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AVICENNA ON GRASPING MATHEMATICAL CONCEPTS

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Abstract

According to Avicenna, some of the objects of mathematics exist and some do not. Every existing mathematical object is a non-sensible connotational attribute of a physical object and can be perceived by the faculty of estimation. Non-existing mathematical objects can be represented and perceived by the faculty of imagination through separating and combining parts of the images of existing mathematical objects that are previously perceived by estimation. In any case, even non-existing mathematical objects should be considered as properties of material entities. They can never be grasped as fully immaterial entities. Avicenna believes that we cannot grasp any mathematical concepts unless we first have some specific perceptual experiences. It is only through the ineliminable and irreplaceable operation of the faculties of estimation and imagination upon some sensible data that we can grasp mathematical concepts. This shows that Avicenna endorses some sort of concept empiricism about mathematics.
Arabic sciences and philosophy   
doi:10.1017/S0957423920000090
AVICENNA ON GRASPING MATHEMATICAL CONCEPTS
  
Department of Philosophy
University of Birmingham
Email: m.s.zarepour@bham.ac.uk
Abstract.           
         
           
         
           
         
          
            
          
            
          
           
          

Résumé.        
         
           
        
            
       
           
        
          
         
         
         
         
        
    
1. INTRODUCTION
          
   ontological     
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   
         
        
     epistemological    
          
         
        
            
         
    a priori      
         

         
      experimental sciences   
         
         
           
        
           
           
  
          
       Studia Islamica  
         
          The unity of sci-
ence in the Arabic tradition        
   Oriens          
         
    Interpreting Avicenna: Critical essays  
           
      Kitāb al-burhān     Raison
et démonstration: Les commentaires médiévaux sur les   
            
           
 Koroth            
              
  Before and after Avicenna: Proceedings of the rst conference of the
Avicenna Study Group         
            
  Interpreting Avicenna: Critical essays    
          
           
          As-
pects of Avicenna         
          
 The unity of science in the Arabic tradition    
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
        
          
         
         
       
         
         
          
            

          
        
   taṣawwur       
taṣdīq        
        
          
           
              
          
           
         
            
         
            
         The healing Al-šifāʾ
Al-manṭiqAl-burhān           
 Al-naǧāt         
        Dānešnāme-ye ʿAlāʾīManṭeq
           
           
             The
journal of philosophy            
         taṣawwur 
taṣdīq           
            Logic and
Aristotle’s rhetoric and poetics in medieval Arabic philosophy    
           
          taṣawwur 
taṣdīq           The
Muslim world      
            
       taṣawwur  taṣdīq    
 Logic and Aristotle’s rhetoric         
      
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
          
         
        
        
          
              
         
          
           
         
          
             
           
   mufāraq       
     maḥsūs   
            
         
       
          
         
   
         
          
      wahm    
          
          
        
         
        
           
       concept empiricism  
          
        tajrīd
         
           
           
          
          
        
           
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
           
           
           
            
          
          
           
          
         
         
        
          
        
           
           
          
         

2. PRELIMINARIES
          
          
         
          
           
           
          Di-
alogue             
           
         
          
Arabic sciences and philosophy        
           
      The Routledge companion to thought exper-
iments        Mathematics and the mind: An
introduction into Ibn Sīnā’s theory of knowledge  
         
         
             
Arabic theology, Arabic philosophy: From the many to the one. Essays in celebration of
Richard M. Frank          
  Oriens       
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
Downloaded from https://www.cambridge.org/core. IP address: 168.151.121.254, on 28 Apr 2021 at 21:24:12, subject to the Cambridge Core terms of use, available
   
         
         
         
            
         
           
           
           
        
            
         
         
            
            
            
         
           
            
           
          
           
           
           
              
            
         
            
          
           
           
         
             
            
         
 ʾaʿdād        maqādīr 
           
The Metaphysics of the Healing        
         ʿaraḍ
 The Metaphysics of the Healing    
   The Metaphysics of the Healing      
     denitions     
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
          
           
         
            
             
             
             
           
           
         
       
         quwwa 
  dunamis         
             
           
           
             
           
             
           
         ontological 
           
 epistemological          
 
           
            
         The Physics of the Healing
        
             
             
             
 
             
           
    Al-šifāʾAl-manṭiqAl-madhal   
          
 The Metaphysics of the Healing       
           Isagoge  
Shifāʾ Journal for the history of Arabic science       
   
             
           
            
        The philosophical re-
view              
            
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
          
           
           
          
 
         
         
           The Cam-
bridge history of medieval philosophy       
          The fac-
ulties: A history    
             
              
    Maqāla al-nafs ʿalā sunna al-ihtiṣār   
            
            
Zeitschrift der Deutschen Morgenländischen Gesellschaft     
     Kitāb al-nafs    Avicenna’s De anima: Being
the psychological part of        
            
      Al-naǧāt       
   Avicenna’s Psychology      
       Al-išārāt wa-l-tanbīhāt bi-šarḥ
al-ṬūsīAl-ṭabīʿīyāt          
      Remarks and admonitions: Physics and meta-
physics             
al-nafs al-nāṭiqa        
           Aḥwāl al-nafs: Risāla
al-nafs wa baqāʾihā wa maʿādihā       
             Avicenna
and the Aristotelian tradition         
            
           
             
             
          
          Interpreting
Avicenna: Science and philosophy in medieval Islam     
            
Avicenna    
  Avicenna’s De anima         
      basīṭ         
          
          
            
        Oxford studies in medieval philosophy
   
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
             
   ẓāhirī       
 ṣuwar         
   bāṭinī       
              
          
        
           
           
            
            
          
             
          
         
             
           
         
           
        
maʿnā         
            volitional
motion          
  Avicenna’s De anima        
quwwa   ḥiss 
 Maʿnā          
          
  Philosophy East and West       
    wahm       
 Dialogue          
  Quaestio        Avicenna on sub-
jectivity: A philosophical study      
        
      maʿnā    intentio   
      maʿnā       
  in           
        maʿnā    
         
 Avicenna’s   in the Latin West     
              
   maʿnā          
           
          
     maʿnā
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
           
          
          
             
           
         
             
        
        
         
          
           
    
            
          
            
               
              
 hayʾa          
                
         
      al-maḥsūs    
              
           
             
         
         
           
   tajrīd      
          
         
           
         
         
           
         
         
  Avicenna’s De anima         
             
             
  
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
    
         
           
 ḥiss muštarak        
            
          
        
         
           
      muṣṣawira  
hayāl          
             
          
          
 mutahayyila        
              
        
            
           
          
          
         
               
            
            
 
            
           
       Intellect and imagination in
medieval philosophy        
             
          
 Harvard theological review      
           
            
    Documenti e studi sulla tradizione losoca medievale
          
         Tópicos     
         
           
             
           
     Oxford studies in medieval
philosophy    
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
        
          
           
    mufakkira    
   wahm       
           
            
         
           
           
           
             
        
           
   ḥāẓa        
          
        
        
         
          
             
            
        Topoi
            
           Philo-
sophical psychology in Arabic thought and the Latin aristotelianism of the thirteenth
century           
 
  Avicenna’s De anima        
              
             
               
              
      Al-naǧāt      
            
 wahm          
            
 Intellect and imagination in medieval philosophy     
  Avicenna on subjectivity  
          
               
             
 Journal of the history of philosophy      
   
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
         
           
          
          
         
      
          
           
         
    ʿaql hayūlānī    
ʿaql bi-l-quwwa     ʿaql bi-l-malaka 
   ʿaql bi-l-ʿl      ʿaql
mustafād        
           
          
         Avicenna’s De anima  
              
       emanationist   
            ʿaql
faʿʿāl          Alfarabi,
Avicenna, & Averroes, on intellect       
Avicenna            
          Debates
in medieval philosophy: Essential readings and contemporary responses  
        abstractionist  
              
            
           
  Aspects of Avicenna       
         
          
         
    Proceedings of the American Catholic Philosoph-
ical Association          
     De anima   Enneads   
     Theories of perception in medieval and early
modern philosophy         
  Philosophers’ imprint        
           

            
               
            
            
    material         
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
        maʿqulāt
ʾūlā            
         
         
           
            
        
           
            
            
         
           
          
         
             
         
          
          
           
           

          
            
            

3. FORMING MATHEMATICAL CONCEPTS
         
         
           
           
        
        The Metaphysics
of the Healing  
         
      al-taʿlīmī al-maḥsūs 
               
 Avicenna’s De anima        Aḥwāl al-nafs
 
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
            
            
al-maḥsūs      taʿlīmī    
         maʿdūd 
              
            
          
           
             
          
      kaṯīr      
        
           
           
          
          
           
      tahayyul   
  taʿaqqul     
          
        
           
          
         
       
           
          
  The Metaphysics of the Healing      
         
 maʿdūd             
           
 mufāraq         
      
             
           
 The Monist           
   Journal of Islamic philosophy     
         
   Journal of the American Philosophical Association  
   
   Al-burhān        Posterior
analytics           
           
   
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
        
         
         
         
       
modest       
          
          
            
       
         
          
       
         
           
            
        
          
         a posteriori 
          
          
          
  
           
       
          
          
           
         
         
       
            
             
             
            
           
      ṭrīyāt    
Philosophy East and West           
            
Philosophy East and West         
      Philosophy East and West
     
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
            
       
           
            
            
      
        
           
           
            
   
          
    maḥsūs        
         
    sensible mathematical objects   
           
 al-maḥsūsāt al-muštaraka     
          
            
             
           
           
          
           
           
             
          
           
           
          
           
            
            
           
          
           
      
      Avicenna’s De anima    
   
          
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
         
         
         
          
          
          
           
           
          
         
           
           
         
          
        
         
          
         
          
           
          

        
          
            
        
              
           
           
          
           
         wahm   
      tujarradu      
            
   Avicenna’s De anima    
             
            
          
               
            
         The Physics of the
Healing    
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
            
          taṣawwuruhū  
             
      
          
       
         
            
         bi-l-tawahhum 
            
           
        
           
        
          
   
            
             
          
             
        
          talḥaqu  
             
          
Conceiving its concept       
        
          The Metaphysics
of the Healing
  Al-madhal          
          
 
   The Metaphysics of the Healing      
   miqdār          
           
          
              
        The Metaphysics of the
Healing          ḥudūd  
              
            
         
  Al-madhal          
        
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
          
yanẓuru al-ʿadad         
         al-ṭabīʿa    
           
        estimative   
       estimation   
          

           
           
             
            
           
          
           
        
         
         
 maʿdūdāt          
         
           
            
            
        
       
         

         
        
 in    al-maḥsūsāt    
  The Metaphysics of the Healing       
 
             
          
             
           
            
          Al-madhal
     The Metaphysics of the Healing    
  Avicenna’s De anima          
    mudrak     al-ḥiss   ṣūra 
      al-wahm    maʿnā
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
        maʿānī
           
 The Salvation       
     in the sensible 
 al-maʿānī al-ghayr al-maḥsūsa al-mawjūda al-maḥsūsāt
al-juzʾīya        
          
     
        
          
         
 in           
           
          
           
            
         
           
           
            
          
           
             
          
         actually 
           
        
          in
         
            
         
        
          
            
             in
             
           in 
       in 
  Al-naǧāt          Al-
išārāt wa-l-tanbīhātAl-ṭabīʿīyāt           
         in  
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
          
            
        
       
         
             
          
        
          
     
           
           
             
          
         
        
         
            
            
          
          
         
        
           
       
            
             
     in     The Metaphysics
of the Healing    
                 
             
             
              
              
         
             
        The Metaphysics of the Healing
             
              
           
           
               
          
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
       
           
            
  
    ʾidrāk      ḥaqīqa  
 mutamaṯṯal         
               
            
            
             
kaṯīr            
           miṯāl
          murtasam  
           
  
          
          
             
          
           
          
            
           
         
           
            
           
         
  
        
          
      all  most   
           kaṯīr 
          
         
           
           
            
          
  Al-išārāt wa-l-tanbīhātAl-ṭabīʿīyāt   
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
            
           
           
          
          
           
            
              
           
         reductio
ad absurdum        
           
         
        
           
        
         
            
           
         
            
         
         
         
       
           
            
           
               
          
          
         
           
  Al-išārāt wa-l-tanbīhātAl-ṭabīʿīyāt     
  Al-išārāt wa-l-tanbīhātAl-manṭiq     
        ʿanqāʾ    
     
             
          
  Archiv für Geschichte der Philosophie    
 
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
        
       
           
      
             
            
             
            
              
         
     
             
          
            
         
         
          
           
         
        taʿrīf  ḥadd
          
           
           
          
        
  Avicenna’s De anima    
          The Metaphysics of the Healing
             
            
               
           
Bulletin de philosophie médiévale        
            
       Bulletin de philosophie médiévale  
               
         reductio ad absur-
dum            
            
             
            
           
             
 
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
            
       
           
        
         
          
        
          
      
          
         
           
            
            
              
           
           
           
           
       
          
            
          
           
         
         
           
       ctionalist-abstractionist
   literalist         
      
4. PERCEIVING PERFECT MATHEMATICAL OBJECTS
           
          
     The Metaphysics of the Healing  
           
           
          
         
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
         
          
           
 
         ʿalā al-naḥw al-
maḥsūs             
   muḍarraṣ
      
            
             
            
         
          
            
           
         
           
          
         
         
             
          
           
           
            
           
          
     literally    
          
        
           
      The Physics of the Healing  
           
           
             
  The Metaphysics of the Healing  
         
     
              
       
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
            
               
          
          
           
          
         
         
          
        n  
          n
           
            
          
             
  
            
p200 =14.142135.. .
      n      
       n
           
                
          

            
              
             
        
      
    

       
  The Physics of the Healing        
             

            
           The
Physics of the Healing             
           
           
       mutakallimūn 
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
             
           
         
          
          
          
          
         
        
          
          
            
        
        
         
      
         
           
          
   
        
        
   
         
        even though these are
never instantiated physically         
       
    even though again in the nitty-gritty world around
us none of these exists      
  bi-l-farḍ     
            
 taṣawwur           
         
            
         

                
   
           
           Al-madhal
                
             
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
         
       
        
          
     perceiving      
   constructing    
            
          
         
             
        
          
          
           
          
         
           
       
           
           
          Archiv für
Geschichte der Philosophie         
         Aristotle transformed:
The ancient commentators and their inuence      
         
          
   Philosophical review       
      Apeiron    
 
        mujarrad   
  The Physics of the Healing         
         abstracta
          
           
             
           
            
          
       
  abstracta  
     māhīya        
             
             
            
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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     
         
         
           
         
        
         
         
          
     as if    
          
       
           
    
5. CONCLUSION
        
         
        
          
          
          
             
         
            
             
             
       how     
            
            
            
          
        
 The Metaphysics of the Healing       
          
         
              
           
 literalism         potentialism     
      
              
              
      Avicenna  
at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S0957423920000090
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   
          
        
          
         
          
           
       
       
       
          
          
        
    
        
          
           
        
          
        
           
           
         
           
        
        
            
 
Acknowledgements.         
Avicenna’s philosophy of mathematics        
            
           
           
           
           
             
           
          
         
           
           
          
      
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