Tarik Lassouaoui
email: tarik.lassaouaoui@insa-lyon.fr

Florin Hutu
email: florin-doru.hutu@insa-lyon.fr

Guillaume Villemaud
email: guillaume.villemaud@insa-lyon.fr

Yvan Duroc
email: yvan.duroc@univ-lyon1.fr

Modulation Depth Enhancement for Randomly Arranged Tags in Passive RFID Tag to Tag Communications

Keywords: backscattering, passive tags, UHF RFID, tag to tag communications

In this paper, the performance evaluation of the tag to tag (T2T) communications using UHF (Ultra High Frequency) passive RFID (Radio Frequency Identification) tags is tackled. The T2T communication system is performed directly between a reader tag (RT) and a listener tag (LT) in the presence of an external electromagnetic (EM) field. Consequently, this kind of communication system implies the use of low RF signal levels, which makes it very sensitive to the noise, to the arrangement between the tags (i.e. spacing, tilt and shift) but also to the position of the source. The performance evaluation of this communication system is based on the study of the modulation depth, an indicator of the separation between the two modulation states and, indirectly, of the T2T communication quality. Considering scenarios in which the two tags are in the same plane but with the RT tilted and staggered with respect to the LT, our study demonstrates that the modulation depth can be kept higher for different separations between the tags by choosing the proper angle of incidence of the RF source with respect to the tag's plane. The proposed approach is studied through electromagnetic simulations and the obtained results are confronted with measurements.

I. INTRODUCTION

The RFID (Radio Frequency IDentification) is a constantly developing technology having numberless applications as logistics, access control, real-time localization or sensor networks [START_REF] Finkenzeller | RFID handbook: fundamentals and applications in contact-less smart cards and identification[END_REF]. Moreover, the development of the IoT (Internet of Things) requires ultra-low power or even battery free devices (sensor, actuators, etc.) and the RFID is an excellent candidate to reduce the energy footprint of the radio communications part. Particularly, the use of an external RF signal in order to carry at a short distance the information between two devices (backscatter principle) can be implemented by the means of RFID [START_REF] Nikitin | RFID paperclip tags[END_REF]. Nowadays, the RFID tags sensitivity remains the key issue since this parameter is defining the communication range.

The concept of tag to tag (T2T) communication has been introduced in [START_REF] Nikitin | Passive tag-to-tag communication[END_REF]. Here, the authors demonstrates experimentally that two parallel passive or semi-passive tags closely spaced can directly communicate even in the absence of an RFID reader. The main idea is to establish a communication between a reader tag (RT) and a listener tags (LT) closely spaced with a reliable communication distance of about 25 millimetres. The envisaged T2T application scenarios demand to increase the communication range. Among these applications, a simple and powerless system allowing the personal protective equipment (PPE) usage monitoring may be implemented. The communication between the tags attached to each safety equipment can only be established when their relative positions have a certain configuration. By verifying the communication between tags, potential misalignment or negligence of the user can be avoided. Consequently, the full understanding of the mutual coupling among the tags and of the intrinsic electromagnetic (EM) characteristics of the antennas is necessary.

The T2T communication performance is dependent of the tag's relative position [START_REF] Nikitin | Performance limitations of passive UHF RFID systems[END_REF], [START_REF] Marrocco | Electromagnetic models for passive tagto-tag communications[END_REF]. Furthermore, the close communication range between tag's antennas brings up an important electromagnetic coupling. Consequently, the tag's antennas characteristics such as the radiation pattern or the input impedance are modified. These alterations have an impact on the T2T's communication performance since the modulation depth is potentially reduced.

To the best of our knowledge, few works have been done about this topic to improve this communication range. In [START_REF] Lassouaoui | Theoretical BER evaluation of passive RFID tag-to-tag communications[END_REF], a simulation framework has been proposed as a tool allowing the performance evaluation of T2T radio links in terms of Bit Error Rate (BER). The analytical formulation of the probability of error in the case of a 2-ASK (Amplitude Shift Keying) modulation for coherent and non-coherent LT's receivers are given. In [START_REF] Karimi | Design of a backscatter-based tag-to-tag system[END_REF], the authors proposed a power efficient receiver working at modulation depth higher than 75% which was In this context, our study focuses on the impact of the different tag's arrangements (e.g. tilt or shift) which impacts the antenna's intrinsic electromagnetic characteristics with a direct consequence on the modulation depth. This study will be supplemented by electromagnetic simulations results and the measurements obtained with printed dipoles. Moreover, the constructive or destructive impact of the orientation of the external source with respect to the two tags is also studied. The rest of the paper is organized as follows. In Section II, the tag's antennas mutual coupling phenomena is described in order to study the LT's input impedance. In Section III, results are exploited in order to express the modulation depth and to study it for different T2T communication scenarios (spacing, tilt and stagger). Finally, the conclusion and the perspectives of this work are given in Section IV.

LISTENER TAG'S IMPEDANCE BEHAVIOR IN PASSIVE T2T COMMUNICATIONS

As stated, a T2T communication system operates at short distance, making it very sensitive to the tags configuration (i.e. spacing, tilt or shift). The electromagnetic coupling between the closely spaced antennas modifies their intrinsic electromagnetic characteristics such as radiation pattern and realised impedance.

The close interaction between the two tags is such that they have to be considered as an array of two coupled antennas characterized by its impedance matrix. The reader tag communicates by switching its load on an Open Circuit (OC) or a Short Circuit (SC). Depending on the RT's load, the LT tag's IC (integrated circuit) observes a particular value of the input impedance Z in according to the tags relative position. It is worth noting here that the gain of the two coupled antennas is also depending on the position of the source which can be a classical RFID reader or any other source providing a power level high enough to activate the tags (source is considered in far field).

A. Theoretical T2T antenna's mutual coupling

Two passive tags are placed in close vicinity one to each other. Each tag has an identical infinitely thin dipole antenna designed such that its intrinsic impedance (i.e. the impedance when the antenna is in free space) is Z 11 = 50 Ω. The currents induced in the two dipoles are assumed to be sinusoidal and at a frequency of 868 MHz. The RT's antenna is positioned at (x RT , y RT ) and its tilt angle with respect to y axis is θ RT . The distance d between the two tags is considered as fractions of the wavelength.

Fig. 2 presents the mutual impedances Z 21 of two coupled thin dipoles as a function of RT's antenna tilt angle θ RT for different spacing distances λ 4 , λ 2 and λ respectively. The theoretical results are calculated by applying the IEMF (Induced Electromotive Force) method [START_REF] Baker | Digital computation of the mutual independance between thin dipoles[END_REF]. As expected, the spacing between the antennas and their tilt change significantly the mutual coupling. The mutual coupling decreases until it is canceled while the distance d between the two antennas and the tilt angle θ RT increases.

B. Listener tag's input impedance

To study the impact of the tags configuration (spacing, tilt and stagger) on the modulation depth D, simulations using CST Microwave studio were performed on a system of two closely coupled dipoles as shown in Fig. 1. The dipoles were 134 mm × 1 mm copper traces on FR4 material (substrate dimensions 150 mm × 10 mm). The dimensions of the antennas have been calculated so as to have a low power reflected from the antenna S 11 ≤ -22 dB at 868 MHz. These simulations are confronted with measurement results using a VNA (Vector Network Analyzer) with the same setup as in Fig. 1.

The input impedance Z in is measured and compared with simulation results for various configurations. The two closely coupled tags are seen as a matrix of two antennas which are considered identical. Assuming the reciprocity between the two tag's identical antennas, we obtain Z 12 = Z 21 and Z 11 = Z 22 .

In the two RT's loads (Z L,ON and Z L,OF F ), the input impedance of the listener tag Z in can be written as follows: The real and imaginary parts of the LT's input impedance Z in are presented in Fig. 3, 4 and 5, 6 for different configurations: one with RT staggered and then tilted respectively. The Z in impedance is given as a function of the distance d between the antennas and with respect to the two RT's commutation states. As can be seen from Fig. 4 and6, the presence of RT's short circuited antenna modifies the input impedance of the LT's antenna. Moreover, we can remark the good agreement between the measurement and the EM simulations. The difference between the measured and simulated values is small, more precisely ∆R in,SC&OC < 2.7 Ω and ∆X in,SC&OC < 9.8 Ω.

Z in,ON/OF F = Z 11 - Z 21 Z 12 Z 22 + Z L,ON/OF F (1)

III. ANALYSIS OF THE MODULATION DEPTH

The T2T communication requires an RF source (dedicated or not) with sufficient power to ensure tags activation. In our study, we considered a source that provides a continuous Wave (CW). The external RF source orientation θ s is considered relative to the z axis as shown in Fig. 1. Due to the commutation state of RT, the amplitude of the signal at the LT tag's input changes between two levels A ON and A OF F . These parameters are defined by the LT's antenna input impedance Z in, ON/OF F and the overall gain of the two antennas G ON/OF F in the direction of the considered farfield source, as shown in [START_REF] Nikitin | RFID paperclip tags[END_REF].

A ON/OF F = A in • G ON/OF F 1 - Z in, ON/OF F -Z 0 Z in, ON/OF F + Z 0 (2)
where A in is the input voltage at LT and G ON/OF F are the voltage gains and Z 0 characteristic impedance. Starting from these two amplitude levels A ON and A OF F , the modulation depth D is defined as in (3):

D = |A ON -A OF F | max(A ON , A OF F ) • 100[%] (3)
This parameter is taken as an indicator of the T2T communication quality. Indeed, the greater the modulation depth is, the easier is for the LT's baseband circuit to distinguish between the two states of the 2-ASK modulation.

As stated, the two tags relative position together with the external source position have an influence on the modulation depth D. In order to study this influence, a first scenario where RT tag is tilted by θ RT = 20 • and a second scenario where RT tag is staggered by 0.2λ were simulated by considering the RF source placed perpendicular to the plane containing the two tags. As can be seen from Fig. 7, the maximum modulation depth is obtained for short distance between the tags. Globally, the modulation depth decreases when the two tags are separated.

In Fig. 7, the modulation depth for different distances between the two tags is given once when the RT tag is tilted by 20 • and then when RT tag is staggered by 0.2λ. The results show the importance of having a parallel arrangement even if RT is staggered in order to maximize the modulation depth. Unlike the tilted case, the staggered arrangement for short separation distances does not have a negative impact on the modulation depth, the coupling in this case being reinforced (D = 84% at d = 0.1λ), thus facilitating the T2T communication.

The position of the source has also an effect on the modulation depth. In Fig. 8, the modulation depths for different source orientation (θ s ) are given. The incident angle of the source can be used to enhance the modulation depth D depending on the separation distance between the tags. Therefore, it is important to know which is the best source position relative to the tag plane that maximizes the modulation depth.

IV. CONCLUSION

Considering a passive T2T communication, this paper investigated the impact of the mutual coupling between the tags, their relative position and the source orientation, notably on the modulation depth. Indeed, higher the modulation depth is, more efficient the T2T communication is in terms of detection and noise robustness. Besides the relative arrangement of tags that modifies input impedances and radiation properties, the orientation of the RF source is also a critical parameter to take into consideration. The presented results describe the variation of the modulation depth for various scenarios. In consequence, with the objective of implementing a tag to tag system, our study would allow predict the T2T communication performance or establish specification guidelines.

In this study, only simple load impedances have been considered (short and open circuits). In perspective, a promising way would be to exploit complex load impedances in order to improve the modulation depth, and so the potential performance of the T2T communications.

Fig. 1 .

 1 Fig. 1. Geometrical configuration of two printed dipoles, RT's antenna is spaced and tilted with respect to LT.

Fig. 2 .

 2 Fig. 2. Real and imaginary part of the mutual impedance Z 21 as function of θ RT tilt angle of the RT's thin dipole for different distances d.

Fig. 3 .

 3 Fig. 3. Comparison between the simulation results with measurements for two tags one staggered by 0.2λ. Input resistance (R in ) and reactance (X in ) as function of distance d between the tags when the RT tag is open circuited.

Fig. 4 .

 4 Fig. 4. Comparison between the simulation results with measurements for two tags one staggered by 0.2λ. Input resistance (R in ) and reactance (X in ) as function of distance d between the tags when the RT tag is short circuited.

Fig. 5 .Fig. 6 .

 56 Fig. 5. Comparison between the simulation results with measurements for two tags one tilted by θ RT = 20 • . Input resistance (R in ) and reactance (X in ) as function of distance d between the tags when the RT tag is open circuited.

Fig. 7 .

 7 Fig. 7. Modulation depth D at LT as function of the distance d between tags. RT is tilted by θ RT = 20 • and RT tag staggered by 0.2λ, the source is on LT side θs = 90 • .

Fig. 8 .

 8 Fig. 8. Modulation depth D enhancement by orientation of the source θs as function of the distance d between tags. Dmax = 51.57% at d = 0.3λ and D min = 21.21% at d = λ.

ACKNOWLEDGMENT

This work was supported by the SPIE ICS-INSA Lyon IoT chair.


Fabrice Pataut
email: patautfabrice@gmail.com

PLATONISM AND MATHEMATICAL EXPLANATIONS: SOME CRITICAL REMARKS ON EXPLANATORY PROOFS AND DEBUNKING ARGUMENTS (II)*

Keywords: Abstract Objects, Clarke-Doane (Justin), Colyvan (Mark), Conservativity, Debunking Arguments, Explanatory Proof, Field (Hartry), Indispensability Argument, Inference to the Best Explanation, Mathematical Explanation, Mathematical Realism, Mathematical Truth, Nominalism, Nominalization, Numbers, Ontological Economy, Platonism, Proof (Classical vs. non Classical or Intuitionistic), Quantification (Objectual vs. Substitutional), Realism, Rizza (Davide), Safety, Sensitivity

Ontological parsimony requires that if we can dispense with A when best explaining B, or when deducing a nominalistically statable conclusion B from nominalistically statable premises, we must indeed dispense with A. When A is a mathematical theory and it has been established that its conservativeness undermines the platonistic force of mathematical derivations (Field), or that a non numerical formulation of some explanans may be obtained so that the platonistic force of the best numerical-based account of the explanandum is also undermined (Rizza), the parsimony principle has been respected.

Since derivations resorting to conservative mathematics and proofs involved in non numerical best explanations also require abstract objects, concepts, and principles under the usual reading of "abstract," one might complain that such accounts turn out to be as metaphysically loaded as their platonistic counterparts. One might then urge that ontological parsimony is also required of these nominalistic accounts. It might, however, prove more fruitful to leave this particular worry to the side, to free oneself, as it were, from parsimony thus construed and to look at other important aspects of the defeating or undermining strategies that have been lavished on the disposal of platonism.

Two aspects are worthy of our attention: epistemic cost and debunking claims. Our knowledge that applied mathematics is conservative is established at a cost, and so is our knowledge that nominalistic proofs play a genuine theoretical role in best explanations. I will suggest that the knowledge one must acquire to show that nominalistic deductions and explanations do indeed play their respective theoretical role involves some question-begging assumptions regarding the nature and validity of proofs.

As for debunking, even if the face value content of either non numerical claims, or conservative mathematical claims, or platonistic mathematical claims didn't figure in our causal explanation of why we hold the mathematical beliefs that we do, construed or understood as beliefs about such contents, or as beliefs held in either of these three ways, we could still be justified in holding them, so that the distinction between nominalistic deductions or non numerical explanations on the one hand and platonistic ones on the other turns out to be spurious with respect to the relevant propositional attitude, i.e., with respect to belief.

Knowledge (II)

Suppose that in order to give a nominalistically respectable explanatory proof you need to know more than you need to know in order to give a mathematically loaded explanatory proof. The extra knowledge isn't mathematical but requires a fair amount of extra-mathematical or extra-numerical assumptions, theories, and techniques to be applied.

How should we assess the merits and demerits of the non mathematical cognitive cost? We may want to measure three things: (i) the sum total of theory one must appeal to, (ii) the length of proofs or formal derivations obtained given (i), and (iii) the range of application of the selected methods. It may be that, in certain cases, the theory appealed to is more complex, or that proofs and derivations happen to be longer and more cumbersome, or that the range of application of the methods is restricted compared to a bona fide mathematical derivation or numerical explanatory proof. In other words, it may turn out that in specific cases we give up our right to epistemic economy ((i) and (ii)), to elegance or beauty ((ii)), to strength ((iii)), or to some combination thereof.

It's important to find out how (i)-(iii) interact in order to assess the merits and demerits of nominalistic proofs. Should these proofs be preferred in spite of these drawbacks? If so, there must be a trade off between costs and rewards that speaks in favor of the nominalistic choice even though the cost happens to be high. The trade off isn't just a question of the psychological or mental collective effort we humans must muster in order to obtain a bona fide nominalistic proof or a nominalistically respectable derivation. We must explain why the effort has the features that it has rather than others it could have but just doesn't have. The assessment of epistemic cost is an objective matter, and the problem in this respect is to determine the means by which one will measure the costs of the trade off. It isn't merely a question of observing and computing how much time has been spent, how much collective stamina has been built up to secure the results, and so on and so forth. It is a question of why it should be so, and in this or that particular way.

Once we know why the nominalistic proof or derivation must be obtained in some particular way, we might perhaps appeal to principles that would justify a conclusion to the effect that the trade off is indeed worthwhile in spite of the cognitive cost; but that is a different matter altogethera matter that must be assessed on independent grounds, i.e., on grounds that must be independent of any consideration that would belittle the cost from the outset, or presuppose that its very consideration is downright irrelevant. Although we don't want to rule out the possibility that some philosophical principle or argument overrides the epistemological drawback or concern, we certainly mustn't put the cart before the horse.

Let us look once again at Field and Rizza's proposals. However different they may be, they share a common feature: In order to avoid a commitment to numbers in the derivation (Field) and in the explanation (Rizza), one must appeal to non numerical axioms whose models have suitable numerical representations. It is essential that the ordinary numerical treatment of the theory one wants to dispense with i.e., the treatment that prima facie commits one to numbers given a face value reading of the theorymay be recovered from a non numerical system of axioms, so that the platonistic reading of the mathematical theory taken at face value is undermined.

In Field's case, we're allowed to use mathematical existence assertions provided that the theory they belong to is conservative (Field 1980: 14). This is established by showing that, for number theory and ZF, consistency implies conservativeness given Robinson's joint consistency theorem [START_REF] Robinson | A Result on Consistency and Its Application to the Theory of Definition[END_REF]. The idea that mathematics applies conservatively to nominalistic versions of physical theories such as Newtownian gravitational theory (Field op. cit.: 19) requires modellability in ZF. What one gets then, in this case, is a platonistic proof that a physical theory "has all the nominalistically statable consequences that the usual platonistic formulation of the […] theory […] has" (Field 1980: 5). What makes it platonistic, presumably, is the recourse to ZF (with the axiom of choice and with or without urelements).

One may, according to Field, use platonistic methods of proof in arguing in favor of nominalism: Since the argument shows that abstract entities are not needed, either for ordinary inferences or for physical science, the Quine-Putnam indispensability argument thereby loses its clout (Field op. cit.: 6).

In Rizza's case, one establishes that mathematics plays an explanatory role "in a non-ontological way" (Rizza 2011: 113) by conceptualizing the structure of an empirical phenomenon so that "the explanation may be presented as a derivation from certain formal conditions" (Rizza: loc. cit.). Obviously, in this case, the epistemological worry has shifted from a focus on non causal abstracta, or on the reliability of beliefs in non causal mathematical objects, to a focus on formal or structural properties of some bona fide non-numerical explanation.

What I want to suggest here is that, even if we grant that platonistic methods are nominalistically respectable (Field), or that an emphasis on a commitment to concepts is preferable to an emphasis on a commitment to objects (Rizza), it remains that the axioms one relies on must possess certain features. Whether they be Suppes's axioms (Rizza) or Zermelo-Fraenkel's (Field), they must allow the recovery of what's usually appealed to in a mathematical practice that is unconcerned with philosophical worries about abstracta (whether they be concepts or objects in the ordinary (non-Fregean) sense).

What are the features that allow and indeed secure the recovery, and how do these axioms inherit them? The features are structural in the following sense. In Suppes's case, a relation that holds between consecutive points in a relational structure whose domain is both finite and non-empty, and on which a quaternary relation is defined, determines congruent intervals. So if one supposes that the lifecycles of Magicicadas do indeed satisfy Suppes's axioms, one may show that the numerical properties one wishes to dispense with are mere counterparts of purely empirical properties of time intervals that an empirically minded nominalist may be satisfied with. The reconstruction of a mathematical or numerical reasoning within a non-numerical, i.e., geometrical theory is possible only provided axioms of measurement theory correctly characterize finite equal-interval structures (see . They do so by fiat, and one has of course, no right to complain that this is how they enable us to secure the much-wanted recovery. Axioms are devised for a purpose, and one may certainly use them in any way one pleases as long as the use is consistent.

Now the fact that one has to establish the dispensability of numerical properties by showing that they have the particular ontological status of counterparts might be a source of worry. The counterpart claim that justifies the dispensability claim that would allow us to shoo platonism away is established given the acceptance of Suppes's axioms. These counterparts are not just "mere" or innocuous, ontologically light counterparts. They inherit a light ontological status because Suppes's axioms have models that may be described or investigated settheoretically as relational structures. The problem, of course, isn't with Suppes's axioms per se, but with the way in which they are used, especially Axiom 4.

As for Field, the conservativity principle guarantees that a nominalistically-stated assertion A* resulting from restricting each quantifier in A with the formula "not M(x)" 1 isn't a consequence of a mathematical theory unless it is logically true. The claim that the mathematical theories one resorts to do indeed satisfy that principle implies that our mathematical claims are true in all possible worlds, i.e., that they are a priori true (Field op. cit.: 13). Field considers the objection that standard, i.e., classical mathematics might after all turn out not to be conservative because it might "conceivably" be inconsistent. The weak reply to this objection is that a proof of the inconsistency of classical mathematics that would motivate a revision of mathematical theories and of mathematical practice is highly unlikely. Of course, Field offers more than a quasi-inductive reason for the belief that classical applied mathematics is consistent.

The key point here is that the conservativity of number theory and ZF is an (almost direct) consequence of their consistency, i.e., of Robinson's joint consistency theorem. It hardly matters whether such theories are true. What matters is that a mathematics that applies conservatively to a nominalized physical theory is modellable in ZF, and that the conservativity claim we need for the vindication of nominalism follows from the consistency of ZF. If conservativity is secured in this way, i.e., if it can be shown that the gap between the consistency of classical mathematics and its conservativity is that narrow, then both consistency and conservativity are indeed sufficient for the purpose of application.

It must be noted that no matter how narrow the gap might be, and although some form of elegance and beauty might be preserved in the cases covered 1 . The predicate "M(x)" means intuitively "x is a mathematical entity." See the first part of this article, sect 2, p. 68. in Field's account (in the arithmetical, the geometrical, and the Newtonian gravitational theory cases), the consistency of set theory that allows us to use mathematics conservatively is something we must assume. We must assume that the usual axiomatizations of set theory are indeed consistent (and we know from Gödel and Church respectively that if ZF were to be proved consistent, it would be incomplete and undecidable). So the avoidance of abstract counterparts that prima facie refer to numbers and the obtaining of "concrete" conclusions using the homomorphisms with those counterparts is secured at a cost. "Descending back to the concrete" requires us to find a consistent mathematical structure that allows us to prove the existence of structurepreserving mappings from nominalistically respectable concreta to the abstract structure.

Nominalistically respectable inferences and derivations are deemed correct and truth-preserving only provided that we may obtain the very same results without ascent into the abstract, so that the secured homomorphisms are just convenient devices. The epistemic cost of not ascending into the abstract must be evaluated in terms of what it must take to prove a "representation theorem." The proof must establish that for any model of the appropriate axiom system, there is a function that satisfies the appropriate "homomorphism conditions," mapping whatever nominalistically respectacle objects one needs in a nominalistically respectable derivation onto whatever platonistically respectacle objects one needs in a downright platonistic derivation. Although it would certainly be unjust to object that the vindication of nominalism thus conceived rests on the assumption that ZF is consistent, or on Gödel and Cohen's relative consistency proofs (because these may be turned into relative conservativeness proofs), it seems fair to note that the epistemic cost of showing that classical mathematics isn't true but merely useful in deducing "concrete conclusions" from carefully crafted nominalistic or nominalized theories is high indeed, both in terms of the derivations themselves and of their theoretical background.

Debunking

What is debunking? To debunk is to expose what was formerly held to be both true and warrant-ed as false and of vastly exaggerated value. When you debunk a belief or a theory, you don't merely expose the pretension of falseness, you somewhat ridicule the belief or the "theory" you take not to be a genuine one by undermining it in a strong way. In the particular case we're looking at, the debunking strategy purports to undermine both the mathematical beliefs and the claim that there are grounds in their favor by exposing their common causal origin. The causal origin disclosed by the explanation of how we come to have such beliefs is supposed to play an epistemic role with respect to both aspects of the undermining. The fact that the beliefs-cumjustifications exhibit a given causal pattern (or perhaps the fact plain and simple that they are caused) is supposed to provide sufficient grounds for rejecting both the beliefs and the justifications in their support as equally unwarranted, that is, as not constituting knowledge.

In particularly cutting versions of the strategy, the argument purports to denounce both as preposterous, perhaps even as irrational, i.e., as clearly not controlled or governed by reason, or by considerations that appeal to objective grounds and rationales. There's a lesson to be learned here, and the epistemic lesson is that one's confidence in the beliefs and their justifications must decrease as much as possible and possibly reach 0 on the scale from 1 (certainty that the beliefs are true and their justifications genuine) to 0 (certainty that the beliefs are false and their justifications spurious). The intermediate degrees of confidence exhibit one's particular sensitivity to the (alleged) force of the debunking argument.

What causes us to have the mathematical beliefs that we have? A number of simplistic answers may come to mind. One factor is their outstanding success. We successfully apply mathematics in a wide array of cases and that gives us reason to believe that the mathematical beliefs we happen to have are both true and warranted. We might argue counterfactually on this particular point: How could false mathematical results be successfully applied? Try to apply 2+2=5 and see what happens.

Another factor is provided by proofs and computations. We obtain the mathematical results we believe to be true and correct by the appropriate mathematical means, and this is how we come to secure our mathematical beliefs. A similar kind of argument might apply: How could we secure mistaken mathematical beliefs with proofs and computations that have stood the test of time? Try to add 2 and 2 with an addition that misbehaves-that behaves like, say, Kripke's quaddition. Unless, of course, proofs and computations are themselves defective and not to be trusted, i.e., unless we are mistaken about what we take to be the appropriate mathematical means to establish our mathematical results.

To address debunking properly, one must therefore distinguish (i) the mathematical beliefs that we have from (ii) the means by which we come to justify the mathematical beliefs that we have, and both from (iii) the philosophical arguments we devise to show that a particular account of our having (i) and of possessing (ii) is the true and correct one.

The debunking strategy targets (i)-(iii) for different reasons, with distinct albeit closely related arguments. One may see how the strategy is supposed to work by expressing the problem in terms of the relation between the content of our beliefs, their causal origin, and our second order views about such contents.

Suppose the content of our mathematical beliefs, say the content of the belief that 2+2=4, plays a role in the explanation that we have of our having that belief. The advocate of the debunking strategy argues that since the best explanation involves causal patterns and evolutionary advantages, the belief is undermined. This is sometimes contended by way of an analogy with moral or evaluative beliefs.

The analogy typically appeals to counterfactuals and works as follows. Just as the evolutionary origin of our evaluative beliefs is supposed to undermine the confidence we have both in the beliefs as such and in whatever we appeal to in order to justify them (maximum utility, religious prescriptions, the pure consideration of a moral law to the exclusion of any consideration that would de facto count as external, etc.), the causal origin of our mathematical beliefs is supposed to undermine the confidence we have both in the beliefs as such and in the particular means we appeal to when justifying our mathematical results. Why? Advocates of the Evolutionary Debunking Argument for moral beliefs argue in the following way: Had the contents of our moral beliefs been false (or perhaps neither true nor false) because of the absence of moral facts or of whatever would make our evaluative beliefs warrentedly true, we still would have the very same beliefs with the very same contents. We have them and judge them to be worthy of our approval not because of their truth or because of any genuine justification we possess in their favor, but because it has benefited our ancestors to have them and because, throughout the causal evolutionary process, we have nolens volens and unbeknownst to us inherited these beliefs for the very same reason-the term "reason" referring in this instance to something purely mechanical or factual, i.e., to something that excludes any contribution on our part in terms of an appeal to grounds, purpose, or justification. These beliefs, whether true, false, or neither, whether justified or not, or beyond justification, have kept providing us with the very same benefits.

So, the argument goes, our evaluative beliefs have no truck with truth. The very fact that their content appears in the causal explanation of why we have them suffices to free these beliefs from the clutches of the semantic notion of truth. Does this evolutionary speculation (or established evolutionary fact), which appeals to benefits as opposed to truth, establish what the advocate of the Evolutionary Debunking Argument would like it to establish, namely that just because natural selection makes us believe certain moral contents independently of their truth or their correspondence to moral facts, these contents are spurious or of no intrinsic value? Does it thereby follow that they are false, or neither true nor false? Are they, just for that reason, not in the market for truth, or not in the market for either truth or falsity?

Hardly so. At this point, a confusion lurks in the argument concerning the distinction between standpoints one may wish to defend regarding (i), (ii), and (iii).

Think in particular of a different but closely related consideration. One could hold a moral view, the view that, say, killing one's offspring is wrong, or hold a mathematical belief, say the belief that 2+2=4, and rest content with evolutionary speculations (or established evolutionary facts). One could have these beliefs and deny that there are either moral facts or mathematical facts that make them true. Should one then move to the level of (ii) to be presented with a causal explanation of the evolutionary advantage of securing the well-being of one's offspring, or of the evolutionary advantage of the mathematical practice that allows us to prove or conclude that 2+2=4 and apply the result of that proof or of that computation, the very same kind of consideration would apply. 2 One could be convinced by the causal explanation and still hold onto one's preferred means of justifying the moral view and to the particular computations or proofs that allow one to establish the arithmetical result.

In other words, one could accept the evolutionary speculation (or established evolutionary theory) and deny that the methods and justifications we appeal to thereby fall into disrepute. The explanation by way of evolutionary advantages doesn't undermine the appeal to say, religious prescriptions or moral law (in the case of moral beliefs), or to arithmetical computations or Peano Arithmetic (in the case of arithmetical beliefs).

As far as morality is concerned, this is of course closely related to the question of relativism in ethics. J. L. Mackie once remarked that one could be thoroughly conventional in one's strict moral views and believe that these are just attitudes others may very well not share (see Mackie 1971: 16). One could also hold a given moral view (either strict or liberal in the sense of open-mindedness and freedom from prejudice or bigotry) and remain agnostic with respect to the evolutionary speculation that would explain why one has that particular view.

In the evaluative case, it might well be psychologically unlikely that one would hold a first order view without being strongly opinionated about the second order one, ie., about the reasons one has to hold the first order view. If you think that killing is wrong whatever the consequences might be, or that slavery is unjust, or that abortion is murder, you usually also believe that you have compelling reasons to view the matters at hand in that particular way, reasons that, more often than not, you are willing to disclose and defend, and that you take to be non negotiable. A strongly held first order view often demands, if only psychologically, or perhaps for sociological or historical reasons, an equally strong second order view with respect to the grounds of the first order view. Now, think of the converse possibility. Mackie argues that one may very well "reject all established morality while believing it to be an objective truth that it was evil or corrupt" (Mackie: loc. cit.). In other words, one may hold a first order view and a second order view that, as it were, don't match psychologically or sociologically (an awkward combination of which there is perhaps no genuine historical example). It looks like, in that case, one shuns the current moral laws or prescriptions of one's community for no particular reason at all, or because one fancies oneself a skeptic about them, but nevertheless agrees that these laws and prescriptions are objectively badalthough agreement at the second order level in fact plays no role in one's first order rejection.

Since the motivation for the rejection isn't based on an argument to the effect that a particular law of prescription is wicked, what is the content of the first order attitude with respect to the second order one?

What is indeed unlikely, at least psychologically in terms of sensibility to the very purpose of morality and as far as choosing certain values rather than others is concerned, is to opt for such a rejection while not caring whether some particular course of action is taken, or some moral judgement voiced for objective reasons rather than for subjective motivations, or because it happens to be customary or fashionable to do so, or on the contrary because once fancies the fashion should be different. In this particular case, the first order rejecter throwing away second order considerations is one dyed-in-the-wool cynic. Diogenes might approve. One rejects all established social norms and behavior not because one believes they are evil or corrupt, but because one happens to dislike them for no particular reason at all. One might have done otherwise and have accepted them after all. Anything goes. Inconsistency lurks in the background.

In the mathematical case, where beliefs do not have the controversial evaluative and subjective component moral beliefs conspicuously have, the situation is of course different. Anyone rejecting a given kind of mathematical practice, e.g., a particular way of proving theorems, may do so for very different kinds of reasons. One may choose, say, to prove theorems of geometry by geometrical means rather than by algebraic ones because of a belief in the purity of methods. But the rejection of certain kinds of proofs might also be motivated by considerations pertaining to the illegitimacy of some of the means by which we come to justify the mathematical beliefs that we have. One may, e.g., decide to reject classical mathematical practice because of a belief that constructive proofs are the only legitimate ones. In this case, the disagreement about the means targets a specific feature of classical proofs, namely the fact that they do not necessarily specify constructions that, at least in principle, we should be able to carry out. The disagreement, which yields a strong repudiation of a given kind of mathematical practice, is thus grounded on a conception of intuitionistic proofs according to which the content of the conclusions of these proofs (i.e., the very content of our mathematical beliefs) must embed the contention that non-constructive proofs may not count as genuine justifications of the very same conclusions.

What is of philosophical interest here, though, is that psychological considerations are nevertheless unhelpful and that they are so for the same reason in both cases, in spite of the fact that psychological considerations and emotive attitudes aren't to be found anywhere in the vicinity of mathematical practice, while the idea of a moral psychology and of a moral behavior brought about by Humean passions makes sense.

As for (iii), why should whatever philosophical argument one may devise to the effect that, say, a platonistic account, or a nominalist one is in order for both (i) and (ii) be threatened by considerations pertaining to debunking?

Since, in the present context, this involves the view that our mathematical beliefs are reliable insofar as they are both safe and sensitive (see Clarke-Doane 2016), let me end this section with a brief comment on truth, safety, and sensitivity. One may consider two epistemological challenges pertaining to mathematical belief formation. One is to show that our mathematical beliefs are safe, i.e., that we could not (at least not so easily) have false mathematical beliefs using the methods we actually use to form these beliefs. Another is to show that our mathematical beliefs are sensitive, i.e., that, had the content of our mathematical beliefs been false, we would have abjured them, using the methods we actually use to form these beliefs. The challenge to be met with respect to the explanation of why we hold the mathematical beliefs that we do, is to show that these beliefs are counterfactually persistent, i.e., that if mathematical truths had been different, or if there had been no mathematical truths at all, our mathematical beliefs would have been correspondingly different.

The construal of the reliability challenge that requires the content of our mathematical beliefs to be both safe and sensitive implies that mathematical truths are both contingent and defeasible. What matters here, the discarding of the necessity of mathematical truths notwithstanding, is that, as far as debunking goes, if it turned out to be impossible, given our cognitive capacities, or because this would involve a contradiction, to show that the content of our mathematical beliefs figure, via some causal explanation or otherwise, into the best explanation of our having them, this would still give us no reason whatsoever to give them up. It would not unless we were also ready to give up the requirement that they must be both safe and sensitive.

Conclusions

It has been argued all along that considerations of parity must play a part in the debate about the role and nature of mathematical explanations vs. the role and nature of non mathematical ones, in particular when they must be counted as being the best, so that competing platonistic and nominalistic devices used in accounts of the usefulness or, on the contrary, of the dispensability of mathematics, may not be treated differently, either epistemically or ontologically. The point was that it must be assessed in both cases which kind of account is more (or, on the contrary, less) ontologically or epistemologically loaded. By Quine's criterion of ontological commitment, any account has commitments. Since both mathematical explanations and proofs, and nominalistic ones, wear such commitments on their sleeves, the fact that we may dispense with number talk, either by way of paraphrase or by way of a conservativeness proof, hardly neutralizes the commitment. We just can't dispense with ontology.

It is sometimes assumed that platonism is a second-best option one accepts as it were by default, a position that would be dismissed just in case a nominalistic alternative to it could be devised. This pre-supposes, of course, that ontological parsimony should be preferred. It isn't clear why it should. In any event, it might be clearer why epistemic parsimony should be preferred. Pragmatic motivations might be called to the rescue: There is no reason to make our life more difficult with certain kinds of derivations or best explanations when an easier course is available. Of course, one might then offer an open question argument: Why should we grant a privilege to epistemological economy as such, or to anything that might imply it has advantages? Perhaps, then, epistemic economy is good sui generis.

In any event, one has no right to complain that there is no privileged neutral point of view from which to judge the advantages and disadvantages of either type of account or explanation. There is no petitio principii at stake in the contention that this is indeed possible. We can't argue that we're always somehow begging the question, either in favor of platonism or in favor of the opposite view. An assessment of the respective merits of these views doesn't necessarily take for granted that platonism offers the only possible account of mathematical truth, or that anti-platonism offers the only possible account of mathematical knowledge.

But sincegiven what has been said in section 5 -we're justified, after all, in holding our mathematical beliefs, the problem must lie elsewhere. We've been concerned with worries that presuppose that classical mathematics and classical reasoning are unproblematic as far as either mathematical knowledge or some nominalistically respectable knowledge that would replace it in best explanations of natural phenomena are concerned.

But suppose that proofs which establish the existence of something by providing the effective means of finding it, as is the case of constructive proofs, provide an insight that classical proofs conspicuously fail to provide. In a nutshell, a constructive proof of a closed statement of the form '(∃x) Fx' with the bound variable ranging over the natural numbers, either proves a specific instance A(n) or provides, at least in principle, an effective means for finding one. Now, a classical logician or mathematician might prefer constructive proofs for particular reasons. But the intuitionistic logician or mathematician believes he has no choice: Proofs must be constructive.

Of course, we're now looking at a kind of worry that is different from the one concerned with the availability of nominalistic derivations and best explanations. We're dealing with disputes that happen to be of a quite different nature. There is, on the one hand, the question whether platonism may be vindicated by an inference to the best explanation of mathematical success or, on the contrary, whether anti-platonism may be vindicated by the possibility of explaining the applicability of mathematics and of providing explanations of natural phenomena by nominalistic means. There is, on the other hand, the question whether classical proofs and classical reasoning should be preferred to constructive ones. This constitutes a second dispute, clearly distinct from the first, and one might complain that the dispute regarding the classical vs. the non classical character of proofs has no bearing on the one we've been concerned with so far.

Let me briefly suggest that it isn't so and that the two disputes aren't as unrelated as it may seem at first blush.

Whatever the theoretical role of a classical explanatory proof may turn out to be, it must indeed be different from that of an intuitionistic explanatory proof, or from that of a proof or derivation which resorts to conservative mathematics only. The role must be played differently, i.e., it must yield a different explanation of a natural phenomena.

This suggests that we want to distinguish two kinds of explanations : the explanation of why classical mathematics is conservative from the explanation of why intuitionistic mathematics is conservative. Likewise, we may want to distinguish the explanation of why a geometrical classical explanation of some natural phenomenon does the job, perhaps even a better job than the numerical one, from the explanation of why a geometrical intuitionistic explanation of the same natural phenomenon has the same advantages.

So what are the question-begging assumptions regarding the nature and validity of proofs that the claim to the effect that nominalistic proofs do play their theoretical role is guilty of?

If it were shown that mathematics isn't conservative, or has no genuine platonistic claims embedded in it, or need not be intuitionistic, the result would have deep implications regarding what we must argue for in order to establish that some suita-ble theoretical role is played. The scrutiny might show that nominalistically respectable proofs, or classically respectable proofs, or conservatively respectable proofs, happen not to be on a par with respect to playing a theoretical role.

This suggests that, because both the validity and the nature of proofs and derivations might differ in each case, the cost of undermining platonism is different from the cost of undermining nominalism or, for want of a better phrase, from that of undermining conservativism.

The differences will be brought about by features that might surface at different levels: at the level of applied number theory and applied geometry, but also at the level of logic. To show that there is indeed a difference and, moreover, that it matters a great deal as far as our problem is concerned, requires further inquiry. What I'm suggesting here is that the difference must matter for reasons that have to do with the most basic features of, respectively, classical proofs and intuitionistic proofs, namely their logic.

Just modify (ii) so that it reads "the means by which we come to justify the moral beliefs that we have" to cover the moral case.