Emiliano Ippoliti

Representation theorems play a pivotal role in mathematical knowledge, both in a technical and in a broader conceptual sense. I argue that these theorems, and especially their construction, are a powerful tool for solving problems and advancing knowledge in mathematics. They create novel objects, and eventually a new theory, in a way only sketched by Lakatos. I consider intra-domain, in particular Cayley’s theorem, as well as inter-domains representation theorems, in particular an examination of the fundamental group in algebraic topology. I maintain that the search for and the construction of a new representation introduce new pieces of information in a very specific way, that
requires assimilation, which might end up with a solution for a problem and a proof of a new theorem. I argue that the work behind the proof of a representation theorem is crucial, since it transforms the mathematical problem in a way that makes it tractable and introduces information into it that it did not contain at the beginning of the process. In this sense, these case studies show how these new objects, as the outcome of an integration of features of multiple objects or fields— ‘hybrids’ [see Grosholz ]—, are a powerful engine for the advancement of mathematical knowledge.
So I will fill some of the gaps in Lakatos account of theoretical novelty in mathematics and I will shed light on new dynamics in the construction of representation, in particular their integration, which enriches Grosholz’s account of productivity in mathematics.

I investigate the construction of the mathematical concept of quaternion from a methodological and heuristic viewpoint to examine what we can learn from it for the study of the advancement of mathematical knowledge. I will look, in particular, at the inferential microstructures that shape this construction, that is, the study of both the very first, ampliative inferential steps, and their tentative outcomes—i.e. small ‘structures’ such as provisional entities and relations. I discuss how this paradigmatic case study supports the recent approaches to problem-solving and philosophy of mathematics, and how it suggests refinements of them. In more detail, I argue that the inferential micro-structures enable us to shed even more light on the informal, heuristic side of mathematical practice, and its inferential and rational procedures. I show how they enable the generation of a problem, the construction of its conditions of solvability, the search for a hypothesis to solve it, and how these processes are representation-sensitive. On this base, I argue that:
(1) the recent development of the philosophy of mathematics was right in moving from Lakatos’ initial investigation of the formal side of a mathematical proof to the investigation of the semi-formal (or informal), heuristic side of the mathematical practice as a way of understanding mathematical knowledge and its advancement.
(2) The investigation of mathematical practice and discovery can be improved by a finer-grained study of the inferential micro-structures that are built during mathematical problem-solving.

I examine the way a relevant conceptual novelty in mathematics, that is, the notion of group, has been constructed in order to show the kinds of heuristic reasoning that enabled its manufacturing. To this end, I examine salient aspects of the works of Lagrange, Cauchy, Galois and Cayley (Sect. 2). In more detail, I examine the seminal idea resulting from Lagrange’s heuristics and how Cauchy, Galois and Cayley develop it. This analysis shows us how new mathematical entities are generated, and also how what counts as a solution to a problem is shaped and changed. Finally, I argue that this case study shows us that we have to study inferential micro-structures (Sect. 3), that is, the ways similarities and regularities are sought, in order to understand how theoretical novelty is constructed and heuristic reasoning is put forward.

In this paper I examine the heuristic view of mathematics focussing on an updated version of it. I discuss how the seminal work of Lakatos has been improved under several respects and I point at four issues that are particularly relevant for the philosophy of mathematics and mathematical practice from a heuristic point of view: 1. the quest for a method of discovery (§2); 2. the construction of heuristic procedures, that is rational and inferential ways of producing a hypothesis to solve a problem (§3); 3. the nature of mathematical objects (§4); 4. the relation between mathematics, evolution, and reality (§5).

The way scientific discovery has been conceptualized has changed drastically in the last few decades: its relation to logic, inference, methods, and evolution has been deeply reloaded. The ‘philosophical matrix’ moulded by logical empiricism and analytical tradition has been challenged by the ‘friends of discovery’, who opened up the way to a rational investigation of discovery. This has produced not only new theories of discovery (like the deductive, cognitive, and evolutionary), but also new ways of practicing it in a rational and more systematic way. Ampliative rules, methods, heuristic procedures and even a logic of discovery have been investigated, extracted, reconstructed and refined. The outcome is a ‘scientific discovery revolution’: not only a new way of looking at discovery, but also a construction of tools that can guide us to discover something new. This is a very important contribution of philosophy of science to science, as it puts the former in a position not only to interpret what scientists do, but also to provide and improve tools that they can employ in their activity.

In this paper I lay out a non-formal kernel for a heuristic logic—a set of rational procedures for scientific discovery and ampliative reasoning—specifically, the rules that govern how we generate hypotheses to solve problems. To this end, first I outline the reasons for a heuristic logic (§1) and then I discuss the theoretical framework needed to back it (§2). I examine the methodological machinery of a heuristic logic (§3), and the meaning of notions like ‘logic’, ‘rule’, and ‘method’. Then I offer a characterization of a heuristic logic (§4) by arguing that a heuristics is ways of building problem-spaces (§4.1). I examine (§4.2) the role of background knowledge for the solution to problems, and how a heuristic logic builds upon a unity of problem-solving and problem-finding (§4.3). I offer a first classification of heuristic rules (§5): primitive and derived one. Primitive heuristic procedures are basically analogy and induction of various kinds (§5.1). Examples of derived heuristic procedures (§6) are inversion heuristics (§6.1), heuristics of switching (§6.2), as long as other kinds of derived heuristics (§6.3). I then show how derived heuristics can be reduced to the primitive one (§7). I examine another classification of heuristics, the generative and selective (§8), and I discuss the (lack of) ampliativity and the derivative nature of selective heuristics (§9). Lastly I show the power of combining heuristics for solving problems (§10).

Theory-building is the engine of the scientific enterprise and it entails both (1) the generation of new hypotheses, (2) their justification, and (3) their selection, as long as collecting data. The orthodox views maintain that there is a clear logical and temporal order, and distinction, between these three stages. As a matter of fact, not only this tenet is defective, but also there is no way to solve these three issues in the way advocated by traditional philosophy of science. In effect, what philosophy of science tells us is that (a) there is not an infallible logic, in the sense of a simple set of logical rules, to justify and confirm a hypothesis, and (b) the process of generation of hypotheses is not unfathomable, but can be rationally investigated, learnt and transmitted. So, as an alterative, I discuss the heuristic approach to theory-building, especially the one based on problems, and I argue that it offers a better way of accounting for theory-building than the traditional ways.

in: Chen, P. - Ippoliti, E. (eds). Methods and Finance. A Unifying View on Finance, Mathematics and Philosophy (2017), pp. 179-194.

The nature of the data in financial systems raises several theoretical and methodological issues, which not only impact finance, but have also philosophical and methodological implications, viz. on the very notion of data. In this paper I will examine several features of financial data, especially stock markets data: these features pose serious challenges to the interpretation and employment of stock markets data, weakening the ‘myth of data’. In particular I will focus on two issues: (1) the way data are produced and shared, and (2) the way data are processed. The first raises an internal issue, while the second an external one. I will argue that the process of construction and employment of the stock markets data exemplifies how data are
theoretical objects and that ‘raw data’ do not exist. Data are not light and ready-to-use objects, but have to be handled conceptually and technically very carefully and they are a kind of ‘dark matter’. Dark data, for the note.

in: Chen, P. - Ippoliti, E. (eds). (2017). Methods and Finance. A Unifying View on Finance, Mathematics and Philosophy , pp. 3-15

The view from outside on finance maintains that we can make sense of, and profit from, stock markets’ behavior, or at least few crucial properties of it, by crunching numbers and looking for patterns and regularities in certain sets of data.
The basic idea is that there are general properties and behavior of stock markets that can be detected and studied through mathematical lens, and they do not depend so much on contextual or domain-specific factors. In this sense the financial systems can be studied and approached at different scales, since it is virtually impossible to produce all the equations describing at a micro level all the objects of the system and their relations. The typical view of the externalist approach is the one provided, for instance, by the application of statistical physics. By focusing on collective behaviour, statistical physics neglects all the conceptual and mathematical intricacies deriving from a detailed account of the inner, individual, and at micro level functions of a system. This chapter examines how the view from outside deals with critical issues such as the mathematical modeling (Sect. 2), the construction and interpretation of data (Sect. 3), and the problem of prediction and performativity (Sect. 4).

in: Magnani, L. -  Casadio, C. (eds.). Model-Based Reasoning in Science and Technology. Logical, Epistemological, and Cognitive Issues. Berlin: Springer, pp. 119-36. ISBN: 978-3-319-38983-7

This paper sets out to show how mathematical modelling can serve as a way of ampliating knowledge. To this end, I discuss the mathematical modelling of time in theoretical physics. In particular I examine the construction of the formal treatment of time in classical physics, based on Barrow's analogy between time and the real number line, and the modelling of time resulting from the Wheeler-DeWitt equation. I will show how mathematics shapes physical concepts, like time, acting as a heuristic means—a discovery tool—, which enables us to construct hypotheses on certain problems that would be hard, and in some cases impossible, to understand otherwise.

in: Sterpetti, F.—Ippoliti, E.—T. Nickels (eds.) Models and Inferences in Science. Berlin: Springer, pp. 141-172. ISBN: 978-3-319-28162-9

The examination of the construction of several approaches put forward to solve problems in topology and knot theory will enable us to shed light on the rational ways of advancing knowledge. In particular I will consider two problems: the classification of knots and the classification of 3-manifolds. The first attempts to tell mathematical knots apart, searching for a complete invariant for them. In particular I will examine the approaches based respectively on colors, graphs, numbers, and braids, and the heuristic moves employed in them. The second attempts to tell 3-manifolds apart, again searching for a complete invariant for them. I will focus on a specific solution to it, namely the algebraic approach and the construction of the fundamental group, and the heuristic moves used in it. This examination will lead us to specify some key features of the ampliation of knowledge, such as the role of representation, theorem-proving and analogy, and will clear up some aspects of the very nature of mathematical objects.

in: Cozzo, C. – Ippoliti, E. (eds.). From a heuristic point of view. Newcastle Upon Tyne: Cambridge Scholars Publishing, 213-239. ISBN: 978-1-4438-5649-2

SUMMARY In this paper I argue for a heuristic point of view about mathematization, in particular about the issue of applicability and effectiveness of mathematics, using Cellucci's characterization of mathematical objects. To this end, I examine the notion of risk and investigate four main approaches, namely the probabilistic, the psychological, the fractal and the evolutionary approaches. I show that the lack of success of the various approaches in the treatment of risk is due to the ways in which they conceptualize and mathematize it. I set out to show that the heuristic point of view can offer a better characterization of risk, which requires a different approach, bottom-up, local and oriented to problem-solving.

In: E. Ippoliti (ed.), Heuristic Reasoning. Berlin: Springer. p. 163-191. ISBN: 978-3-319-09158-7.

In this paper I argue that the effective way to account for the behavior of
stock market prices is to put in use a dynamic approach—bottom-up, local, nonaxiomatic, heuristic. To this end, I provide an analysis of the generation of four
main hypotheses used to explain stock market prices (SMP). In particular I show
how the means of generating these hypotheses is essential to assessing their efficiency and plausibility. In formulating a hypothesis, a selection of features of SMP
is made for incorporation in a theory. This selection may be expressed mathematically
in most of the cases. An examination of these means of generation can show
us why some of these hypotheses are successful and efficient and some not, and
can also shed light on the extent to which a particular hypothesis can be usefully
applied. Thus the study of the means of generation of hypotheses will offer us a
guide to formulating new hypotheses in a reliable and cogent fashion.

Introduzione di Carlo Cellucci.

Le odierne introduzioni alla logica mirano generalmente a fornire i primi elementi della logica matematica, il tipo di logica creato da Gottlob Frege (1848-1925). Questo libro è invece un’introduzione a una logica di tipo differente.

In un certo senso, questo libro ritorna agli scopi della logica fissati da Aristotele. Tra le finalità che esso assegna alla logica, infatti, vi sono innanzitutto quelle di sviluppare capacità argomentative e di valutare gli argomenti che vengono usati nel dibattito pubblico. Il testo insegna a giudicare le ragioni che possono esserci per credere in un’affermazione e per farla accettare anche ad altri interlocutori, perché la considera una capacità essenziale per la crescita e lo sviluppo di una società aperta e democratica.
In secondo luogo, tra le finalità che questo libro assegna alla logica, vi sono anche quelle, corrispondenti agli scopi indicati da Aristotele, di essere utile per la costruzione delle teorie. Costruire teorie è il cuore del processo conoscitivo, e in particolare di quello scientifico.

Le inferenze sono definite ampliative quando permettono di estendere – ampliare – le nostre conoscenze, nel senso che i dati e le informazioni contenute nella conclusione non sono contenute nelle premesse, come avviene nel caso delle inferenze non-ampliative (ossia quelle deduttive). Esse sono il principale motore di avanzamento e crescita della conoscenza, in particolare di quella matematica.
Il testo, usando alcuni esempi tratti dalla teoria dei numeri, la teoria dei nodi e la teoria delle trecce, fornisce un’analisi di queste forme inferenziali esaminandone tre articolazioni concettuali particolarmente significative quali la visualizzazione, l’analogia e le rappresentazioni  multiple, e ne discute proprieta', statuto e  alcune loro rilevanti implicazioni epistemologiche, in particolare la loro relazione con la verita'.
Il testo sostiene la tesi che le conoscenze originate dalle inferenze ampliative hanno in un senso preciso lo stesso statuto di quelle generate dalle inferenze non-ampliative

La plausibilità, quale strumento per costruire e valutare ipotesi e guidare l'azione, è un concetto che nasce da un'esigenza fondamentale non solo dell'impresa conoscitiva, ma dell'uomo in quanto tale: quella di limitare e gestire l'incertezza e la provvisorietà che contraddistinguono ogni aspetto della sua esistenza.
Il libro, mediante alcuni richiami storici, fornisce un'analisi delle teorie e dei modelli della plausibilità, sia nella sua versione probabilistica sia in quella non-probabilistica, sviluppando una prospettiva che propone un superamento dell'opposizione tra vero e plausibile.

Economia, Etica e Finanza: Riflessioni Multidisciplinari.

Lunedì 27 (15:00-19:00)
Martedì 28 Maggio 2019 (10:30-13:30; 15:30:19:00),
Aula Onida (3° piano)
Facoltà di Economia -  Sapienza Università di Roma,
Via del Castro Laurenziano 9.
Dipartimento di matematica Sapienza Universtà di Roma - Aula Picone venerdì 25 Gennaio ore 16 alle 18.30

L'impiego sempre più diffuso di algoritmi sui mercati finanziari ha modificato profondamente le loro dinamiche. Nel mio intervento discuterò se e come questo cambiamento abbia spinto i mercati finanziari verso una dimensione sempre più 'performativita' (nel senso di Callon e MacKenzie). Partirò da un caso studio di interazione tra algoritmi finanziari, e discuterò le connessioni che rivela tra machine, manipolazione e performatività dei mercati finanziari e la possibilità di una 'finanza inversa

In my paper, I examine the role that models play, and the relation between models and data, in financial systems, in particular in stock markets. I discuss several dangerous liaisons between models and data both from a theoretical and a practical viewpoint, and I consider their effects on the behaviour of the financial systems and their actors.
I will examine two issues in particular.
First, these relations and liaisons defy the way traditional philosophy of science accounts for models and the relation between models and data, as stock markets exhibit several dynamics and features that do not fit them. For instance, they challenge the ontological issue about models (the debate about the fictional character of models), or the way model and phenomena and connected, and consequently undermine classical taxonomies as ‘models of data’, ‘model of phenomena’, ‘models of theory’.
Second, these relations and liaisons open the way to possible exploitations and manipulation of stock market’s dynamics by means of appropriate use of models and data, including ‘reverse finance’, which should be closely analysed in order to contribute to better functioning and serving of the financial systems.

Nell'ultima esposizione del tirocinio, il gruppo Humans ha sicuramente fatto tesoro degli incontri precedenti, strutturando la propria presentazione in modo da evitare alcune mancanze e concentrandosi sui punti deboli delle tesi altrui. Encomiabile l'impostazione del discorso: il team ha adottato un approccio filosofico-scientifico, fornendo definizioni puntuali ed esplicitando, con un vero e proprio elenco, le assunzioni di partenza per la costruzione della propria tesi e dello scenario prospettato, oltre ad offrire una dettagliata analisi dei meccanismi cognitivi ed emozionali tipici dell'umano. Questa analisi è stata utilizzata come punto fondamentale per l'elaborazione delle argomentazioni del gruppo, e si sono rivelate utili alla autoregolamentazione sociale e finanziaria. Un solo appunto va fatto riguardo le assunzioni stesse, e riguarda le assunzioni 5 e 6 dell'elenco: 5-Il mercato finanziario, in quanto attività umana, deve avere come fine ultimo il benessere collettivo e non il profitto 6-Non vi è una relazione di causalità necessaria tra profitto e benessere Certo è che, fra i concetti base per la comprensione della tesi Humans, vi è quello di "benessere", definito dal team stesso anche come scopo ultimo di qualsiasi attività umana, quindi del mercato finanziario, in quanto rientrante in quest'ultima descrizione. Seppur accennata e, a livello generale, comprensibile, la definizione di benessere è stata però tracciata piuttosto vagamente, mentre sarebbe stato necessario esplicarla maggiormente per poter delineare in modo chiaro una sorta di "etica finanziaria", scopo che non era, in realtà, proprio del gruppo, ma nella cui direzione esso si è effettivamente mosso. In particolare, il modo in cui il concetto di benessere è stato affrontato ci è sembrato una identificazione implicita, impropria e scorretta con il concetto di felicità, difficilmente definibile ma certamente non univocamente identificabile con quello di benessere. Quest'ultimo può infatti essere declinato in diversi sensi e contesti, di cui quello economico è solo uno e anche molto distante da quello etico. Inoltre ci si è concentrati sul concetto di benessere come collettivo, intendendo con ciò una situazione in cui tutte le singole preferenze individuali vengono rispettate. In realtà il rapporto fra benessere collettivo ed individuale è ben più complesso. A questo proposito, è interessante trattare del teorema di Arrow. Requisiti ragionevoli per un sistema di preferenze equo: 1. universalità o dominio non ristretto: la funzione di scelta sociale dovrebbe creare un ordinamento delle preferenze sociali deterministico e completo, a partire da qualsiasi insieme iniziale di preferenze individuali 2. non imposizione (o sovranità del cittadino): qualsiasi possibile preferenza sociale deve essere raggiungibile a partire da un appropriato insieme di preferenze individuali (ogni risultato deve poter essere raggiunto in qualche maniera) 3. non dittatorialità: la funzione di scelta sociale non deve semplicemente seguire l'ordinamento delle preferenze di un individuo o un sottoinsieme di individui, al contempo ignorando le preferenze degli altri 4. monotonicità, o associazione positiva tra i valori individuali e sociali: se un individuo modifica il proprio ordinamento di preferenze promuovendo una data opzione, la funzione di scelta sociale deve promuovere tale opzione o restare invariata, ma non

Ogni dibattito della serie head-to-head (h2h) propone un confronto tra due esperti che hanno posizioni divergenti su uno specifico tema filosofico o scientifico. Ogni esperto tenterà di dimostrate la forza della sua posizione in relazione a quella dell’altro. L’obiettivo è quello di stimolare la discussione e la comprensione della questione in gioco.

Il confronto avviene nel modo seguente:

- 5 minuti breve introduzione del tema
- 15 minuti esposizione di A
- 15 minuti esposizione di B
- 45 minuti domande e risposte con il pubblico*
- 25 minuti dibattito A-B

* il moderatore accetterà solo domande—non interventi. In caso può porre lui stesso domande o chiedere delucidazioni.

Science & Philosophy Colloquia is a series of seminars aiming at deepening some of the most puzzling issues put forward in the course of philosophical and scientific thought, and their main historical-conceptual branches.

These issues are approached through the contribution of leading figures both in philosophy and various scientific disciplines.

Alla base di molta ricerca psicologica c’è la tesi della realizzabilità multipla. In base a questa, uno stesso tipo di stato mentale può essere realizzato in tipi diversi di stati fisici, siano essi cerebrali o no. Tale tesi nasce da una reazione alla teoria dell’identità tra mente e cervello, che invece ipotizza che a un certo tipo di stato mentale corrisponda un tipo di stato cerebrale.

Ma quante implicazioni metafisiche sottostanno alla tesi della realizzabilità
multipla?

Sosterrò che, sorprendentemente, queste riguardano anche il concetto di
persona.

Molti scienziati lamentano la completa irrilevanza della filosofia della scienza per la scienza. Dopo aver esaminato gli argomenti principali portati a favore di questa tesi da autorevoli scienziati, si discuteranno possibili risposte a queste critiche. Persino se si ammettesse che la filosofia della scienza non contribuisce direttamente al progresso interno delle conoscenze scientifiche, si potrebbe ribattere che il suo scopo non sia propriamente questo. Una seconda risposta mostra come la filosofia anche in passato abbia avuto un ruolo euristico molto importante, avendo posto problemi concettuali che sono stati poi affrontati e in qualche caso risolti con metodi scientifici: lo studio sperimentale della nonseparabilità quantistica è una conseguenza delle preoccupazioni filosofiche di Einstein dovute all’incompletezza della meccanica quantistica. Una terza risposta insiste sull’importanza che la filosofia della scienza ha nel fornire un’immagine rivedibile ma per quanto possibile unitaria delle conoscenze scientifiche raggiunte in una certa epoca, attraverso l’esame di concetti quali riduzionismo, sopravvenienza, legge di natura, causalità, etc. Infine, si discuterà l’importanza del metodo storico-critico che i filosofi della scienza hanno adottato e possono adottare per una valutazione adeguata delle teorie scientifiche contemporanee.

I first review the innovative representations of time in the work of Galileo, Descartes and Newton, and then turn to the debate over whether time is absolute (to be defined analytically) or relative (to be defined referentially) between Newton and Clark on the one hand, and Leibniz on the other. In this process, Leibniz offers an interesting answer to the question, how shall we combine two kinds of discourse - empirical compilation and theoretical analysis - in the new science? Leibniz calls on metaphysics, in particular the Principle of Sufficient Reason, to regulate a science that must be both empirical and rational. The correlation of precise empirical description with the abstract conception of science more geometrico is guaranteed by the thoroughgoing intelligibility and perfection of the created world. Leibniz encourages us to work out our sciences through successive stages, moving back and forth between concrete taxonomy and abstract systematization, a method we see in his investigations of mechanics and planetary motion, and his mathematical-metaphysical account of time, which was subtler and more multivalent than that of Newton. In the last section of my talk, I will discuss the role of metaphysics in science and logic.