent evolution, when similar features evolve independently in different	species	as	a	result	of	similar	evolutionary	pressures.	For	example, insects, birds	and	bats	developed	wings	that	help	them	to	escape	predators	or	pursue	prey. INC is a good candidate for convergent evolution: a trait that emerged in diverse clades	as	a	result	of	similarly	evolutionary	pressures.	An	alternative	explanation	for INC is homology, when traits that evolve through convergent evolution share a similar genetic regulatory	apparatus. Examples include the	Pax6	gene,	which	helps regulate vision in mollusks, vertebrates, and insects, and the FoxP2 gene, which regulates	human language	development	and song	production in songbirds (Scharff and 2011). However, even in these cases of deep homology there is considerable independent evolution to accommodate anatomical differences (e.g., the eye structure	of insects	versus	mammals).	Moreover, if	the	structures in	question	were not adaptive, it is unlikely that these deep homologies would have occurred. For example,	the	Pax6	gene	regulates	the	prenatal	development	of	eyes,	such	as	the	iris, and	its	function	can	be	explained	by	the	fact	that	seeing	is	adaptive.	Thus,	even	if	a deep homology underlies numerical competence in these	widely divergent clades, the	similarities	between	them	remain	striking. Some	of	the	convergence	in	numerical	cognition	across	clades	likely	has	to	do with constraints in computation and memory, including the Weber-Fechner signature	and	the	limitations	of	the	OFS.	Nothing	of	mathematical	interest	happens when natural numbers > 3, it is just a limitation of	working	memory.	Why	would animals	be	better at	discriminating smaller	numerosities, and	why	would the ratio difference	be	more	relevant than	the	absolute	difference?	The	difference	between small numbers is often more ecologically significant than that between large numbers, for	example,	to	a	hungry	foraging	monkey, it is	more	relevant	to	see	the difference	between	a	patch	with	one fruit versus two fruit than it is to	be	able to distinguish	between	11	and	12	fruits.	This	ecological	function	of	numerical	cognition leads me to posit the following claim: INC presents substantial evidence for mathematical realism. It indicates that animals are tracking something in the environment	(numerosities),	and	realism	is	the	best	explanation	for	numerosities. In	an	earlier	paper	(De	Cruz	2016),	I	outlined	an	indispensability	argument	for mathematical realism from numerical cognition. I proposed that the best explanation for numerosities involves numbers-animals make representations of magnitude	in	the	way	they	do	because	they	are	tracking	structural	(or	other	realist) properties of numbers. This fits in an ongoing discussion on whether physical phenomena have genuine mathematical explanations. Baker (2005, 2015) has argued that this is the case, citing such cases as the	primeness	of the life-cycle	of insects that are members of the genus	Magicicada and structural properties of honeycombs. In the	case	of	Magicicada, their life cycles	are	either	13	or	17	years. These consist of a long	phase they spend	as larvae	underground	and	a	brief adult phase	spent	above	ground	when	they	reproduce.	The	primeness	of	their life	cycles makes	it	less	likely	that	the	life	cycles	of	predatory	species	would	intersect	with	them, 11 thus increasing their reproductive success. Primeness is a mathematical property that	plays	a	relevant	role	in	the	biological	explanation	for	why	their	life	cycles	have these durations. Such examples are used to bolster the case for platonism about mathematical	objects. I will not here reiterate these arguments, but instead will consider ClarkeDoane's (2014,	manuscript)	more recent challenge.	Now, the	mathematical beliefs Clarke-Doane	targets	are	those	of	professional	mathematicians,	such	as	the	axioms of	set	theory,	rather	than	more	elementary	beliefs	such	as	that	7	is	prime,	or	that	2	+ 2 = 4. Indeed, he is happy to concede that the latter	would be safe, just like the belief that	burning	babies for fun is	wrong is true for	any	moral	non-error theorist (Clarke-Doane 2014). However, the evolutionary challenge against mathematical realism targets those more basic beliefs too, just like evolutionary debunking arguments	against	moral realism	challenge fundamental	moral	beliefs such	as that pain	is	bad. With INC	we	have	a	clear	disanalogy	between	mathematics	and	morality:	the proto-moral	beliefs	of	different	species	are	divergent,	whereas	numerical	cognition is invariant across species. This	makes numerical cognition less susceptible to the contingency	challenge	that	has	been	proposed	against	moral	realism.	In	the	case	of moral	realism,	one	can	see	how	our	beliefs	would	easily	have	been	different if	our evolutionary	history	had	gone	a	different	way.	Our	moral	beliefs could	have	easily been	false	(assuming	a	non-naturalist	form	of	moral	realism1),	but	our	mathematical beliefs	could	not	have	been.	This	is	because	evolution	has	shaped	our	minds	(as	well as	those	of	bees,	crows,	rhesus	monkeys,	chicks,	angelfish,	etc.) to	track	numerical information. Similarities in numerical cognition across a wide range of unrelated species require some explanation, and mathematical realism can provide this explanation straightforwardly, namely	what animals are tracking are	mathematical truths/structures.	I	am	not	arguing	that	antirealists	cannot	explain	INC.	Nevertheless, the	anti-realist	would	need	to	explain	why	unrelated	animals	such	as	salamanders, bees,	crows,	angelfish	and	rhesus	monkeys	(and	of	course	humans),	would	be	able	to track discrete quantities in their environment, would be able to do so across modalities,	and	would	use	this	information	to	inform	their	adaptive	choices.	INC	thus shifts	the	burden	of	proof	in	the	direction	of	the	antirealist. One can, of course, resort to highly contrived scenarios	where animals have 1	Some naturalistic forms of	moral realism are less susceptible to the contingency challenge,	in	particular	the	neo-Aristotelian	approach	to	morality	(as	e.g.,	outlined	by Foot,	2001).	According to	neo-Aristotelians,	what	counts	as	a	good	human life	and human flourishing is the truth-maker of moral claims. Humans have, as evolved creatures, certain limitations on the conditions that will make them thrive and flourish. The role of the ethicist is to find out how to fulfill these conditions. Fitzpatrick (2000)	has	challenged the	neo-Aristotelian	account	by	pointing	out that not	all	evolved	features	lead	to	flourishing,	for	example,	male	elephant	seals	fight	to gain	control	of	large	harems,	which	makes	evolutionary	sense	but	does	not	seem	to contribute to their wellbeing. However, as Lott (2008) has countered, the neoAristotelian	approach	does	not look	at	animals from	the	outside,	but instead from the inside of life-forms. In that respect, it would seem that it is "good" for bee queens	to	kill	their	fertile	daughters,	to	harken	back	to	Darwin's	example. 12 adaptive responses, such as choosing the most numerous shoal or cache of mealworms,	without tracking	mathematical truths. Plantinga's (1993) evolutionary argument against naturalism famously argued that animals can have the right adaptive behaviors without truth-tracking beliefs, e.g., a hominin who runs away from	a tiger (adaptive	response),	but	does	so	because	he	believes the tiger is	cute and	he	wants	to	pet	it,	but	he	also	believes	that	the	best	way	to	pet	it	is	to	run	away from it (maladaptive	belief).	While	such	scenarios	are	metaphysically	possible (and some	have	outlined	them	for	the	case	of	numerical	beliefs,	e.g.,	Clarke-Doane	2012), they	are	not	very	plausible.	An	error-theorist	would	have	to	come	up	with	a	scenario for	each	case	of	evolved	numerical	cognition	(which,	to	the	best	of	our	knowledge, has occurred independently at least in insects, birds, mammals, and fish) where somehow	wrong	or	irrelevant	mathematical	beliefs	would	lead	to	the	right	adaptive responses. At present, there is no satisfying positive case for mathematical antirealism	that	accounts	for	INC	without	resorting	to	arcane	scenarios. 5.	Which	form	of	realism	does	the	animal	cognition	literature	support? The	Benacerraf-Field	challenge to	mathematical realism	asks realists to	explain the reliability of mathematical beliefs. This is not as demanding as outlining a causal account	(which	would	be	impossible	under	some	forms	of	realism	in	any	case),	but requires	us	to	show	that	our	evolved	mathematical	beliefs	are	safe	from	error.	I	have argued	in	previous	work	(De	Cruz	2016)	that	Shapiro's	(1997)	ante	rem	structuralism is	a	possible	candidate	in	the	light	of	evolution.	One	reason	to	look	more	closely	into realist	structuralist	accounts	is	that	authors	in	this	field,	such	as	Shapiro	(1997)	have made	substantial	efforts to	explain	how their	account	would	work in	a	naturalistic framework.	Moreover,	ante	rem	structuralism	provides	a	straightforward	account	of reference and semantics, and can provide an account of mathematical structure irrespective of the agent cognizing it, which	makes the approach suitable for our explanation of numerical cognition across species. To summarize, ante rem structuralism	holds	that	non-applied	mathematics	is	concerned	with	structures	that are conceived	of	as	abstract	entities (platonic	universals), i.e., structures that	exist independently	and	prior	to	any	instantiations	of	them.	Ante	rem	structuralists	do	not specify the	precise	nature	of these	entities,	but rather focus	on	the	role they	play. Numbers are positions in a certain structure, and can be discerned in the environment as patterns. The bootstrapping account (Carey 2009) can offer a glimpse of how we can have mathematical beliefs that are safe, and that track mathematical structures. According to this account, young children learn to recognize the 1, 2 and 3 pattern, thanks to their OFS, which allows for exact discrimination	of	numerosities	up	to	3.	Since	the	OFS	is	very	precise,	learning	the	1,	2 and	3	pattern	is	a	reliable	process	(at	least	in	neurotypical	children	who	do	not	suffer from dyscalculia). The children learn the remaining natural numbers through a process of induction. This part of the learning process is stable thanks to the abundant cultural scaffolding (e.g., counting songs) and feedback (e.g., parents correcting their child, or helping their child to count a given collection of items) children	receive (see	also	De	Cruz	2018). In this	way,	an	ante	rem structuralist	can explain	the	reliability	of	our	natural	number	concepts	to	track	mathematical	truths. Another	plausible	realist	(non-platonist,	in	this	case)	account	that	is	compatible 13 with INS, is Millean empiricism. Kitcher (1984) revived this position, arguing that mathematical epistemology should seek inspiration from how children learn arithmetic. It thus fits well in a naturalistic account of mathematical cognition. A closely	related	view	is	Aristotelian	realism,	recently	defended	by	Franklin	(2014).	Mill (1843,	165)	proposed	that	numbers	are	properties	of	physical	aggregates: When	we	call	a	collection	of	objects	two,	three,	or	four,	they	are	not	two, three,	or	four	in	the	abstract;	they	are	two,	three,	or	four	things	of	some particular	kind;	pebbles,	horses,	inches,	pounds'	weight.	What	the	name of	number	connotes is, the	manner in	which	single	objects	of the	given kind	must	be	put	together,	in	order	to	produce	that	particular	aggregate. In	this	view,	numerical	cognition	detects	high-level,	general	properties	of	aggregates, e.g.,	an	angelfish	that	chooses	a	shoal	of	seven	fish	over	three	fish is	detecting	the high-level general property	of aggregates that 7 > 3.	According to	Mill,	we	do	not need	to	invoke	the	existence	of	3	and	7,	separate	from	their	concrete	instantiations in the physical	world.	Millean empiricism	does not presuppose platonist ontology, but it is nevertheless a realist ontology (see	Balaguer 1998, chapter 5), because it regards	the	laws	of	arithmetic	as	highly	general	laws	of	nature. A common objection to Millean empiricism is that aggregates do not have determinate	number	properties. For	example, a	group	of lions can	be	divided into many different parts, for instance, it is composed of 7 lions, 28 legs, etc. Kessler (1980)	responds	to	this	problem	by	arguing	that	in	Mill's	account,	numbers	are	not properties	of	aggregates,	but	relations	that	hold	between	aggregates	(e.g.,	the	pride) and properties of those aggregates (e.g., individual lions). Infants and animals are successful at finding the relevant properties of aggregates in numerical tasks, for instance, they can compare the number of speakers they see	with the number of voices they hear (Jordan & Brannon 2006). When they are presented with a collection	of	objects	(e.g.,	an	array	of	dots)	infants	seem	to	be	less	able	to	detect	a decrease	or	increase	in	the	individual	objects'	size,	than	they	are	to	detect	a	change in numerosity. They need as much as a four-fold change in size to notice it, as revealed by a longer looking time. This suggests that once infants attend to numerosity,	they	disregard	the	physical	particulars	of	the	items	that	constitute	them (Cordes	&	Brannon	2011).	In	line	with	Millean	empiricism,	they	can	make	high-level generalizations about numerosities that go beyond the physical properties of aggregates.	Given that the	world	at	our scale	mostly	consists	of separable	objects, there	may	have	been	an	evolutionary	advantage	of	making	high-level	generalizations about numerosities, along the lines of separable objects as	we and other animals encounter	them	in	daily	life	(see	also	Dehaene	2011,	231). Ante	rem	structuralism	and	Millean	empiricism	are	two	realist	ontologies	that are compatible with the evolved features of numerical cognition. Both meet the Benacerraf-Field	challenge	of	explaining	the	reliability	of	numerical	representations. For the structuralist account, direct interaction with structures is not required to know numbers, and for Millean empiricism, numerosities form a high-level generalization	of	the	properties	of	discrete	middle-sized	objects. 6.	Conclusion 14 In this	paper, I	have	argued that	mathematics	and	morality	are	disanalogous in	an important respect. Mathematical beliefs seem to be less contingent upon our peculiar	evolutionary	history than	moral	beliefs	are. I	have	presented	evidence for invariantism	in	numerical	cognition:	numerical	cognition	occurs	across	many	animal clades, including	insects,	fish,	amphibians,	birds	and	mammals,	and	is,	according	to the	dominant	theories	on	numerical	cognition,	subserved	by	two	systems:	the	ANS, which deals with larger collections of items through approximation, and an exact system	for	small	numerosities	up	to	3	or	4	(the	OFS).	Numerical	information	plays	a crucial role in	animal	decision-making.	Animals	across	widely	different	clades	show similar cognitive limitations and strategies in dealing with numbers, including an ability to deal with numbers across	modalities. Neural evidence suggests	multiple instances	of	convergent	evolution.	If	animal	minds	have	hit	upon	these	solutions	so many times independently, this would be a formidable coincidence which antirealists	would	need	to	explain.	Of	course,	INC	also	requires	an	explanation	under the assumption of mathematical realism. In particular, the Benacerraf-Field challenge asks the mathematical realist to explain the reliability of mathematical beliefs. If this were in principle impossible to achieve, this would undermine our mathematical	beliefs,	according	to	Field (1989).	The	Benacerraf-Field	challenge	can be	cashed	out	in	terms	of	safety:	the	realist	needs	to	show	that	we	could	not	easily have	had	false	mathematical	beliefs. I showed that ante rem structuralism and Millean empiricism provide a solution	to	the	Field-Benacerraf	challenge:	they	can	explain	the	reliability	of	animal numerical beliefs, and thus by extension of human	mathematical beliefs that are based upon them, such as the belief that 11 follows 10, or that 7 is prime. My argument does not provide an evolutionary justification of more formal mathematical	beliefs,	such	as	those	involved	in	set	theory. Acknowledgments Many thanks to Johan De Smedt, Justin Clarke-Doane, Brendan Larvor, Jan Verpooten, Anne Jacobson, two anonymous reviewers, and the editors of this volume	for	their	comments	to	an	earlier	version	of	this	paper. References Agrillo,	C.,	Dadda,	M.,	Serena,	G.,	&	Bisazza,	A.	(2008).	Do	fish	count?	Spontaneous discrimination	of	quantity	in	female	mosquitofish.	Animal	Cognition,	11(3),	495-503. Baker, A. (2005). Are there genuine mathematical explanations of physical phenomena?	Mind,	114,	223–238. Baker, A. (2015). 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